A sequential method for a class of box constrained quadratic programming problems

Abstract

The aim of this paper is to propose an algorithm, based on the optimal level solutions method, which solves a particular class of box constrained quadratic problems. The objective function is given by the sum of a quadratic strictly convex separable function and the square of an affine function multiplied by a real parameter. The convexity and the nonconvexity of the problem can be characterized by means of the value of the real parameter. Within the algorithm, some global optimality conditions are used as stopping criteria, even in the case of a nonconvex objective function. The results of a deep computational test of the algorithm are also provided

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.…”

“…Generally speaking, the correctness of these algorithms is given by the fact that the optimal solution of P (if it exists) is also an optimal level solution and that all of the optimal level solutions are visited. The convergence of these algorithms is given by some assumptions on the objective function, assumptions aimed to guarantee that ξ m > ξ and that a basis is visited at most a finite number of times (for the problems studied in [8][9][10][11][13][14][15][16][17][18] it is proved that a basis can be visited just once); in this light the convergence follows noticing that a polyhedron has just a finite number of basis. Clearly, since these are simplex-like algorithms, they have the same complexity of simplex algorithm, that is to say that in the worst case all of the vertices must be visited.…”

“…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]).…”