Improvement and implementation of some algorithms for nonconvex optimization problems

Abstract

As is well-known, many problems of practical origin can be formulated as nonconvex optimization problems, such as concave minimization problems, bilinear programming problems, complementarity problem, d.c. programming problems., problems with reverse convex constraints... (see [9] for details). A lot of methods for solving these problems have been developed in recent years. Most of them use concave minimization aigorithms as one of main subroutines. For concave minimization problems the outer approximation algorithm is easy to implement but its efficiency crucially depends upon the concrete procedures for determining the vertices of a polytope (bounded convex polyhedron) which is obtained by adding a new linear constraint to a given polytope with known vertices. In this paper we shall present an improved algorithm for solving this subproblem which allows to handle easily degeneracy and show that a similar problem for unbounded convex polyhedral sets can be transformed into the case of polytopes without producing an additional variable. As a new result, the linear complementarity problem can be reduced to concave minimization over polytopes (rather than over convex polyhedral, possibly unbounded sets). We shall also report some numerical experiments with the following algorithms for nonconvex optimization problems on the basic of the improvement : Outer approximation algorithm and normal conical algorithm for concave minimization over potytopes. Decomposition algorithms for solving concave programs with special structure, including the lay-out planning problem with concave cost. Solution of bilinear programming problems by reduction to concave minimization. Finite algorithm for solving the linear complementarity problem.