Note on combinatorial optimization with max-linear objective functions

AbstractWe consider combinatorial optimization problems with a feasible solution set S⊆{0,1}n specified by a system of linear constraints in 0–1 variables. Additionally, several cost functions c1,…,cp are given. The max-linear objective function is defined by f(x):=max{c1x,…,cpx: x∈S}; where cq:=(cq1,…,cqn) is for q=1,…,p an integer row vector in Rn.The problem of minimizing f(x) over S is called the max-linear combinatorial optimization (MLCO) problem.We will show that MLCO is NP-hard even for the simplest case of S⊆{0,1}n and p=2, and strongly NP-hard for general p. We discuss the relation to multi-criteria optimization and develop some bounds for MLCO

Semi-obnoxious location models: a global optimization approach

In the last decades there has been an increasing interest in environmental topics. This interest has been reflected in modeling the location of obnoxious facilities, as shown by the important number of papers published in this field. However, a very common drawback of the existing literature is that, as soon as environmental aspects are taken into account, economical considerations (e.g. transportation costs) are ignored, leading to models with dubious practical interest. In this paper we take into account both the environmental impact and the transportation costs caused by the location of an obnoxious facility, and propose as solution method of the well-known BSSS, with a new bounding scheme which exploits the structure of the problem.Dirección General de Investigación Científica y Técnic

Reoptimization in lagrangian methods for the quadratic knapsack problem

International audienceThe 0-1 quadratic knapsack problem consists in maximizing a quadratic objective function subject to a linear capacity constraint. To solve exactly large instances of this problem with a tree search algorithm (e.g. a branch and bound method), the knowledge of good lower and upper bounds is crucial for pruning the tree but also for fixing as many variables as possible in a preprocessing phase. The upper bounds used in the best known exact approaches are based on Lagrangian relaxation and decomposition. It appears that the computation of these Lagrangian dual bounds involves the resolution of numerous 0-1 linear knapsack subproblems. Thus, taking this huge number of solvings into account, we propose to embed reoptimization techniques for improving the efficiency of the preprocessing phase of the 0-1 quadratic knapsack resolution. Namely, reoptimization is introduced to accelerate each independent sequence of 0-1 linear knapsack problems induced by the Lagrangian relaxation as well as the Lagrangian decomposition. Numerous numerical experiments validate the relevance of our approach

Combinatorial Optimization and Integer Programming

Solution techniques for combinatorial optimization and integer programming problems are core disciplines in operations research with contributions of mathematicians as well as computer scientists and economists. This article surveys the state of the art in solving such problems to optimality

Combinatorial optimization under ellipsoidal uncertainty

We study combinatorial problems with ellipsoidal uncertainty in the objective function concerning their theoretical and practical solvability. Ellipsoidal uncertainty is a natural model when the coefficients are normally distributed random variables. Robust versions of typical combinatorial problems can be very hard to solve compared to their linear versions. Complexity and approaches differ fundamentally depending on whether uncorrelated or correlated uncertainty occurs. We distinguish between these two cases and consider first the unconstrained binary optimization under uncorrelated ellipsoidal uncertainty. For this we develop an algorithm which computes an optimal solution by merely sorting the variables and, correspondingly, has a running time of O(n log n). The algorithm is based on the diminishing returns-property, which is characteristic for submodular functions. We introduce a new and a more general p-norm-uncertainty and show that with only slight modifications the sorting algorithm can be easily applied. We also extend the algorithm to general integer variables, which in this case only leads to a pseudo-polynomial time. The next step to the general case is investigation of problems with arbitrary combinatorial sets X ⊆ {0, 1}n under uncorrelated ellipsoidal uncertainty. For this case we embed the O(n log n)-algorithm for the unconstrained binary problems into a Lagrangean decomposition approach. The approach separates the objective function from the combinatorial structure applying Lagrangean relaxation to some artificial connecting constraints. This creates two subproblems, one of which is the linear version of the combinatorial problem and the other one is just the unconstrained binary uncorrelated problem, which can be solved using the O(n log n)-algorithm. The solutions of the subproblems are used to obtain primal and dual bounds which are used in a branch and bound-approach. The approach shows an excellent performance in practice. In the correlated case already the unconstrained binary problem turns out to be strongly NP-hard. Here we also define a branch and bound-approach, now with lower bounds determined by underestimation of the given ellipsoid with certainly defined axis-parallel ellipsoids. We use this idea to extend the decomposition approach to general combinatorial problems under correlated uncertainty. In contrast to the uncorrelated case the uncertain subproblem of the decomposition is here strongly NP-hard in itself. We solve it approximately using the developed underestimators which are determined in a preprocessing step. The approach offers room for improvement concerning in the primal extent a faster computation of the underestimators, which is done by solving semidefinite programs

Analysis of large scale linear programming problems with embedded network structures: Detection and solution algorithms

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Linear programming (LP) models that contain a (substantial) network structure frequently arise in many real life applications. In this thesis, we investigate two main questions; i) how an embedded network structure can be detected, ii) how the network structure can be exploited to create improved sparse simplex solution algorithms. In order to extract an embedded pure network structure from a general LP problem we develop two new heuristics. The first heuristic is an alternative multi-stage generalised upper bounds (GUB) based approach which finds as many GUB subsets as possible. In order to identify a GUB subset two different approaches are introduced; the first is based on the notion of Markowitz merit count and the second exploits an independent set in the corresponding graph. The second heuristic is based on the generalised signed graph of the coefficient matrix. This heuristic determines whether the given LP problem is an entirely pure network; this is in contrast to all previously known heuristics. Using generalised signed graphs, we prove that the problem of detecting the maximum size embedded network structure within an LP problem is NP-hard. The two detection algorithms perform very well computationally and make positive contributions to the known body of results for the embedded network detection. For computational solution a decomposition based approach is presented which solves a network problem with side constraints. In this approach, the original coefficient matrix is partitioned into the network and the non-network parts. For the partitioned problem, we investigate two alternative decomposition techniques namely, Lagrangean relaxation and Benders decomposition. Active variables identified by these procedures are then used to create an advanced basis for the original problem. The computational results of applying these techniques to a selection of Netlib models are encouraging. The development and computational investigation of this solution algorithm constitute further contribution made by the research reported in this thesis.This study is funded by the Turkish Educational Council and Mugla University