Benders decomposition uses a strategy of "learning from one's mistakes." The aim of this paper is to extend this strategy to a much larger class of problems. The key is to generalize the linear programming dual used in the classical method to an "inference dual." Solution of the inference dual takes the form of a logical deduction that yields Benders cuts. The dual is therefore very different from other generalized duals that have been proposed. The approach is illustrated by working out the details for propositional satisfiability and 0-1 programming problems. Computational tests are carried out for the latter, but the most promising contribution of logic-based Benders may be to provide a framework for combining optimization and constraint programming methods. *

We describe the design and implementation of a finite domain constraint solver embedded in a Prolog system using an extended unification mechanism via attributed variables as a generic constraint interface. The solver is essentially a scheduler for indexicals, i.e. reactive functional rules encoding local consistency methods performing incremental constraint solving or entailment checking, and global constraints, i.e. general propagators which may use specialized algorithms to achieve a higher degree of consistency or better time and space complexity. The solver has an open-ended design: the user can introduce new constraints, either in terms of indexicals by writing rules in a functional notation, or as global constraints via a Prolog programming interface. Constraints defined in terms of indexicals can be linked to 0/1-variables modeling entailment; thus indexicals are used for constraint solving as well as for entailment testing. Constraints can be arbitrarily combined using the propositional connectives by automatic expansion to systems of reified constraints.

Optimization and constraint satisfaction methods are complementary to a large extent, and there has been much recent interest in combining them. Yet no generally accepted principle or scheme for their merger has evolved. We propose a scheme based on two fundamental dualities: the duality of search and inference, and the duality of strengthening and relaxation. Optimization as well as constraint satisfaction methods can be seen as exploiting these dualities in their respective ways. Our proposal is that rather than employ either type of method exclusively, one can focus on how these dualities can be exploited in a given problem class. The resulting algorithm is likely to contain elements from both optimization and constraint satisfaction, and perhaps new methods that belong to neither.

Efforts aimed at combining operations research and constraint programming have become increasingly prominent and successful in the last few years. It is now widely recognized that integration, e.g., inference in the form of constraint propagation and relaxation in the form of linear programming, can yield substantial results. In this paper, we argue the benefits of constraint programming's global constraints as a basis for such an integration and discuss the advantages along with some examples. We illustrate the integration on the global cardinality structure, on piecewise linear functions, on variable subscripts, on the cycle structure and on resource constraints. Each example is completed with a case study.