The Potential of Constraint Programming

By -
The Potential of Constraint Programming

In the 1990s, constraint programming features, including powerful constraint-solving algorithms, were successfully incorporated into a number of general-purpose programming languages, as well as several special-purpose programming languages. This brought computer science closer and closer to the Holy Grail of computer programming, namely, allowing the user to simply state the problem and then the computer will solve it.

As word of this exciting development began to spread beyond the computer science community, researchers in operations research began to realize the great potential of integrating constraint programming with the traditional techniques of integer programming (and other areas of mathematical programming as well). The much greater flexibility in expressing the constraints of the problem should greatly increase the ability to formulate valid models for complex problems. It also should lead to much more compact and straight- forward formulations. In addition, by reducing the size of the feasible region that needs to be considered while efficiently finding solutions within this region, the constraint-solving algorithms of constraint programming might help accelerate the progress of integer programming algorithms in finding an optimal solution.

Because of their substantial differences, integrating constraint programming with integer programming is a very difficult task. Since integer programming does not recognize most of the constraints of constraint programming, this requires developing computer- implemented procedures for translating from the language of constraint programming to the language of integer programming and vice versa. Good progress is being made, but this undoubtedly will continue to be one of the most active areas of OR research for some years to come.

To illustrate the way in which constraint programming can greatly simplify the formulation of integer programming models, we now will introduce two of the most important “global constraints” of constraint programming. A global constraint is a constraint that succinctly expresses a global pattern in the allowable relationship between multiple variables. Therefore, a single global constraint often can replace what used to require a large number of traditional integer programming constraints while also making the model considerably more readable. To clarify the presentation, we will use very simple examples that don’t require the use of constraint programming to illustrate global constraints, but these same types of constraints also can readily be used for some much more complicated problems.

Comments

Post a Comment

Popular posts from this blog

DUALITY THEORY:THE ESSENCE OF DUALITY THEORY

By -
Image
THE ESSENCE OF DUALITY THEORY Given our standard form for the primal problem at the left (perhaps after conversion from another form), its dual problem has the form shown to the right. Thus, with the primal problem in maximizatio n form, the dual problem is in minimization form instead. Furthermore, the dual problem uses exactly the same pa r amete r s as the primal problem, but in different locations, as summarized below. 1. The coefficients in the objective function of the primal problem are the right-hand sides of the functional constraints in the dual problem. 2. The right-hand sides of the functional constraints in the primal problem are the coef- ficients in the objective function of the dual problem. 3. The coefficients of a variable in the functional constraints of the primal problem are the coefficients in a functional constraint of the dual problem. To highlight the comparison, now look at these same two problems in matrix notation (as introduced at the beginn...
Read more

NETWORK OPTIMIZATION MODELS:THE MINIMUM SPANNING TREE PROBLEM

By -
Image
THE MINIMUM SPANNING TREE PROBLEM The minimum spanning tree problem bears some similarities to the main version of the shortest-path problem presented in the preceding section. In both cases, an undi r ecte d and connected network is being considered, where the given information includes some mea- sure of the positive length (distance, cost, time, etc.) associated with each link. Both problems also involve choosing a set of links that have the shortest total length among all sets of links that satisfy a certain property. For the shortest-path problem, this property is that the chosen links must provide a path between the origin and the destination. For the minimum spanning tree problem, the required property is that the chosen links must provide a path between ea c h pair of nodes. The minimum spanning tree problem can be summarized as follows: 1. You are given the nodes of a network but not the links. Instead, you are given the po- tential links and the positive length for each ...
Read more

INTEGER PROGRAMMING:THE BRANCH-AND-CUT APPROACH TO SOLVING BIP PROBLEMS

By -
TH E BRANCH-AND-CUT APPROACH TO SOLVING BIP PROBLEMS Integer programming has been an especially exciting area of OR since the mid-1980s be- cause of the dramatic progress being made in its solution methodology. Background To place this progress into perspective, consider the historical background. One big break- through had come in the 1960s and early 1970s with the development and refinement of the branch-and-bound approach. But then the state of the art seemed to hit a plateau. Rel- atively small problems (well under 100 variables) could be solved very efficiently, but even a modest increase in problem size might cause an explosion in computation time be- yond feasible limits. Little progress was being made in overcoming this exponential growth in computation time as the problem size was increased. Many important problems arising in practice could not be solved. Then came the next breakthrough in the mid-1980s, with the introduction of the b r an c h- and-cu t approach to sol...
Read more