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Logic, Optimization, and Constraint Programming

Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA jh38@andrew.cmu.edu

John N. Hooker

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ecause of their complementary strengths, optimization and constraint programming can be pro?tably merged. Their integration has been the subject of increasing commercial and research activity. This paper summarizes and contrasts the characteristics of the two ?elds; in particular, how they use logical inference in different ways, and how these ways can be combined. It sketches the intellectual background for recent efforts at integration. It traces the history of logic-based methods in optimization and the development of constraint programming in arti?cial intelligence. It concludes with a review of recent research, with emphasis on schemes for integration, relaxation methods, and practical applications. ( Optimization; Constraint Programming; Logic-Based Methods; Arti?cial Intelligence )

1.

Optimization and constraint programming are beginning to converge, despite their very different origins. Optimization is primarily associated with mathematics and engineering, while constraint programming developed much more recently in the computer science and arti?cial intelligence communities. The two ?elds evolved more or less independently until a very few years ago, but they have much in common and are applied to many of the same problems. Both have enjoyed considerable commercial success. Most importantly for present purposes, they have complementary strengths, and the last few years have seen growing efforts to combine them. Constraint programming, for example, offers a more ?exible modeling framework than mathematical programming. It not only permits more succinct models, but the models allow one to exploit problem structure and direct the search. It relies on such logicbased methods as domain reduction and constraint propagation to accelerate the search. Conversely, optimization brings to the table a number of specialized techniques for highly structured problem classes, such

0899-1499/02/1404/0295$5.00 1526-5528 electronic ISSN

Introduction

as linear programming problems or matching problems. It also provides a wide range of relaxations, which are often indispensable for proving optimality. They may be based on polyhedral analysis, in which case they take the form of cutting planes, or on the solution of a dual problem, such as the Lagrangean dual. The recent interaction between optimization and constraint programming promises to change both ?elds. It is conceivable that portions of both will merge into a single problem-solving technology for discrete and mixed discrete/continuous problems. The aim of the present paper is to survey current activity in this area and sketch its intellectual background. It begins with a general summary of the characteristics of optimization and constraint programming: what they have in common, how they differ, and what each can contribute to a combined approach. It then brie?y recounts developments that led up to the present state of the art. It traces the history of boolean and logic-based methods for optimization, as well as the evolution of logic programming to constraint logic programming and from there to the constraint programming “toolkits” of today.

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HOOKER Logic, Optimization, and Constraint Programming

It concludes with a summary of some current work. Since a proper tutorial in these ideas would require a much longer paper, only a cursory explanation of key concepts is provided. A more complete treatment is provided by Hooker (2000) as well as other sources cited in the context of particular topics discussed below. The emphasis in this paper is on exact methods, because this is where most of the collaboration is now occurring. However, optimization and constraint programming also share a strong interest in heuristic methods. Interaction between the two ?elds in this area is hard to characterize in general, but both are unaware of much recent research conducted by the other. This may present a second opportunity for cross-fertilization in the near future.

a brief moment of fame in the 1970s but can be seen as an early form of today’s constraint propagation methods. 2.2. Two Main Differences As the names might suggest, constraint programming seeks a feasible solution, and optimization seeks an optimal solution. But this is a super?cial distinction, as optimization is easily incorporated into the algorithms of constraint programming by gradually tightening a bound on the objective function value. The main differences between optimization and constraint programming lie elsewhere. 2.2.1. Programming Versus “Programming.” One key distinction between the two ?elds is that constraint programming formulates a problem within a programming language. The formulation generally has a declarative look, because one of the historical goals of the ?eld is to allow one to integrate constraints into programming. However, the modeling language can give the modeler considerable control over the search procedure. Thus the “programming” in constraint programming actually refers to computer programming, unlike the “programming” in mathematical programming, which recalls George Dantzig’s historical application of linear models to military logistics. 2.2.2. Constraint Propagation Versus Relaxation. Another distinction is that constraint programming uses logical inference in different ways than optimization uses it. It uses inference to reduce the search space directly through such techniques as domain reduction and constraint propagation. Optimization uses inference (perhaps in the form of cutting planes) primarily to create better relaxations, which accelerate the search indirectly. Although optimization is not fundamentally distinguished by the fact that it optimizes, the presence of cost and pro?t functions (whether they occur in an objective function or constraints) has nonetheless taken the ?eld in a different direction than constraint programming. Cost and pro?t functions tend to contain many variables, representing the many activities that can incur cost or contribute to pro?t. This often

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2.

Optimization and constraint programming are similar enough to make their combination possible, and yet different enough to make it pro?table. 2.1. Two Areas of Commonality There are at least two broad areas of commonality between optimization and constraint programming. One is that both rely heavily on branching search, at least when an exact method is required and the problem is combinatorial in nature. The other is less obvious: both use logical inference methods to accelerate the search. Constraint programming uses logical inference in the form of domain reduction and constraint propagation. In optimization, logical inference takes forms that are not ordinarily recognized as such, including cutting plane algorithms and certain duality theorems. A cutting plane, for example, is a logical implication of a set of inequalities. Preprocessing algorithms for mixed integer programming can also be viewed as limited forms of domain reduction and constraint propagation. When one takes a more historical view, the commonality becomes even more visible. Boolean methods for operations research, which are based on logic processing, date from the 1950s. Implicit enumeration methods for integer programming enjoyed only

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makes domain reduction and constraint propagation ineffective. The domain of a variable is the set of possible values it can take. Domain reduction uses restrictions on the domain of one variable in a constraint to deduce that the other variables can take only certain values, if the constraint is to be satis?ed. If the constraint does not contain too many variables, this can signi?cantly reduce domains. The smaller domains are passed to other constraints, where they are further reduced, thus implementing a form of constraint propagation. If all goes well, a combination of branching search and constraint propagation eventually reduce each variable’s domain to a single value, and the corresponding solution can be checked for feasibility. When a constraint contains a cost or pro?t function of many variables, however, domain reduction is ineffective. The optimization community escapes this impasse by using relaxation techniques, generally continuous relaxations. The community perhaps moved away from the early implicit enumeration techniques, which resemble constraint propagation, because its business and engineering applications require measurement of cost and pro?t. The constraint satisfaction community, by contrast, developed to a large degree through consideration of problems whose constraints are binary; that is, they contain only two variables each. Many combinatorial problems discussed in the arti?cial intelligence literature are of this sort. A standard example is the n-queens problem, which asks one to place n queens on a chessboard so that no one attacks another. Restricting the domain of one variable in a binary constraint can substantially reduce the domain of the other variable, and propagation tends to be effective. It is not surprising that domain reduction and constraint propagation methods have evolved from a ?eld that historically emphasized binary constraints. These methods can work well when there are no sums over cost or pro?t, and when the objective is a criterion such as a min/max or minimum makespan. It is sometimes said that constraint programming is more effective on “tightly constrained” problems, and optimization more effective on “loosely constrained” problems. This is misleading, because one

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can typically make a problem more tightly constrained (even infeasible) by tightening the bound on a single constraint or the objective function. If the constraint or objective function contains many variables, this is likely to make the problem harder for constraint programming, not easier. 2.3. Opportunities for Integration Many if not most practical problems have constraints that require relaxation as well as constraints that propagate well. For this reason alone, it is natural to use a combination of optimization and constraint satisfaction methods. In addition, the versatile modeling framework of constraint satisfaction leads to models that are more readable and easier to debug than traditional optimization models. Conversely, powerful optimization methods can solve structured subproblems more ef?ciently than constraint programming methods. 2.3.1. Global Constraints. Still another inducement to integration is that constraint programming’s global constraints provide a practical way for techniques from both ?elds to exploit problem structure. When formulating a problem, one can often identify groups of constraints that exhibit a “global” pattern. For instance, a set of constraints might require that jobs be scheduled so that they do not overlap (or consume too many resources at any one time). These constraints can be represented by a single global constraint, in this case a cumulative constraint. An equally popular global constraint is all-different, which requires that a set of variables all take different values. It can be used to formulate the traveling salesman problem, for example, with a single constraint, rather than exponentially many as in the most popular integer programming model. Let xj denote the j th city visited by the salesman, and let cij be the distance from city i to city j . Then the traveling salesman problem can be written minimize subject to

n j =1

cxj xj +1 xn

(1)

all-different x1

where city n + 1 is identi?ed with city 1.

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Note the use of variables as indices in the objective function of (1). Variable indices are a powerful modeling device disallowed in traditional mathematical programming but accessible to constraint programming methods. They can be implemented with another global constraint, element. An expression of the form ux , where x is a discrete variable, is replaced by a new variable z, and the constraint element x u1 uk z

is added to the problem. If each uj is a constant, the element constraint sets z equal to the xth constant in the list u1 uk . If each uj is a variable, it imposes the constraint z = uj , where j is the current value of x. There are ef?cient domain reduction (or “?ltering”) procedures for the element, cumulative, and alldifferent constraints. The algorithms for the element constraint are straightforward, whereas those for the cumulative constraint are numerous, complex, and largely proprietary. The all-different constraint provides a good example of how optimization can contribute to constraint programming. Suppose that the current domain of x1 and of x2 is 1 2 , and the domain of x3 is 1 2 3 . If all-different x1 x2 x3 is to be satis?ed, x1 and x2 must take the values 1 and 2. This forces x3 = 3, and the domain of x3 is reduced to 3 . This reduction is accomplished in general by applying an algorithm for maximum cardinality bipartite matching and then using a theorem of Berge (1970) to identify arcs that cannot be part of any matching (Régin 1994). 2.3.2. Exploiting Global Structure. Such global constraints as element, cumulative, and all-different tell the solver how to exploit structure. Standard constraint propagation methods are likely to be ineffective when applied “locally” to the individual constraints represented by the global constraint. But the global constraint invokes a specialized domain reduction algorithm that exploits the global pattern. Thus constraint programming identi?es structure in a subset of constraints, rather than in a subset of problems, as is typical of optimization. The idea of associating a procedure with a constraint is a natural outgrowth of the constraint programmer’s computer science background. In a computer program, a procedure is associated with every statement.

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Global constraints also take advantage of the practitioner’s expertise. The popular global constraints evolved because they formulate situations that commonly occur in practice. An expert in the problem domain can recognize these patterns in practical problems, perhaps more readily than the developers of a constraint programming system. By using the appropriate global constraints, a practitioner who knows little about solution technology automatically invokes the best available domain reduction algorithms that are relevant to the problem. Global constraints can provide this same service to optimization methods. For example, a global constraint that represents inequality constraints with special structure can bring along any known cutting planes for them. Much cutting plane technology now goes unused in commercial solvers because there is no suitable framework for identifying when it applies. Existing optimization solvers identify some types of structure, such as network ?ow constraints. But they do so only in a limited way, and in any event it is inef?cient to obliterate structure by modeling with an atomistic vocabulary of inequalities and equations and then try to rediscover the structure automatically. The vast literature on relaxations can be put to work if practitioners write their models with as many global constraints as possible. This means that modelers must change their modeling habits, but they may welcome such a change. In the current state of the art, a global constraint may have no known relaxation. In this case the challenge to optimization is clear: use the well-developed theory on this subject to ?nd one.

3.

Boolean and logic-based methods have a long history in discrete optimization. Their development might be organized into two overlapping stages. One is the classical boolean tradition, which traces back to the late 1950s and continues to receive attention, even though it has never achieved mainstream status to the same extent as integer programming. The second stage introduced logical ideas into integer and mixed integer programming. This research began with the

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Logic-Based Methods in Optimization

HOOKER Logic, Optimization, and Constraint Programming

implicit enumeration and disjunctive programming ideas of the 1970s and likewise continues to the present day. A more detailed treatment of this history is provided by Hooker (2000). 3.1. Boolean Methods George Boole’s small book, The Mathematical Analysis of Logic (1847), laid the foundation for all subsequent work in computational logic. In the 1950s and 1960s, the new ?eld of operations research was quick to notice the parallel between his twovalued logic and 0-1 programming (Fortet 1959, 1960; Hansen et al. 1993). The initial phase of research in boolean methods culminated in the treatise of Hammer and Rudeanu (1968). Related papers from this era include those of Hammer et al. (1963); Balas and Jeroslow (1972); Granot and Hammer (1971, 1975); and Hammer and Peled (1972). These and subsequent investigators introduced three ideas that today play a role in the integration of optimization with constraint programming: (a) nonserial dynamic programming and the concept of induced width, (b) continuous relaxations for logical constraints, and (c) derivation of logical propositions from constraints. 3.1.1. Nonserial Dynamic Programming. Nonserial dynamic programming is implicit in Hammer and Rudeanu’s “basic method” for pseudoboolean optimization (i.e., optimization of a real-valued function of boolean variables). Bertele and Brioschi (1972) independently developed the concept in the late 1960s and early 1970s, but boolean research took it an important step further. When Crama et al. (1990) revisited the basic method more than twenty years after its invention, they discovered that it can exploit the structure of what they called the problem’s co-occurrence graph. The same graph occurs in the constraint programming literature, where it is called the dependency graph (or primal graph). Crama et al. found that the complexity of the algorithm depends on a property of the graph known in the arti?cial intelligence community as its induced width, which is de?ned for a given ordering of the variables. Consideration of the variables in the right order may result in a small induced width and faster solution.

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To elaborate on this a bit, the dependency graph G for an optimization problem contains a vertex for each variable xj and an edge i j whenever xi and xj occur in the same constraint, or the same term of the objective function. To de?ne the width of G, suppose that one removes vertices from G in the order n n?1 1. If dj is the degree of j at the time of its removal, the width of G with respect to the ordering 1 n is the maximum of dj over all j . Now suppose that just before vertex j is removed, one adds edges to G as needed to link all vertices that are connected to j by an edge. The induced width of G with respect to the ordering 1 n is the maximum of dj over all j . A small induced width indicates that the variables are loosely coupled and the problem is therefore easier, at least when solved by nonserial dynamic programming. Nonserial dynamic programming allows the state in a given stage of the recursion to be a function of the control and the state of the system in several previous stages, rather than only the immediately preceding stage as in classical dynamic programming. In particular, the state in stage j depends on the stages i for which i is adjacent to j in the problem’s dependency graph G at the time of j ’s removal. The next state therefore never depends on more than w previous states, if w is the width of G, which means that the complexity of the algorithm is at worst exponential in w . It is clearly advantageous to order the variables so that w is small, although ?nding the best ordering is an NP-hard problem. This method is readily applied to pseudo-boolean optimization. Any (linear or nonlinear) 0-1 programming problem with equality constraints can be viewed as the unconstrained minimization of some pseudo-boolean function, which is a real-valued function of boolean (two-valued) variables. Such a function can always be written in the form f x =

K k =1 k j ∈ Jk

sjk

(2)

where each sjk is xj or 1 ? xj . An edge connects xi and xj in the dependency graph for the problem if xi and xj occur in the same term of (2). Methods that are essentially nonserial dynamic programming

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have also surfaced in the analysis of Markov trees (Schaefer 1978; Shafer et al. 1987), facility location (Chhajed and Lowe 1994), and Bayesian networks (Lauritzen and Spiegelhalter 1988). A graph with induced width k is a partial k-tree, studied by Arnborg et al. (1987), Arnborg and Proskurowski (1986), and others. 3.1.2. Continuous Relaxations. A second contribution of Boolean research was to provide early examples of continuous relaxations for a logical condition. Perhaps the best-studied relaxation is the roof dual of a pseudo-boolean optimization problem, which is equivalent to a specially structured linear programming problem. Hammer et al. (1984) introduced this concept for quadratic pseudoboolean optimization and demonstrated its linear programming formulation, along with several other properties. Lu and Williams (1987) extended these ideas to general pseudoboolean polynomials, as did Hansen et al. (1984). Balas and Mazzola (1980a, 1980b) studied a family of bounding functions of which the roof bound is one sort. The roof dual turns out to be an instance of the well-known Lagrangean dual, applied to a particular integer programming formulation of the problem. Adams and Dearing (1994) demonstrated this for the quadratic case, and Adams et al. (1990) generalized the result. Today the formulation of continuous relaxations for logical and other discrete constraints is an important research program for the integration of optimization and constraint programming. 3.1.3. Implied Constraints. A third development in boolean research was the derivation of logical implications from 0-1 inequality constraints. The derived implications can be used in either of two ways. They have been primarily used as cutting planes that strengthen the continuous relaxation of the inequalities, as for example in Balas and Mazzola (1980a, 1980b). Of greater interest here, however, is the purely logical use of derived implications. One can solve a 0-1 problem by reducing its constraints to logical form so that inference algorithms can be applied. The most advanced effort in this direction is Barth’s (1995) pseudo-boolean solver. The early boolean literature shows how to derive logical clauses from linear or nonlinear 0-1 inequalities. A clause is a disjunction of literals, each of which

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is a logical variable xj or its negation ?xj . For instance, the clause x1 ∨ x2 ∨ ?x3 states that x1 is true or x2 is true or x3 is false (where the “or” is inclusive). A simple recursive algorithm of Granot and Hammer (1971) obtains all nonredundant clauses implied by a single linear 0-1 inequality. Granot and Hammer (1975) also stated an algorithm for deriving all clausal implications of a single nonlinear inequality. Barth took this a step further by deriving cardinality clauses from 0-1 inequalities. A cardinality clause says that at least k of a set of literals are true (k = 1 in an ordinary clause). Cardinality clauses tend to capture numerical ideas more succinctly than ordinary clauses and yet retain many of their algorithmic advantages. Barth’s (1995) derivation is based on a complete inference method for 0-1 inequalities (Hooker 1992) and takes full advantage of the problem structure to obtain nonredundant clauses ef?ciently. Another motivation for deriving constraints is to make a problem more nearly “consistent” (discussed below), so as to reduce backtracking. Constraints derived for this reason can be contrasted with cutting planes, which are derived with the different motivation of strengthening a continuous relaxation of the problem. To appreciate the difference, note that facetde?ning inequalities are in some sense the strongest cutting planes but are not necessarily the most useful derived implications for logic-based methods. A facet-de?ning cut can be strictly dominated, in a logical sense, by an implied inequality that is not facetde?ning (Hooker 2000). Consider for instance the set of 0-1 points de?ned by x1 + x2 ≥ 1 and x2 + x3 ≥ 1. Both inequalities de?ne facets of the convex hull of the feasible points. However, the implied inequality x1 + 2x2 + x3 ≥ 2 dominates both facet-de?ning inequalities. It dominates x1 + x2 ≥ 1, for example, because it logically implies x1 + x2 ≥ 1 (i.e., all 0-1 points satisfying the former satisfy the latter), and it excludes a 0-1 point 1 0 0 that x1 + x2 ≥ 1 does not exclude. 3.2. Logic in Mixed Integer Programming A second phase of research into logic-based methods, beginning in the 1970s, brought logical ideas into integer programming.

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HOOKER Logic, Optimization, and Constraint Programming

3.2.1. Implicit Enumeration. In the early days of integer programming, problems were often solved by “implicit enumeration.” This is a branching method that uses “preprocessing” to ?x variables or simplify the problem, but it typically does not use the continuous relaxations associated with branch-and-bound methods. As examples one might cite Hansen’s work on boolean problems (1969, 1970), or Gar?nkel and Nemhauser’s (1970) solution of a political districting problem. Implicit enumeration can be viewed as an early form of today’s constraint propagation methods. For reasons already discussed, the integer programming community soon moved away from implicit enumeration and toward branch-and-bound and branch-and-cut methods. 3.2.2. Disjunctive Programming. Balas (1974, 1975, 1977, 1979) introduced disjunctive programming, which optimizes a linear objective function subject to logical disjunctions of linear inequality systems, each of which takes the form, A1 x ≤ a1 ∨ · · · ∨ Ak x ≤ ak (3)

space is the feasible set of some mixed integer model if and only if it is the union of ?nitely many polyhedra, all of which have the same set of recession directions. He proved a similar result for mixed integer nonlinear programming (Jeroslow 1987, 1989). His analysis provides a general tool for obtaining continuous relaxations for nonconvex regions of continuous space, which again may or may not be practical in a given case. 3.3. Links Between Logic and Mathematical Programming Williams (1976, 1977, 1987, 1991, 1995) was among the ?rst to point out parallels between logic and mathematical programming (see also Williams and Yan 2001a). Laundy (1986), Beaumont (1990), and Wilson (1990, 1995, 1996) also contributed to this area. 3.3.1. Connections with Resolution. To take one parallel, the well-known resolution method for logical inference can be viewed as Fourier-Motzkin elimination plus rounding. Fourier-Motzkin elimination was one of the earliest linear programming algorithms, proposed for instance by Boole (Hailperin 1976, 1986). Given two logical clauses for which exactly one variable xj occurs positively in one and negatively in the other, the resolvent of the clauses is the clause consisting of all the literals in either clause except xj and ?xj . For example, the resolvent of the clauses x1 ∨ x2 ∨ ?x3 ?x1 ∨ x2 ∨ x4 (5)

A 0-1 formulation of the disjunction can be written, A i x i ≤ a i yi y ∈ 0 1

k

i=1

k (4)

x = x1 + · · · + xk

The continuous relaxation of this formulation (obtained by replacing each yi ∈ 0 1 by 0 ≤ yi ≤ 1) provides a convex hull relaxation of the feasible set of (3). It does so at the cost of adding variables yi and vectors xi of variables. Nonetheless (4) provides a tool for obtaining continuous relaxations of logical constraints, including constraints other than simple disjunctions. Sometimes the additional variables can be projected out, or the formulation otherwise simpli?ed. In the meantime Jeroslow brought his background in formal logic to integer programming (e.g., Balas and Jeroslow 1972). He proved what is perhaps the only general theorem of mixed integer modeling: that representability by a mixed integer model is the same as representability by disjunctive models of the form (4). From this he derived that a subset of continuous

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is x2 ∨ ?x3 ∨ x4 . The logician Quine (1952, 1955) showed that repeated application of this resolution step to a clause set, and to the resolvents generated from the clause set, derives all clauses that are implied by the set. To see the connection with Fourier-Motzkin elimination, write the clauses (5) as 0-1 inequalities: x 1 + x 2 + 1 ? x3 ≥ 1 1 ? x1 + x2 + x4 ≥ 1 where 0-1 values of xj correspond to false and true, respectively. If one eliminates x1 , the result is the 01 inequality x2 + 1 1 ? x3 + 1 x ≥1 , which dominates 2 2 4 2 1 x2 + 1 ? x3 + x4 ≥ 2 . The right-hand side can now be rounded up to obtain the resolvent of (5). As this

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example might suggest, a resolvent is a rank 1 Chvátal cutting plane. Further connections between resolution and cutting planes are described by Hooker (1988, 1989, 1992). For example, deriving from a clause set all clauses that are rank 1 cuts is equivalent to applying the unit resolution algorithm or the input resolution algorithm to the clause set (Hooker 1989). (In unit resolution, at least one of the two parents of a resolvent contains a single literal. In input resolution, at least one parent is a member of the original clause set.) Resolution is also related to linear programming duality, as shown by Jeroslow and Wang (1989) in the case of Horn clauses, and generalized to gain-free Leontief ?ows, as shown by Jeroslow et al. (1989). Horn clauses are those with at most one positive literal and are convenient because unit resolution can check their satis?ability in linear time. Jeroslow and Wang pointed out that linear programming can also check their satis?ability. Chandru and Hooker (1991) showed that the underlying reason for this is related to an integer programming rounding theorem of Chandrasekaran (1984). They used this fact to generalize Horn clauses to a much larger class of “extended Horn” clauses that have the same convenient properties, a class that was further extended by Schlipf et al. (1995). Connections such as these suggest that optimization can help solve logical inference problems, as well as the reverse. In fact there is a stream of research that does just this, summarized in a book of Chandru and Hooker (1999). Recent work in this area began with Blair et al.’s (1988) use of branch-and-bound search to solve the satis?ability problem. The link between resolution and cutting planes described above leads to a specialized branch-and-cut method for the same problem (Hooker and Fedjki 1990). Recent work shows that integer programming and Lagrangean relaxation can yield a state-of-the-art method for the satis?ability and incremental satis?ability problems (Bennaceur et al. 1998). The earliest application of optimization to logic, however, seems to be Boole’s application of linear programming to probabilistic logic (Boole 1952; Hailperin 1976). (It was in this connection that he used elimination to solve linear programming problems.) Optimization can also solve inference problems in ?rst order predicate logic, modal logic, such belief

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logics as Dempster-Shafer theory, and nonmonotonic logic. 3.3.2. Mixed Logical/Linear Programming. Mixed logical/linear programming (MLLP) might be de?ned roughly as mixed discrete/linear programming in which the discrete aspects of the problem are written directly as logical conditions rather than with integer variables. For example, disjunctions of linear systems are written as (2) rather than with the convex hull formulation (3) or the big-M formulation A i x ≤ a i + M i 1 ? yi y∈ 0 1

k

i=1

k

(6)

A logical formulation can be more natural and require fewer variables, but it raises the question as to how a relaxation can be formulated. The traditional integer programming formulation comes with a readymade relaxation, obtained by dropping the integrality requirement for variables. Solution of the relaxation provides a bound on the optimal value that can be essential for proving optimality. Beaumont (1990) was apparently the ?rst to address this issue in an MLLP context. He obtained a relaxation for disjunctions (2) in which the linear systems Ai x ≤ b i are single inequalities ai x ≤ bi . He did so by projecting the continuous relaxation of the bigM formulation (6) onto the continuous variables x. This projection simpli?es to a single “elementary” inequality

k ai bi x≤ +k?1 Mi Mi i =1 i =1 k

(7)

An inexpensive way to tighten the inequality is presented by Hooker and Osorio (1999). Beaumont also identi?ed some valid inequalities that are facetde?ning under certain (strong) conditions. A generalization of Beaumont’s approach is to introduce propositional variables and to associate at least some of them with linear systems. Logical constraints can then express complex logical relationships between linear systems. One can also process the logical constraints to ?x values, etc., as proposed by Hooker (1994) and Grossmann et al. (1994). In the latter, Grossmann et al. designed chemical processing networks by associating a propositional variable xj

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HOOKER Logic, Optimization, and Constraint Programming

with processing units. When xj is true, processing unit j and its associated arcs are present in the network. When xj is false, ?ow through unit j is forced to zero. Purely logical constraints are written to ensure that a unit is not installed unless units that supply its feedstock and receive its output are also present. A number of processing network design, process scheduling, and truss structure design problems have been solved with the help of logic (Bollapragada et al. 2001; Cagan et al. 1997; Pinto and Grossmann 1997; Raman and ¨ Grossmann 1991, 1993, 1994; Turkay and Grossmann 1996). Some of these problems are nonlinear and are attacked with mixed logical/nonlinear programming (MLNLP). A key advantage of MLNLP is that situations in which an activity level drops to zero can be distinguished as logically distinct states with different associated equations, thus avoiding the singularities that tend to occur in traditional models. A general approach to MLLP was developed in the mid-1990’s (Osorio and Hooker 1996; Hooker and Osorio 1999). In particular, these authors critiqued the role of integer variables in optimization and suggested guidelines for when a logical formulation is better. They, along with Little and Darby-Dowman (1995), proposed incorporating constraint programming methods into mathematical programming.

4.

Constraint programming can be conceived generally as the embedding of constraints within a programming language. This combination of declarative and procedural modeling gives the user some control over how the problem is solved, even while retaining the ability to state constraints declaratively. It is far from obvious how declarative and procedural formulations may be combined. In a procedural code, for example, it is common to assign a variable different values at various points in the code. This is nonsense in a declarative formulation, since it is contradictory to state constraints that assign the same variable different values. The developments that gave rise to constraint programming can in large part be seen as attempts to address this problem. They began with logic programming and led to a

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Constraint Programming and Constraint Satisfaction

number of alternative approaches, such as constrainthandling rules, concurrent constraint programming, constraint logic programming, and constraint programming. Constraint programming “toolkits” represent a somewhat more procedural version of constraint logic programming and are perhaps the most widely used alternative. Two main bodies of theory underlie these developments. One is the theory of ?rst-order logic on which logic programming is based, which was later generalized to encompass constraint logic programming. The other is a theory of search developed in the constraintsatisfaction literature, which deals with such topics as consistency, search orders, the dependency graph, and various measures of its width. Lloyd’s text (1984) provides a good introduction to logic programming, which is further exposited in Clocksin and Mellish (1984) and Sterling and Shapiro (1986). Tsang (1993) provides excellent coverage of the theory of constraint satisfaction. Van Hentenryck (1989) wrote an early exposition of constraint logic programming, while Marriott and Stuckey’s text (1998) is a valuable resource for recent work in constraint programming and constraint logic programming. Chapters 10–11 of Hooker (2000) summarize these ideas. 4.1. Programming with Constraints Several schemes have been proposed for embedding constraints in a programming language: constraint logic programming, constraint programming “toolkits,” concurrent constraint programming, and timeindex variables. All of these owe a conceptual debt to logic programming. 4.1.1. Logic Programming. One of the central themes of logic programming is to combine the declarative and the procedural. As originally conceived by Kowalski (1979) and Colmerauer (1982), a logic program can be read two ways: as a series of logical propositions that state conditions a solution must satisfy, and as instructions for how to search for a solution.

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To take a very simple example, rules 1 and 2 in the following logic program can be read as a recursive de?nition of an ancestor: 1. ancestor X Y ← parent X Y . 2. ancestor X Z ← parent X Y , ancestor Y Z . 3. parent a b . 4. parent b c .

The search next tries to unify the original goal with the consequent of rule 2 (right branch of root node). Uni?cation is possible, and it poses the subgoals parent a Y and ancestor Y c . Both subgoals eventually succeed with Y = b , which establishes ancestor a c . 4.1.2. Constraint Logic Programming. The power of recursion allows one to de?ne a wide range of constraints in a logic program, but solution can be inef?cient, particularly where numerical operators are involved. To solve many practical problems, one must exploit the special properties of constraints, for instance, by using linear solvers or interval arithmetic for inequalities and specialized propagation schemes for all-different constraints. Constraint logic programming (CLP) addresses this problem. Its key idea is to regard the uni?cation step of logic programming as a special case of constraint solving. The search tree of Figure 1 becomes the search tree of Figure 2. For instance, unifying ancestor a c with ancestor X Z is tantamount to solving the equation a c = X Z for X and Z . This is illustrated in the right descendant of the root node in the ?gure. The process continues as before, except that constraints are accumulated into a constraint store as subgoals are discharged. At the leaf nodes only

(8)

Here X Y and Z are variables, and a b and c are constants. Rule 1 says that X is an ancestor of Y if X is a parent of Y . One can deduce from statements 1 through 4 that a is c ’s ancestor. This is the declarative reading of the program. The procedural reading sees the program as directing a search. Suppose for example that one poses the goal ancestor a c ; that is, the goal of showing that a is an ancestor of c . The search procedure ?rst tries to interpret ancestor a c as an instance of the consequent (i.e., the then-part) of rule 1. (This is illustrated by the left branch from the root node in Figure 1.) It is in fact an instance, because one can substitute a for X and c for Y . This substitution uni?es ancestor a c with ancestor X Y . It poses the subgoal parent a c . If this subgoal succeeds, then the original goal succeeds, and a is found to be an ancestor of c . To make parent a c succeed, it is necessary to unify it with the consequent of some rule. (Rules 3 and 4 have no antecedent or if-part, and so their consequents are always true.) In this case uni?cation is not possible, and the subgoal parent a c fails.

Figure 1

Search Tree for Finding Ancestors: Each Node Shows Open Goals or Subgoals That Have Yet to Be Achieved

Figure 2

Search Tree in Which Uni?cation Is Interpreted as Constraint Solving

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constraints remain, and the node succeeds if they are soluble. In a pure logic programming language such as the early PROLOG (Colmerauer’s abbreviation for programmation en logique), the constraint store contains only equations that unify predicates. One obtains CLP by expanding the repertory of constraints and variables. Marriott and Stuckey (1998) identify the ?rst CLP system to be Colmerauer’s (1982) language PROLOG II, in which uni?cation requires solution of disequations as well as equations. Jaffar and Stuckey (1986) showed that the theory of pure logic programming can be extended to PROLOG II. Jaffar and Lassez (1987) pointed out that PROLOG II is a special case of a general scheme in which uni?cation is viewed as a constraint-solving problem. The term “constraint logic programming” originates from this paper. Several CLP systems quickly followed. Colmerauer and his colleagues added constraint solving over strings, boolean variables, and real linear arithmetic in PROLOG III (Colmerauer 1987, 1990). Jaffar and Michaylov (1987) and Jaffar et al. (1992) added real arithmetic to obtain CLP( ). Dincbas et al. (1988) and Aggoun et al. (1987) added constraints over ?nite domains, including domains of integers, in their system CHIP. Although uni?cation is not a hard problem in classical PROLOG, it can quickly become hard when one adds such constraints as inequalities over integers or constraints over ?nite domains. For this reason the constraint solver may be incomplete; it may fail to ?nd a solution even if one exists. More often constraint solvers narrow the range of possible solutions through domain reduction. In such cases the constraint store does not contain the hard constraints but only such very simple ones as “in-domain” constraints; i.e., constraints stating that each variable must take a value within a certain domain. Domainreduction algorithms add to the constraint store by deriving new in-domain constraints. If the constraint solvers and domain reduction fail to ?nd a solution (i.e., fail to reduce the domains to singletons), one can branch further by splitting variable domains. The constraint solvers are reapplied after branching, and the domains further reduced.

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This in turn reduces the amount of further branching that is necessary. Constraint programming “toolkits” are based on CLP but do not require the model to be written in a strict logic programming framework. Early toolkits include the CHARME system of Oplobedu et al. (1989) and the PECOS language of Puget (1992), both of which evolved from CHIP. Puget (1994) and Puget and Leconte (1995) later developed the initial ILOG Solver, which embeds constraints in the objectoriented language C++. The toolkit provides a library of C++ objects that implement many of the same constraint propagation algorithms found in CLP systems. Constraints are de?ned by using the abstraction and overloading facilities of C++. 4.1.3. Links with Optimization. Linear and even nonlinear programming have played a role in constraint programming systems for some years. They appear in such systems as CHIP (Aggoun et al. 1987), the ILOG Solver (Puget 1994) and PROLOG III and IV (Colmerauer 1990, 1996). Beringer and De Backer (1995) used linear programming to tighten upper and lower bounds on continuous variables. Solnon (1997) proposed that a linear programming solver minimize and maximize each variable, to obtain bounds on it, at each node of a branch-and-bound tree. Using a somewhat different approach, McAloon and Tretkoff (1996) developed a system 2LP that allows one to invoke linear programming in a script that implements logicbased modeling. 4.2. Theories of Search The constraints community has developed at least two related theories of search. One, which comprises much of the constraint satisfaction ?eld, examines factors that govern the amount of backtracking necessary to complete a branching search. It is fully explained in Tsang’s text (1993) and summarized from a mathematical programming perspective by Hooker (1997). Another explores the idea of constraint-based search, which combines inference and branching. It is exposited in Chapter 18 of Hooker (2000). 4.2.1. Constraint Satisfaction. The fundamental concept of constraint satisfaction is that of a consistent constraint set, which is not the same as a satis?able

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or feasible constraint set. A consistent constraint set is fully rami?ed in the sense that all of its implications are explicitly stated by constraints. To be more precise, let the vector x = x1 xn be arbitrarily partitioned x = x1 x2 . Then the assignment x1 = v1 is a partial assignment (or compound label). By convention, a partial assignment x1 = v1 can violate a constraint only when x1 contains all the variables that occur in the constraint. Let D1 be the cartesian product of the domains of the variables in x1 , and similarly for D2 . Then if v1 ∈ D1 , the partial assignment x1 = v1 is redundant for a constraint set C if it violates no constraints in C but cannot be extended to a feasible solution. That is, x1 = v1 violates no constraint in C , but x1 x2 = v1 v2 is infeasible for all v2 ∈ D2 . A constraint set is consistent if there are no redundant partial assignments. In other words, any redundant partial assignment is explicitly ruled out because it violates some constraint. It is not hard to see that one can ?nd a feasible solution for a consistent constraint set, if one exists, without backtracking. The concept of consistency also provides a theoretical link between the amount of backtracking and the branching order, a link that has not been achieved in the optimization literature. A constraint set C is kconsistent if for any partition x = x1 x2 in which x1 contains k ? 1 variables, and for any xj in x2 , any partial assignment x1 = v1 that violates no constraints in C can be extended to an assignment x1 xj = v1 vj that violates no constraints in C , where vj ∈ Dj . A constraint set is strongly k-consistent if it is t -consistent for t = 1 k. Suppose that one seeks a solution for a strongly k-consistent constraint set C by branching on the variables in the order x1 xk . Freuder (1982) showed that no backtracking will occur if the dependency graph for C has width less than k with respect to the ordering x1 xn . Consistency is closely related to logical inference. For instance, applying the resolution algorithm to a set of logical clauses makes the set consistent. If the algorithm is modi?ed so that it generates only resolvents with fewer than k literals, it makes the clause set strongly k-consistent. In general, drawing inferences from a constraint set tends to make it more nearly consistent and to reduce backtracking. This contrasts with mathematical programming, where inferences in

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the form of cutting planes are drawn to tighten the continuous relaxation of the problem. Cutting planes can have the ancillary effect of making the constraint set more nearly consistent, although the optimization literature has never formally recognized the concept of consistency. There are other results that connect backtracking with the search order. Dechter and Pearl (1988) showed that a given search order results in no backtracking if the constraint set has adaptive consistency (a kind of local consistency) with respect to that ordering. The bandwidth of a constraint set’s depenxin dency graph with respect to an ordering xi1 is the maximum of j ? k over all arcs ij ik in the dependency graph. The bandwidth with respect to an ordering is the maximum number of levels one must backtrack on encountering an infeasible solution during a tree search that branches on variables in that same order. The bandwidth is an upper bound on the induced width (Zabih 1990), and a minimum bandwidth ordering can be computed by dynamic programming (Gurari and Sudborough 1984; Saxe 1980). Because commercial solvers process constraints primarily by reducing variable domains, they tend to focus on types of consistency that relate to individual domains. The ideal is hyperarc consistency, which is achieved when the domains have been reduced as much as possible. Thus a constraint set is hyperarc consistent when any individual assignment xj = v that violates no constraint is part of some feasible solution. Hyperarc consistency does not imply consistency; it implies 2-consistency but is not equivalent to it. When all the constraints are binary (contain two variables), hyperarc consistency reduces to arc consistency, which in this case is equivalent to 2-consistency. Domain-reduction procedures often do not achieve hyperarc consistency. A popular weaker form of consistency is bounds consistency, which applies to integervalued variables. A constraint set is bounds consistent when any integer-valued variable assumes the smallest value in its domain in some feasible solution and assumes the largest value in its domain in some feasible solution. Bounds consistency can be achieved by interval arithmetic, which is a standard feature of constraint programming systems but is especially important in such nonlinear equation solvers as Newton

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(Van Hentenryck and Michel 1997; Van Hentenryck et al. 1998). Domain reduction can be viewed as the generation of “in-domain” constraints that restrict the values of variables to smaller domains. The resulting set of in-domain constraints is in effect a relaxation of the problem. Constraints generated by some other type of consistency maintenance can conceivably issue a stronger relaxation that consists of more interesting constraints. 4.2.2. Constraint-Based Search. Depth-?rst branching and constraint-based search represent opposite poles of a family of search methods. Depth-?rst branching incurs little overhead but is very in?exible. Once it begins to explore a subtree, it must search the entire subtree even when there seems little chance of ?nding a solution in it. Constraint-based search can be much more intelligent, but the mechanism that guides the search exacts a computational toll. After exploring an initial trial solution, it generates a constraint, called a nogood, that rules out the trial solution just explored (and perhaps others that must fail for the same reason). At any point in the search, a set of nogoods have been generated by past trial solutions. The next candidate solution is identi?ed by ?nding one that satis?es the nogoods. One might, for example, optimize the problem’s objective function subject to the nogoods, which may result in a more intelligent search. Benders decomposition, a well-known optimization method, is a special case of constraint-based search in which the nogoods are Benders cuts. Constraint-based search requires solution of a feasibility problem simply to ?nd the next trial solution to examine. One way to avoid solving such a problem is to process the nogood set suf?ciently to allow discovery of the next candidate solution without backtracking. A depth-?rst search can be interpreted as applying a very weak inference method to the nogoods, which suf?ces because the choice of the next solution is highly restricted. By strengthening the inference method, the freedom of choice can be broadened, until ?nally arriving at full constraint-based search. Such dependency-directed backtracking strategies as backjumping, backmarking and backchecking (Gaschnig 1977, 1978) are intermediate methods of this kind. More advanced

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methods include dynamic backtracking and partialorder dynamic backtracking (Ginsberg 1993; Ginsberg and McAllester 1994; McAllester 1993). All of these search methods represent a compromise between pure branching and pure constraint-based search. As shown by Hooker (2000), they can be organized under a uni?ed scheme in which each search method is associated with a form of resolution.

5.

Recent work on the boundary of optimization and constraint programming consists largely of three activities. One is the proposal of schemes for combining them. Another is the formulation of relaxations for predicates found in constraint programming models. A third is the adaptation of hybrid methods for a variety of practical applications, many of them in scheduling or design. A brief overview of some of this research is provided by Williams and Wilson (1998), and a detailed discussion by Hooker (2000). 5.1. Schemes for Integration Previous sections reviewed several ways in which logic can assist optimization, and in which optimization can play a role in constraint programming. Several more recent schemes have been proposed for integrating optimization and constraint programming on a more equal basis. These schemes might be evaluated by how well they implement the types of mutual reinforcement that were cited above as arguments for a hybrid method: ? From constraint programming: the use of global constraints to exploit substructures within a constraint set, and the application of ?ltering and constraint propagation to global constraints. The global constraints should include not only those currently used in constraint programming, but constraints that represent highly-structured subsets of inequality and equality constraints that typically occur in mathematical programming. ? From optimization: The association of relaxations (as well as ?ltering algorithms) with global constraints, and the use of specialized algorithms to solve relaxations and subproblems into which a problem is

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decomposed. The relaxations should include not only those currently used in mathematical programming, but new relaxations developed for popular global constraints in the constraint programming literature. 5.1.1. Double Modeling. One straightforward path to integration is to use a double modeling approach in which each constraint is formulated as part of a constraint programming model, or as part of a mixed integer model, or in many cases both. The two models are linked and pass domain reductions and/or infeasibility information to each other. Rodo?ek et al. (1997) and Wallace et al. (1997), for example, implemented this idea. They adapted the constraint logic programming system ECLiPSe so that linear constraints could be dispatched to commercial linear programming solvers (CPLEX and XPRESS-MP). Several investigations have supported the double modeling approach. Heipcke (1998, 1999) proposed several variations on it. Darby-Dowman and Little (1998) studied the relative advantages of integer programming and constraint programming models. Focacci et al. (1999a, 1999b) addressed the dif?culties posed by cost and pro?t functions with “costbased domain ?ltering.” It adapts to constraint programming the old integer programming idea of using reduced costs to ?x variables. A double modeling scheme can be implemented with ILOG’s OPL Studio (Van Hentenryck 1999), a modeling language that can invoke both constraint programming (ILOG) and linear programming (CPLEX) solvers and pass some information from one to the other. Double modeling occurs in all of the integration schemes discussed here and is perhaps best viewed as a ?rst step toward more speci?c schemes. 5.1.2. Branch and Infer. Bockmayr and Kasper (1998) proposed an interesting perspective on the integration of constraint programming with integer programming, based on the parallel between cutting planes and inference. It characterizes both constraint programming and integer programming as using a branch-and-infer principle. As the branching search proceeds, both methods infer easily-solved primitive constraints from nonprimitive constraints and pool the primitive constraints in a constraint

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store. Constraint programming has a large repertory of nonprimitive constraints (global constraints, etc.) but only a few, weak primitive ones: equations, disequations, and in-domain constraints. Integer programming enjoys a much richer class of primitive constraints, namely linear equalities and equations, but it has only one nonprimitive constraint: integrality. Bockmayr and Kasper’s scheme does not so much give directions for integration as explain why more explicit integration schemes are bene?cial: they enrich constraint programming’s primitive constraint store, thus providing better relaxations, and they enlarge integer programming’s nonprimitive constraint vocabulary, thus providing a more versatile modeling environment. 5.1.3. Integrated Modeling. It is possible for the very syntax of the problem constraints to indicate how constraint programming and optimization solvers are to interact. One scheme for doing so, introduced by Hooker and Osorio (1999) and elaborated by Hooker at al. (2000), is to write constraints in a conditional fashion. The model has the form minimize subject to f u g i x → Si u all i

In the conditional constraints gi x → Si u , gi x is a logical formula involving discrete variables x, and S u is a set of linear or nonlinear programming constraints with continuous variables u. The constraint says that if gi x is true, then the constraints in Si u are enforced. In degenerate cases a conditional can consist of only a discrete part ?gi x or only a continuous part Si u . The search proceeds by branching on the discrete variables; for instance, by splitting the domain of a variable xj . At each node of the search tree, constraint propagation helps reduce the domains of xj ’s, perhaps to the point that the truth or falsehood of gi x is determined. If gi x is true, the constraints in Si u become part of a continuous relaxation that is solved by optimization methods: minimize f u subject to Si u all i for which gi x is true

The relaxation also contains cutting planes and inequalities that relax discrete constraints. Solution of

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the relaxation provides a lower bound on the optimal value that can be used to prune the search tree. To take a simple example, consider a problem in which the objective is to minimize the sum of variable and ?xed cost of some activity. If the activity level u is zero, then total cost is zero. If u > 0, the ?xed cost is d and the variable cost is cu. The problem can be written minimize z subject to x → z ≥ cu + d 0 ≤ u ≤ M ?x → z = u = 0 (9)

the variables and their domains (de?ned by a declaration window), and one that holds a relaxation collectively generated by global constraint windows. The search window essentially implements a recursive call that may result in an exhaustive search (e.g., branching) or heuristic algorithm (e.g., local search). Examples of integrated modeling appear in Sections 5.3.1 and 5.3.2 below. 5.1.4. Benders Decomposition. Another promising framework for integration is a logic-based form of Benders decomposition, a well-known optimization technique (Benders 1962; Geoffrion 1972). The variables are partitioned x y , and the problem is written, minimize f x (11) subject to h x all i gi x y ? that minThe variable x is initially assigned a value x imizes f x subject to h x . This gives rise to a feasibility subproblem in the y variables: ? y gi x all i

where x is a propositional variable. One could also write the problem with a global constraint that might be named inequality-or: minimize z subject to inequality-or z ≥ cu + d x ?x 0≤u≤M

u=0 z=0

(The constraint associates propositions x ?x respectively with the two disjuncts.) The inequality-or constraint can now trigger the generation of a convex hull relaxation for the disjunction: z≥ c+ d x M 0≤x≤M (10)

These constraints are added to the continuous relaxation at the current node if x is undetermined. In general, the antecedent gi x of a conditional might be any constraint from a class that belongs to NP, and the consequent Si x any set of constraints over which one can practically optimize. Global constraints can be viewed as equivalent to a set of conditional constraints. These ideas are developed further by Hooker (2000) and Hooker et al. (2000, 2001). In a more general integrated modeling scheme proposed by Hooker (2001), a model consists of a sequence of “modeling windows” that correspond to global constraints, variable declarations, or search instructions. Each window is associated with a modeling language that is convenient for its purposes. The windows are implemented independently and are linked by only two data structures: one that holds

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The subproblem is attacked by constraint program? , then ming methods. If it has a feasible solution y ? y ? is optimal in (11). If there is no feasible solux tion, then a Benders cut Bx ? x is formulated. This is a ? and perhaps by many constraint that is violated by x other values of x that can be excluded for a similar reason. In the K th iteration, the master problem minimizes f x subject to h x and all Benders cuts that have been generated so far. minimize subject to f x hx Bx k x

k=1

K ?1

The master problem would ordinarily be a problem for which optimization methods exist, such as a ? of mixed integer programming problem. A solution x the master problem is labeled xK , and it gives rise to the next subproblem. If the subproblem is infeasible, one generates the next Benders cut BxK x . The procedure terminates when the subproblem is feasible, or when the master problem becomes infeasible. In the latter case, (11) is infeasible.

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The logic-based Benders decomposition described here was developed by Hooker (1995, 2000), Hooker and Yan (1995), and Hooker and Ottosson (1998) in a somewhat more general form in which the subproblem is an optimization problem. Just as a classical Benders cut is obtained by solving the linear programming dual of the subproblem, a generalized cut can be obtained by solving the inference dual of the subproblem. Jain and Grossmann (2001) found that a logic-based Benders approach can dramatically accelerate the solution of a machine scheduling problem, relative to commercial constraint programming and mixed integer solvers. Their work is described in Section 5.3.3. Hooker (2000) observed that the master problem need only be solved once if a Benders cut is generated for each feasible solution found during its solution. Thorsteinsson (2001) obtained an additional order-of-magnitude speedup for the Jain and Grossmann problem by implementing this idea, which he called branch and check. Classical Benders cuts can also be used in a hybrid context, as illustrated by Eremin and Wallace (2001). 5.2. Relaxations A key step in the integration of constraint programming and optimization is to ?nd good relaxations for global constraints and the other versatile modeling constructs of constraint programming. 5.2.1. Continuous Relaxations. There are basically two strategies for generating continuous relaxations of a constraint. One is to introduce integer variables as needed to write an integer programming model of the constraint. Then one can relax the integrality constraint on the integer variables. This might be called a lifted relaxation. Specialized cutting planes can be added to the relaxation as desired. The integer variables need not serve any role in the problem other than to obtain a relaxation; they may not appear in the original model or play a part in branching. If a large number of integer variables are necessary to write the model, one may wish to write a relaxation using only the variables that are in the original constraint. This might be called a projected relaxation. It conserves variables, but the number of constraints could multiply.

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Disjunctive programming and Jeroslow’s representability theorem, both mentioned earlier, provide general methods for deriving lifted relaxations. For example, Balas’ disjunctive formulation (4) provides a convex hull relaxation for disjunctions of linear systems. The big-M formulation (6) for such a disjunction, as well as many other big-M formulations, can be derived from Jeroslow’s theorem. This lifted relaxation can be projected onto the continuous variables x to obtain a projected relaxation. In the case of a disjunction of single linear inequalities, the projected relaxation is simply Beaumont’s elementary inequality (7). In addition, one can derive optimal separating inequalities for disjunctions of linear systems (Hooker and Osorio 1999), using a method that parallels cut generation in the lift-and-project method for 0-1 programming (Balas et al. 1996). This is one instance of a separating constraint, a key idea of integer programming that may be generalizable to a broader context. Many logical constraints that do not have disjunctive form are special cases of a cardinality rule: If at least k of x1 of y1 xm are true, then at least

yn are true.

Yan and Hooker (1999) describe a convex hull relaxation for propositions of this sort. It is a projected relaxation because no new variables are added. Williams and Yan (2001b) describe a convex hull relaxation of the at-least predicate, at-leastm x1 xn = k

xn which states that at least m of the variables x1 take the value k. Another example is the convex hull relaxation (10) of the ?xed charge problem (9). It is also a projected relaxation because it contains only the continuous variables u z. When a ?xed charge network ?ow problem is relaxed in this manner, the relaxation is a minimum cost network ?ow problem (Kim and Hooker 2001). It can be solved much more rapidly than the traditional relaxation obtained from the 0-1 model, which has no special structure that can be exploited by linear solvers. Piecewise linear functions can easily be given a convex hull relaxation without adding variables. Such

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a relaxation permits both a simpler formulation and faster solution than using mixed integer programming with specially ordered sets of type 2 (Ottosson et al. 1999). Réfalo (1999) shows how to use the relaxation in “tight cooperation” with domain reduction to obtain maximum bene?t. xn can be The global constraint all-different x1 given a convex hull relaxation. For simplicity let the n . The relaxation is domain of each xj be 1 based on the fact that the sum of any k distinct integers in 1 n must be at least 1 + 2 + · · · + k. As shown by Hooker (2000) and Williams and Yan (2001a), the following is a convex hull relaxation: 1 xj = n n + 1 2 j =1

j ∈J n

variables be positive. These relaxations can be useful when imposing type 1 and type 2 specially ordered constraints without the addition of integer variables. Hooker and Yan (2001) recently proposed a continuous relaxation for the cumulative constraint, one of the key global constraints due to its importance in scheduling. Suppose jobs 1 n start at times tn . Earliest start times a = a1 an and t = t1 latest start times b = b1 bn are given by the domains of t . Each job j has duration dj and consumes resources at the rate rj . The constraint cumulative t d r L ensures that the jobs running at any one moment do not collectively consume resources at a rate greater than L. An important special case occurs when r= 1 1 . In this case at most L jobs may be running at once, a constraint that is imposed in L-machine scheduling problems. Sophisticated (and often proprietary) domain reduction algorithms have been developed to reduce the intervals within which the start time tj of each job can be scheduled. A relaxation can be obtained by assembling cuts of the following forms. If a given subset of jobs j1 jk are identical (i.e., they have the same earliest start time a0 , duration d0 and resource consumption rate r0 ), then the following is a valid cut 1 t j 1 + · · · + t j k ≥ P + 1 a 0 + P 2k ? P + 1 Q d 0 2 where Q = L/r0 and P = k/Q ? 1. The cut is facetde?ning if there are no deadlines. More generally, the following is a valid cut for any subset of jobs j1 jk : tj1 + · · · + tjk ≥

k i =1

xj ≥

1 J 2

J +1

all J ? 1

n with J < n

Unfortunately the relaxation is rather weak. An important relaxation is the one for element x uk z , because it implements variable u1 indices. If each ui is a variable with bounds 0 ≤ ui ≤ mi , the following relaxation is derived (Hooker 2000; Hooker et al. 1999) from Beaumont’s elementary inequalities: 1 z≤ mi 1 z≥ mi ui +k?1 mi ui ?k+1 mi

i ∈Dx

i ∈Dx

i ∈Dx

i ∈Dx

where Dx is the current domain of x. If each ui satis?es 0 ≤ ui ≤ m0 , then Hooker (2000) shows that the convex hull relaxation of the element constraint simpli?es to

i ∈Dx

k?i+

1 ri 1 ? d 2 L 2 i

ui ? K ? 1 m 0 ≤ z ≤

i ∈Dx

ui

These relaxations can be very useful in practice, particularly when combined with domain reduction. De Farias et al. (2001) have developed relaxations, based on a lifting technique of integer programming, for constraints on which variables may be positive. For instance, one might require that at most one variable of a set be positive, or that only two adjacent

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Possibly stronger cuts can be obtained by applying a fast greedy heuristic. Lagrangean relaxation can also be employed in a hybrid setting. Sellmann and Fahle (2001) use it to strengthen propagation of knapsack constraints in an automatic recording problem. Benoist et al. (2001) apply it to a traveling tournament problem. It is unclear whether this work suggests a general method for integrating Lagrangean relaxation with constraint propagation.

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5.2.2. Discrete Relaxations. Discrete relaxations have appeared in the optimization literature from time to time. An early example is Gomory’s (1965) relaxation for integer-programming problems. It would be useful to discover discrete relaxations for constraint programming predicates that do not appear to have good continuous relaxations, such as alldifferent. Research in this area has scarcely begun. One approach is to solve a relaxation dual, which can be viewed as a generalization of a Lagrangean or surrogate dual. Given a problem of minimizing f x subject to constraint set S , one can de?ne a parameterized relaxation: minimize f x subject to S (12)

dual discussed earlier has a dependency graph with induced width of one, because it is a linear inequality. Dechter (1999, 2000) used a similar relaxation in “bucket-elimination” algorithms for solving in?uence diagrams; these algorithms are related to the nonserial dynamic programming methods for Bayesian networks mentioned earlier. The relaxation just described is ineffective for alldifferent, but there are other possibilities. One can relax the traveling salesman problem, for example, as follows (Hooker 2000). Here f x = j cxj xj +1 and S xn . Let S = and contains all-different x1 f x =

j

cxj xj +1 +

j

xj

?

j

j

Here ∈ is the parameter, S ? S for all ∈ , and f x ≤ f x for all x satisfying S x and all ∈ . In a Lagrangean relaxation, is a vector of nonnegative real numbers, S is a set of inequalities gi x ≤ 0, and f x = f x + i i gi x , where the sum is over inequalities i in S \ S . In a surrogate relaxation (Glover 1975), f x = f x and S = g x ≤ 0 , where the sum is over all inequalities i i i i in S . For any ∈ , the optimal value of (12) is a lower bound on the minimum value of f x subject to x ∈ S . The relaxation dual is the problem of ?nding the tightest possible bound over all ; it is the maximum of subject to ∈ . One strategy for obtaining a discrete relaxation is to solve a relaxation dual when is a discrete variable. For example, one can create a relaxed constraint set S by removing arcs from the dependency graph G for S , resulting in a sparser graph G . The parameter might be the set of arcs in G . The set might contain ’s for which G has small induced width. The relaxation could then be solved by nonserial dynamic programming. An arc xi xj can be removed from G by projecting each constraint C of S onto all variables except xi , and onto all variables except xj , to obtain new constraints. The new constraints are added to S and C is deleted. This approach is explored by Hooker (2000). It can be augmented by adding other relaxations that decouple variables. The relaxation obtained by the roof

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where consists of vectors of integers, perhaps primes, and is a constant. The second term vanishes when x satis?es the all-different constraint. Classical dynamic programming can compute , and a heuristic method can be applied to the dual problem of maximizing with respect to . 5.3. Applications Applications to processing network design, lot sizing, and machine scheduling are discussed below. The ?rst two illustrate the integrated modeling approach, and the third the Benders approach. Subsequently a literature survey of other applications is presented. 5.3.1. Processing Network Design. An early application of integrated modeling (Section 5.1.3) was to processing network design problems in the chemical industry (Grossmann et al. 1994; Raman and Grossmann 1994). It illustrates the use of conditional and global constraints as well as the concept of a “don’t-be-stupid” constraint. Figure 3 displays a small instance of a processing network design problem. The object is to determine which unit to include in the network so as to maximize net income (revenue minus cost). Each processing unit i incurs a ?xed cost di and delivers revenue di ui , where the variable ui represents the ?ow volume entering the unit. The revenue is normally positive for the terminal units (units 4-6) because their output is sold, and it is normally negative for the remaining units.

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in practice be replaced by inequality-or global constraints as discussed in Section 5.1.3, in order to obtain the continuous relaxation described there. One can accelerate the search by making some simple observations. It is clearly pointless to install unit 1 unless one installs unit 2 or unit 3. This can be written y 1 → y 2 ∨ y3 Similarly there is no point to installing unit 2 unless one installs units 1 and 6, or unless one installs unit 4 or unit 5. Rules of this sort can be listed as follows and added to the model: ? ? y 3 → y4 y1 → y2 ∨ y3 ?y → y y 3 → y 5 ∨ y6 ? ? ? 2 1 ? ? (14) cnf ? y2 → y4 ∨ y5 y 4 → y 2 ∨ y3 ? ? ? ? y2 → y6 y 5 → y 2 ∨ y3 ? y3 → y1 y 6 → y 2 ∨ y3 These “don’t-be-stupid” rules are not implied by the original model, because they exclude feasible solutions. There is nothing infeasible about installing a unit that is necessarily idle. Yet computational experiments show (Grossmann et al. 1994) that without don’t-be-stupid constraints, a branch-and-bound search (traditional or hybrid) can examine and reject a large number of solutions in which useless units are installed. A large number of don’t-be-stupid rules can be used when they are processed symbolically rather than added to the relaxation. After one or more variables yi are ?xed by branching, logical inference methods can infer that other variables must be ?xed, even in cases where integer-programming methods would not detect a ?xed variable. This reduces the number of solutions examined, which can be very useful in engineering design problems where the continuous relaxation can be nonlinear and hard to solve. A global constraint “cnf” is used in (14) to signal the solver to apply a logical inference procedure, such as the resolution method. A similar consideration applies to symmetrybreaking rules, which can be used in some problems to help avoid enumerating equivalent solutions. They can appear in large numbers if processed symbolically.

313

Figure 3

Superstructure for a Processing Network Design Problem

The model can be written, maximize

i

ri u i ?

subject to u = Ax bu = Bx y i → zi = di ?yi → ui = 0 u≤c u x≥0

i

zi

a b c d e f

all i all i

(13)

In the objective function (a), zi is the ?xed cost actually incurred for unit i. Constraint (b) is a set of linear equations that compute the ?ow into each unit. Speci?cally, these equations are u1 = x12 + x13 u2 = x12 u3 = x13 u4 = x24 + x34 u5 = x25 + x35 u6 = x26 + x36

where xij is the ?ow from unit i to unit j . Constraint (c) computes the ?ows out of each intermediate unit. In this case there are only two intermediate units (2 and 3), and (c) reduces to 0 6u2 = x24 + x25 0 4u2 = x26 0 7u3 = x34 0 3u3 = x35 + x36

Note that the two ?ow volumes out of unit 2 must bear a ?xed proportion, and similarly for unit 3. Constraint (f) enforces a capacity constraint. Constraints (d) and (e) would be written as big-M constraints in traditional integer programming. They state that if unit i is installed (yi = true), then ?xed cost di is incurred, whereas if it is omitted (yi = false), the unit can accept no ?ow. These constraints would

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5.3.2. Lot Sizing. A lot-sizing problem discussed by Wolsey (1998) illustrates some of the considerations that arise when designing a relaxation in the context of integrated modeling. (We modify the example slightly.) Several products i must be shipped in speci?ed quantities dit on each day t . However, at most one product can be manufactured on a given day, so that inventory must be accumulated. The unit daily holding cost for product i is hi , and qij is the cost of switching the manufacturing process from product i to product j (qii = 0). A product may be manufactured for a fraction of a day or for several days in a row, and in fact there may be a minimum run length Ri . The objective is to minimize the total cost of holding inventory and switching from one product to another. Let yt be the product manufactured on day t , with yt = null if nothing is manufactured. Note that yt need not have a numerical value. Let xit be the quantity of product i manufactured on day t . The integrated model, below, is considerably simpler than the 0-1 model presented by Wolsey. minimize

t

the minimum run length. Although it has a conditional form, it is purely discrete and is a degenerate conditional in the sense de?ned above. The objective function and constraints (b), (d) and (e) naturally form a continuous relaxation because they are the consequents of degenerate conditionals. Constraint (c) is implemented with an element constraint as follows. Replace (c) with vt ≥ zt and impose the constraint element yt yt?1 Q zt where Q is an array of elements qij . The constraint has the effect of setting z = qyt?1 yt . Constraints (f) and (g) are not represented in the relaxation. The relaxation that results is not as tight as the integer programming relaxation, but it is much smaller due to the absence of triply-subscripted variables and can be solved more rapidly. Constraint propagation applied to (f) and (g) helps to keep the search tree in check. Ef?cient domain reduction is available for the element constraints, and it also becomes part of the propagation procedure. 5.3.3. Machine Assignment and Scheduling. A machine assignment and scheduling problem of Jain and Grossmann (2001) can be interpreted as illustrating the Benders framework of Section 5.1.4. In computational tests it was solved much more rapidly than pure 0-1 programming or constraint programming models. The problem may be stated as follows. Each job j is assigned to one of several machines i that operate at different speeds. Each assignment results in a processing time Dij and incurs a processing cost Cij . There is a release date Rj and a due date Sj for each job j . The objective is to minimize processing cost while observing release and due dates. To formulate the problem, let xj be the machine to which job j is assigned and tj the start time for job j . It also convenient to let tj xj = i denote the tuple of start times of jobs assigned to machine i, arranged in increasing order of the job number. The problem can

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ut + v t

i

a all t b c d e f g

subject to ut ≥

hi sit

vt ≥ qyt?1 yt all i t all i t si t?1 + xit = dit + sit 0 ≤ xit ≤ C sit ≥ 0 all i t yt = i → xit = 0 yt ?1 = i → yt+1 = ··· = yt+Ri ?1 = i yt = i

In the objective function (a), ut represents the holding cost incurred on day t , and vt the switchover cost. Constraints (b) and (c) de?ne these costs, where sit is the stock level of product i on day t . Constraint (c) is particularly interesting because it uses a variable index. It de?nes the switchover cost for today (day t ) to be the cost qyt?1 yt of switching from product yt?1 yesterday to product yt today. The traditional model uses 0-1 variables with three subscripts to de?ne this cost, resulting in a very large number of 0-1 variables and a correspondingly large continuous relaxation. Constraint (d) is an inventory balance constraint, and conditional (f) requires the output of a product to be zero if there is no setup for it. Constraint (g) enforces

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HOOKER Logic, Optimization, and Constraint Programming

be written minimize

j

Cx j j

a b c all i d (15)

all j subject to tj ≥ Rj tj + Dxj j ≤ Sj all j cumulative tj xj = i Dij xj = i e 1

machine i cannot all be scheduled on that machine. In fact, going beyond Jain and Grossmann (2001), there may be a subset J of these jobs that cannot be scheduled on machine i. This gives rise to a Benders cut stating that at least one of the jobs in J must be assigned to another machine.

j ∈J

The objective function (a) measures the total processing cost. Constraints (b) and (c) observe release times and deadlines. The cumulative constraint (d) ensures that jobs assigned to each machine are scheduled so that they do not overlap. (Recall that e is a vector of ones.) The problem has two parts: the assignment of jobs to machines, and the scheduling of jobs on each machine. The assignment problem is treated as the master problem and solved with mixed integer programming methods. Once the assignments are made, the subproblems are dispatched to a constraint programming solver to ?nd a feasible schedule. If there is no feasible schedule, a Benders cut is generated. Variables x and t go into the master problem. Finite domain variables t that are linked with t appear in the subproblem. The problem is therefore written as follows for Benders decomposition. minimize

j

xj = i

(18)

Let xk be the solution of the kth master problem, I k the set of machines i in the resulting subproblem for which the schedule is infeasible, and Jki the infeasible subset. The master problem can now be written, minimize

j

Cx j j (19) K

subject to

tj ≥ R j all j t j + Dx j j ≤ S j all j xj = i i ∈ Ik k = 1

j ∈Jki

The master problem can be reformulated for solution with conventional integer programming technology. Let xij be a 0-1 variable that is 1 when job j is assigned to machine i. The master problem (19) can be written minimize

i j

Cx j j

a b c all i d e f (16)

Cij xij all j i ∈ Ik k = 1

j

a b c K all i d e f

subject to tj ≥ Rj all j all j tj + Dxj j ≤ Sj cumulative tj xj = i Dij xj = i e 1 tj ≥ Rj all j tj + Dxj j ≤ Sj all j link tj tj all j

subject to

all j tj ≥ Rj tj + Dij xij ≤ Sj

j ∈Jki j

1 ? xij ≥ 1

j

i

Dij xij ≤ max Sj ? min Rj all i j

xij ∈ 0 1

The link constraint requires that Lj ≤ tj ≤ Uj , where Lj is the smallest and Uj the largest time in the current domain of tj . ? , the subproblem is If x has been ?xed to x tj ≥ Rj all j t j + Dx ≤ Sj all j ?j j ?j = i cumulative tj x (17) ?j = i e 1 all i Dij x

Constraints (e) are valid cuts added to strengthen the continuous relaxation. They simply say that the total processing time on each machine must ?t between the earliest release time and the latest deadline. 5.3.4. Other Applications. A wide variety of applications demonstrate the potential of combining optimization with constraint programming or logic-based methods. Successful applications to chemical processing network design, already mentioned, are described by Grossmann et al. (1994), Hooker and Osorio (1999), Raman and Grossmann (1991,

315

The subproblem can be decomposed into smaller problems, one for each machine. If a smaller problem is infeasible for some i, then the jobs assigned to

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HOOKER Logic, Optimization, and Constraint Programming

¨ 1994) and Turkay and Grossmann (1996). A hybrid approach also advances the state for the art for truss structure design (Bollapragada et al. 2001). In transportation, hybrid methods have been applied to vehicle routing with time windows (Focacci et al. 1999b; Caseau et al. 2001), vehicle routing combined with inventory management (Lau and Liu 1999), crew rostering (Caprara et al. 1998; Junker et al. 1999), the traveling tournament problem (Benoist et al. 2001), and the classical transportation problem with piecewise linear costs (Réfalo 1999). Other applications include inventory management (Rodo?ek et al. 1997), of?ce cleaning (Heipcke 1999), product con?guration (Ottosson and Thorsteinsson 2000), generalized assignment problems (DarbyDowman et al. 1997), multidimensional knapsack problems (Osorio and Glover 2001), automatic recording of television shows (Sellmann and Fahle 2001), and resource allocation in ATM networks (Lauvergne et al. 2001). Most applications seem to have been to scheduling. These include machine scheduling (Heipcke 1998; Raman and Grossmann 1993), sequencing with setups (Focacci et al. 1999a), hoist scheduling (Rodo?ek and Wallace 1998), employee scheduling (Partouche 1998), dynamic scheduling (El Sakkout et al. 1998), and lesson timetables (Focacci et al. 1999a). Production scheduling applications include scheduling with resource constraints (Pinto and Grossmann 1997) and with labor resource constraints in particular (Heipcke 1999), two-stage process scheduling (Jain and Grossmann 2001), machine allocation and scheduling (Puget and Lustig 1999), production ?ow planning with machine assignment (Heipcke 1999), scheduling with piecewise linear costs (Ottosson et al. 1999), scheduling with earliness and tardiness costs (Beck 2001), and organization of a boat party (Hooker and Osorio 1999; Smith et al. 1996). These applications only begin to tap the potential of integrated methods. New ones are appearing as this article is written.

6.

Several research needs surfaced in the course of the foregoing discussion. One might begin with two broad research goals:

316

Future Research

? Develop a robust and widely acceptable integration scheme for optimization and constraint programming. Perhaps the ?rst step is to take the integrated modeling and decomposition schemes (discussed earlier) deeper into practical applications to see how they fare. ? Develop a generic modeling language that can serve as the basis for one or more viable commercial products that integrate optimization and constraint programming. The language should (a) strike a balance between the procedural and the declarative, (b) indicate by its syntactic structure how optimization and constraint programming should interact, and (c) allow the user to encode knowledge of problem structure, for instance by using global constraints. Some more speci?c research projects might also be mentioned. ? Combine integrated modeling and decomposition into a single framework. ? Develop a repertory of useful global constraints for each of several application domains. They should succinctly capture characteristic modeling situations of the domain and be amenable to logic processing and/or relaxation. ? Take advantage of the optimization community’s experience with continuous relaxations to develop such relaxations for constraint programming predicates. ? Explore discrete relaxations for constraint programming predicates, such as those based on discrete relaxation duals. ? Move from the exclusive focus on domain reduction to other types of consistency maintenance as well. Rather than generate in-domain and other very simple constraints, generate a wider variety of constraints that comprise tighter and yet soluble relaxations. ? Generalize the integer programming idea of a separating constraint to a broader context. There is something to be said for isolated research groups working in parallel. They may develop complementary approaches that at some point can be pro?tably combined. This seems to be the situation with optimization and constraint programming. Their ignorance of each other over a period of years was perhaps a good thing. It is hard to pick the right time for interaction to begin, but in any event it has begun.

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Accepted by John W. Chinneck; received April 2000; revised November 2001; accepted November 2001.

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