The goal of this paper is to develop models and methods that use complementary strengths of Mixed Integer Linear Programming (MILP) and Constraint Programming (CP) techniques to solve problems that are otherwise intractable if solved using either of the two methods. The class of problems considered in this paper have the characteristic that only a subset of the binary variables have non-zero objective function coefficients if modeled as an MILP. This class of problems is formulated as a hybrid MILP/CP model that involves some of the MILP constraints, a reduced set of the CP constraints, and equivalence relations between the MILP and the CP variables. An MILP/CP based decomposition method and an LP/CP-based branch-and-bound algorithm are proposed to solve these hybrid models. Both these algorithms rely on the same relaxed MILP and feasibility CP problems. An application example is considered in which the least-cost schedule has to be derived for processing a set of orders with release and due dates using a set of dissimilar parallel machines. It is shown that this problem can be modeled as an MILP, a CP, a combined MILP-CP OPL model (Van Hentenryck 1999), and a hybrid MILP/CP model. The computational performance of these models for several sets shows that the hybrid MILP/CP model can achieve two to three orders of magnitude reduction in CPU time.

. We present how to apply Constraint Based Column Generation to a large class of subproblems, namely Constrained Knapsack Problems (CKP). They evolve e.g. from Cutting Stock Problems (see [7]) with additional constraints on the cutting patterns. Focussing on Constrained Knapsack Problems, we developed a new reduction algorithm for KP. It is being used as propagation routine for the CKP with O(n log n) preprocessing time and O(n) time per call. This sums up to an amortized time of O(n) for (log n) calls. Keywords: Constrained Based Column Generation, Constrained Knapsack Problems, Cutting Stock Problems, Reduction Algorithms. 1 Introduction Recently, a new framework for the integration of CP and OR within column generation approaches was developed, the so called Constraint Based Column Generation [11]. It describes a generic way of how to treat arbitrary constraints for the constrained subproblem in the column generation phase. The approach has been successfully used for the C...

Integer programming and constraint (logic) programming are two traditional techniques for solving combinatorial optimization problems; the former based on linear programming relaxations and the latter on constraint propagation. Attempts to combine them have mainly focused on incorporating either technique into the framework of the other traditional models have been left intact. We argue that a rethinking of our modeling traditions is necessary to achieve the greatest benet of such an integration. We propose a declarative modeling framework in which the structure of the constraints indicates how LP and CP can interact to solve the problem. 1 Introduction Linear programming (LP) and constraint propagation (CP) are two complementary techniques with potential for integration to benet the solution of combinatorial optimization problems. Integer programming (IP) has been successfully applied to a range of problems, such as capital budgeting, bin packing and traveling salesman pr...

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.

We present Branch-and-Check, a hybrid framework integrating Mixed Integer Programming and Constraint Logic Programming, which encapsulates the traditional Benders Decomposition and Branch-and-Bound as special cases. In particular we describe its relation to Benders and the use of nogoods and linear relaxations. We give two examples of how problems can be modelled and solved using Branch-and-Check and present computational results demonstrating more than order-of-magnitude speedup compared to previous approaches. We also mention important future research issues such as hierarchical, dynamic and adjustable linear relaxations.

Constraint Programs and Mixed Integer Programs are closely related optimization problems originating from different scientific areas. Today’s state-of-the-art algorithms of both fields have several strategies in common, in particular the branch-and-bound process to recursively divide the problem into smaller subproblems. On the other hand, the main techniques to process each subproblem are different, and it was observed that they have complementary strengths. We present the programming framework Scip that integrates techniques from both fields in order to exploit the strengths of both, Constraint Programming and Mixed Integer Programming. In contrast to other proposals of recent years to combine both fields, Scip does not focus on easy implementation and rapid prototyping, but is tailored towards expert users in need of full, in-depth control and high performance.

The complementing strengths of Constraint (Logic) Programming (CLP) and Mixed Integer Programming (IP) have recently received signicant attention. Although various optimization and constraint programming packages at a rst glance seem to support mixed models, the modeling and solution techniques encapsulated are still rudimentary. Apart from exchanging bounds for variables and objective, little is known of what constitutes a good hybrid model and how a hybrid solver can utilize the complementary strengths of inference and relaxations. This paper adds to the eld by identifying constraints as the essential link between CLP and IP and introduces an algorithm for bidirectional inference through these constraints. Together with new search strategies for hybrid solvers and cut-generating mixed global constraints, solution speed is improved over both traditional IP codes and newer mixed solvers. Keywords: Mixed Integer Programming, Constraint Logic Programming, Integration, Mixed Global Contraints, Dynamic Linear Relaxations, Inference. AMS Subject classication: 68N99,68Q99,68T99,90C05,90C11,90C27. 1.

We present a general framework for augmenting instances of the Disjunctive Temporal Problem (DTP) with finite-domain constraints. In this new formalism, the bounds of the temporal constraints become conditional on the finite-domain assignment. This hybridization makes it possible to reason simultaneously about temporal relationships between events as well as their nontemporal properties. We provide a special case of this hybridization that allows reasoning about a limited form of spatial constraints; namely, the travel time induced by the locations of a set of activities. We develop a least-commitment algorithm for efficiently finding solutions to this combined constraint system and provide empirical results demonstrating the effectiveness of our approach.

Integer and combinatorial optimization problems constitute a major challenge for algorithmics. They arise when a large number of discrete organizational decisions have to be made, subject to constraints and optimization criteria. This thesis

Constraint programming offers modeling features and solution methods that are unavailable in mathematical programming but are often flexible and efficient for scheduling and other combinatorial problems. Yet mathematical programming is well suited to declarative modeling languages and is more efficient for some important problem classes. This raises this issue as to whether the two approaches can be combined in a declarative modeling framework. 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 influenced the modeling framework of mathematical programming. Inequality constraints, for example, are ubiquitous not only ...