A Declarative Modeling Framework that Integrates Solution Methods

Constraint programming o ers modeling features and solution methods that are un available in mathematical programming but its models are not declarative This raises this issue as to whether the two approaches can be combined in a declarative modeling frame work This paper proposes a general declarative modeling system in which the conditional structure of the constraints shows how to integrate any checker and any special purpose solver In particular this integrates constraint programming and optimization methods because the checker can consist of constraint propagation methods and the solver can be a linear or nonlinear programming routine Solution technology has strongly in uenced the modeling framework of mathematical pro gramming Inequality constraints for example are ubiquitous not only because they are useful for problem formulation but because solvers can deal with them Linear programming is teth ered to a highly structured modeling language because the solver requires it Its historical popularity has been possible only because of George Dantzig s discovery that its restricted vocabulary is surprisingly versatile in applications Nonetheless mathematical programmers are becoming more aware of the limitations of inequality based modeling This may be due primarily to the recent commercial success of constraint programming and its richer modeling resources Constraint programming systems permit not only such logical constructions as disjunctions and implications but they include all di erent constraints and other highly useful predicates that are foreign to a mathematical programming environment Variables need not even have numerical values and they can appear in subscripts The power of this modeling framework is dramatically illustrated by the traveling salesman problem which requires exponentially many constraints in its most popular integer programming formulation In a constraint programming milieu it can be written with a single all di erent constraint if variable subscripts are permitted in the objective function Two things about the conventional modeling habits of mathematical programmers how ever are arguably very right One is that their models are fully declarative meaning that they state the problem without describing the solution procedure A linear or integer programming model can be passed to any number of solvers without modi cation Constraint programming systems by contrast typically require the user to write high level C code or some other script that invokes customized procedures A second strength of mathematical programming is that the very structure of the modeling language anticipates the solution task Although the human modeler may think little about algorithmic matters the syntax of the language forces the model into the solver s mold This is not to say that constraint programming permits formulations that are unsupported by solvers It avoids this however only to the extent that the modeling language is procedural in the rst place An all di erent constraint for example is not so much a constraint as a procedure that propagates the constraint If a set of variables are required to assume di erent values and the rst ve variables have already been assigned values then the propagation procedure removes those ve values from the domain of each of the remaining variables i e from the set of values the variable is allowed to take Although mathematical programming enjoys advantages in its declarative approach it would nonetheless bene t from the larger modeling repertory of constraint programming and the solution techniques that go with it The issue is whether there a principled way for a modeling system to integrate optimization and constraint satisfaction methods without obliging the modeler to think about the solution algorithm More generally is there a general way in which a modeling framework can be sensitive to available solution technology while remaining declarative This paper is intended to address these questions A simple example can illustrate how a modeling language might take into account the strengths and weakness of available solution methods Suppose that the language consists of systems of linear inequalities and logical formulas It permits an expression like