A parametric approach for solving a class of generalized quadratic-transformable rank-two nonconvex programs

Abstract

The aim of this paper is to propose a solution algorithm for a particular class of rank-two nonconvex programs having a polyhedral feasible region. The algorithm is based on the so-called ‘‘optimal level solutions’’ method. Various global optimality conditions are discussed and implemented in order to improve the efficiency of the algorithm

A unifying approach to solve a class of rank-three programs involving linear and quadratic functions

“…Such an approach has been applied to study various classes of problems in [4,7,22,[25][26][27]. Later, the optimal level solution method has been applied to study more general classes of problems, providing also computational results obtained by extensive numerical experiences [8][9][10][11][13][14][15][16][17][18][19].…”

“…Clearly, in order to have a working algorithm some properties need to be verified by the objective function of the problem, that is to say properties which guarantee the existence of a basis providing a value ξ m > ξ . See for example [8][9][10][11][13][14][15][16][17][18] for some particular classes of problems studied with this parametric approach.…”

“…As it has been shown in [8][9][10][11][13][14][15][16][17], it is possible to greatly improve the performance of the solution algorithm by implicitly visiting some of the feasible levels. Specifically speaking, it is possible to avoid the visit of the feasible levels which are not able to improve the incumbent optimal level solution.…”

“…These examples point out the wideness of the class of problems P which, at the best of our knowledge, have never been studied in the literature by means of a unifying framework. Notice also that even in the particular case φ(ξ) = ξ the problems considered in this paper differ from the rank-two ones studied in [8][9][10][11]15]. The solution method proposed to solve this class of problems is based on the so called "optimal level solutions" method (see [6,9,[13][14][15][16][17]26]).…”

“…Notice also that even in the particular case φ(ξ) = ξ the problems considered in this paper differ from the rank-two ones studied in [8][9][10][11]15]. The solution method proposed to solve this class of problems is based on the so called "optimal level solutions" method (see [6,9,[13][14][15][16][17]26]). It is known that this is a parametric method, which finds the optimum of the problem by determining the minima of particular subproblems.…”