• Keine Ergebnisse gefunden

3. State of the Art 41

3.5. Hybrid LP-based Approaches

integrative combinations, one algorithm is embedded into another algorithm. According to this classification, our technique fits into the collaborative combination class, i.e., the two-stage algorithm: linear programming first, and then local search to the optimal solution of the LP.

The special combinations, the hybridizations of LP and the metaheuristic, are proposed for several combinatorial optimization problems. In these combinations the fractional so-lutions of LP are rounded into integral soso-lutions by some rounding techniques, and the integral solutions guide heuristic procedures.

In contrast, in our approach the underlying LP generates directly a feasible solution of the primary problem, which guides the local search. We briefly describe several hybrid LP-based heuristic approaches.

French and Wilson [61] proposed an LP-based heuristic algorithm for generalized assign-ment problem (GAP) with special ordered sets of type two [26], the extension of GAP. The algorithm is an extensive adaptation of heuristic algorithm for GAP. Their heuristic method consists of two phases. In the first phase the series of LP-relaxations is solved iteratively within heuristic procedure. A feasible solution obtained in the first phase is improved by a local search in the second phase of the algorithm. The rational valued variables of the LP-solution, which satisfy certain criterion are fixed at “0”, and this information is used to generate the LP in the next iteration.

Klose [100] proposed an LP-based heuristic for the two-stage capacitated facility location problem. An LP formulation, which is a relaxation of the original problem, is iteratively refined using known valid inequalities and facets for various relaxations of the problem. The method solves a series of LP-relaxations. At each iteration, a feasible solution is obtained from a fractional solution of LP by applying (simple) heuristic.

Sridhar et al. [150] proposed a heuristic method for a capacitated network design prob-lem. An aim of the problem is to find a feasible solution, a topology of the network. Again, they solve an LP relaxation of an MIP formulation of the initial problem, and the fractional solution of LP is used by some heuristic to build a feasible solution. The method uses subsequentially several heuristics to improve the feasible solution.

Umetani et al. [154] proposed a local search approach based on LP for a special version of the one dimensional cutting stock problem, which they have stated as pattern restricted problem. In the algorithm, a certain LP relaxation is solved. The optimal solution of this LP is rounded to get a feasible solution of primary problem.

3.5. HYBRID LP-BASED APPROACHES 59

Alvarez-Valdes et al. [4] developed an LP-based heuristic method for solving a two-dimensional cutting stock problem. The algorithm is based on the Gilmore and Gomory column generation scheme [67, 68] and consists of two steps, an iterating step and a rounding step. In each iteration of the first step, a certain LP relaxation of the problem is solved and the LP-solution is used to generate the subproblems (the subproblems can be solved exactly, or by applying heuristic methods). Depending on the solutions of subproblems, the iterating step continues, or stops. If the iterating procedure continues, the solutions of subproblems are used to generate the LP relaxation of the primary problem in the next iteration. In the second step, the feasible solution of the primary problem is obtained by rounding the LP-solution.

Vasquez and Hao proposed LP based heuristic approach for the 0-1 multidimensional knapsack problem [155]. The algorithm involves a subsequent run of the LP solver and the heuristic method. The algorithm solves a series of certain LP relaxations of the primary problem. The fractional solutions of the LPs are rounded by some rounding scheme. The obtained integral solution is taken as a start solution for tabu search.

Chapter 4

Local Search Starting From an LP Solution

In this chapter we introduce the LP-based technique that generates the start solutions for local search to themax-di-cutproblem with source and sink, themax-k-satproblem, wherek≥2, and thelongest directed pathproblem with source and sink. Clearly, the proposed method applies to the max-2-sat, max-3-sat and max-4-sat problems, since these are the variations of general problem where k= 2, k= 3 and k= 4, respectively.

4.1. Introduction

The main goal of our technique is to solve the given problem fast. The success of the simple local search crucially depends on the start solutions. Consequently, providing the local search with good start solutions that are near to optimal solutions could lead to a fast and efficient algorithm. We present an LP based technique that generates start solutions of apparently high quality. This LP based technique might apply to a variety of optimization problems but adaptation to a given problem does not seem to be entirely trivial. Therefore, we must particularly consider the underlying problem to develop this technique.

It is a requirement for our method to work that there be an LP such that the optimal basis solutions to the LP be feasible solutions to the original problem. This does not mean that this LP describes the original problem. We also demonstrate where it is promising to look for such an LP: take a quadratic programming (QP) formulation and try to develop an LP that meets this requirement.

In the following sections we present our LP based technique for themax-di-cutproblem with source and sink, the max-k-sat problem, where k≥2, and the longest directed

61

path problem with source and sink. We consider the nonlinear formulations of the max-di-cut and max-k-sat, k ≥2, problems and apply small modifications to transform the non-linear ingredients into linear ones. In each case, we define the modification such that the vertices of the resulting LP are integral, and that the simplex method will not end up at infinity (although the polyhedron itself will be unbounded in general). In case of the max-di-cutwe substitute the quadratic objective function by linear function, and in case of the max-k-satwe substitute the quadratic constraints by linear constraints to get the desired LPs. In case of the longest directed path we obtain the required LP by alternative way – incorporating flow consistency inequalities. We use the longest directed path problem as an example to corroborate that (i) our technique is not suitable for arbitrary problem, but, (ii) our technique achieves near optimal solutions to themax-di-cutproblem with source and sink and the variations of themax-k-satproblem, wherek∈ {2,3,4}. The two-phase algorithm that results from our LP-based local search technique is as follows:

1) solve the corresponding LP;

2) run the local search starting at the LP solution.