Monique Guignard

Lagrangean decomposition: A model yielding stronger lagrangean bounds

Given a mixed-integer programming problem with two matrix constraints, it is possible to define a Lagrangean relaxation such that the relaxed problem decomposes additively into two subproblems, each having one of the two matrices of the original problem as its constraints. There is one Lagrangean multiplier per variable. We prove that the optimal value of this new Lagrangean dual dominates the optimal value of the Lagrangean dual obtained by relaxing one set of constraints and give a necessary condition for a strict improvement. We show on an example that the resulting bound improvement can be substantial. We show on a complex practical problem how Lagrangean decomposition may help uncover hidden special structures and thus yield better solution methodology.

Generalized Kuhn–Tucker Conditions for Mathematical Programming Problems in a Banach Space

Generalized Kuhn–Tucker conditions stated in this paper correspond to the optimality conditions for mathematical programming problems in a Banach space. Constraint qualifications given before can be regarded as special cases of the present constraint qualification introduced to prove the necessity. Pseudoconvexity of the constraint set rather than convexity is required for sufficiency. In case this hypothesis fails to be satisfied, second order optimality conditions are sufficient for an isolated local optimum.

School redistricting: embedding GIS tools with integer programming

The paper deals with a school redistricting problem in which blocks of a city must be assigned to schools according to diverse criteria. Previous approaches are reviewed and some desired properties of a good school districting plan are established. An optimization model together with a geographic information system environment are then proposed for finding a solution that satisfies these properties. A prototype of the system is described, some implementation issues are discussed, and two real-life examples from the city of Philadelphia are studied, one corresponding to a relatively easy to solve problem, and the other to a much harder one. The trade-offs in the solutions are analysed and feasibility questions are discussed. The results of the study strongly suggest that ill-defined spatial problems, such as school redistricting, can be addressed effectively by an interaction between objective analysis and subjective judgement.

A direct dual method for the mixed plant location problem with some side constraints

The mixed plant location problem (mixed in the sense of allowing capacitated as well as uncapacitated plants) is a difficult and important mixed integer problem. We give a direct dual method, consisting of several phases (each of which appears essential for some data), to resolve a strong relaxed form of the problem with additional constraints over the integer variables (user specified, or derived from the data themselves). When all features of the algorithm are employed, there appears to be no difficulty with problems of 100 plants, even in an inefficient computer implementation. The primal solutions which we derive from the orthogonality conditions and a simple greedy heuristic are almost always much better than those we obtain from a standard relaxed problem in the Lagrangean sense. With an enumeration code in an efficient implementation we would expect to be capable of resolving very large problems (of perhaps up to 500 or 1000 plants) to within practically well acceptable tolerances.

A level-2 reformulation-linearization technique bound for the quadratic assignment problem

Abstract This paper studies polyhedral methods for the quadratic assignment problem. Bounds on the objective value are obtained using mixed 0–1 linear representations that result from a reformulation–linearization technique (rlt). The rlt provides different “levels” of representations that give increasing strength. Prior studies have shown that even the weakest level-1 form yields very tight bounds, which in turn lead to improved solution methodologies. This paper focuses on implementing level-2. We compare level-2 with level-1 and other bounding mechanisms, in terms of both overall strength and ease of computation. In so doing, we extend earlier work on level-1 by implementing a Lagrangian relaxation that exploits block-diagonal structure present in the constraints. The bounds are embedded within an enumerative algorithm to devise an exact solution strategy. Our computer results are notable, exhibiting a dramatic reduction in nodes examined in the enumerative phase, and allowing for the exact solution of large instances.

A Lagrangian relax-and-cut approach for the sequential ordering problem with precedence relationships

The sequential ordering problem with precedence relationships was introduced in Escudero [7]. It has a broad range of applications, mainly in production planning for manufacturing systems. The problem consists of finding a minimum weight Hamiltonian path on a directed graph with weights on the arcs, subject to precedence relationships among nodes. Nodes represent jobs (to be processed on a single machine), arcs represent sequencing of the jobs, and the weights are sums of processing and setup times. We introduce a formulation for the constrained minimum weight Hamiltonian path problem. We also define Lagrangian relaxation for obtaining strong lower bounds on the makespan, and valid cuts for further tightening of the lower bounds. Computational experience is given for real-life cases already reported in the literature.

Logical Reduction Methods in Zero-One Programming—Minimal Preferred Variables

In the first part of this paper, the concept of logical reduction is presented Minimal preferred variable inequalities are introduced, and algorithms are given for their calculation. A simple illustrative example is carried along from the start, further examples are provided later. The second part of the paper describes certain properties of the generated logical inequalities. It then explains some of the decreases of computational effort which may be achieved by the use of minimal preferred inequalities and outlines a number of concrete applications with some numerical results. Finally, a number of more recent concepts and results are discussed, among them the notion of “probing” and a related zero-one enumeration code for large scale problems under the extended control language of MPSX/370.

A Problem of Forest Harvesting and Road Building Solved Through Model Strengthening and Lagrangean Relaxation

We consider a problem of forest planning on pine plantations over a two to five year horizon. Basic decisions concern the areas to harvest in each period, the amount of timber to produce to satisfy aggregate demands for log exports, sawmills and pulp plants, and the roads to build for access and storage of timber. A linear programming model with 0--1 variables describes the decision process. Solution strategies involve strengthening of the model, lifting some of the constraints, and applying Lagrangean relaxation. Results on real planning problems show that even as these problems become more complex, the proposed solution strategies lead to very good solutions, reducing the residual gap for the most difficult data set from 162% to 1.6%, and for all data sets to 2.6% or less.

A Lagrangean dual ascent algorithm for simple plant location problems

Abstract We propose to strengthen the separable Lagrangean relaxation of the Simple Plant Location Problem (SPLP) by using Benders inequalities generated during a Lagrangean dual ascent procedure. These inequalities are expressed in terms of the 0–1 variables only, and they can be used as knapsack constraints in the pure integer part of the Lagrangean relaxation. We show how coupling this technique with a good primal heuristic can substantially reduce integrality gaps.

Technical Note-An Improved Dual Based Algorithm for the Generalized Assignment Problem

The generalized assignment problem GAP determines the minimum cost assignment of n jobs to m agents such that each job is assigned to exactly one agent, subject to an agents capacity. Existing solution algorithms have not solved problems with more than 100 decision variables. This paper designs an optimization algorithm for the GAP that effectively solves problems with up to 500 variables. Compared with existing procedures, this algorithm requires fewer enumeration nodes and shorter running times. Improved performance stems from: an enhanced Lagrangian dual ascent procedure that solves a Lagrangian dual at each enumeration node; adding a surrogate constraint to the Lagrangian relaxed model: and an elaborate branch-and-bound scheme. An empirical investigation of various problem structures, not considered in existing literature, is also presented.