Spectral projected gradient and variable metric methods for optimization with linear inequalities

Abstract

A family of variable metric methods for convex constrained optimization was introduced recently by Birgin, Martinez and Raydan. One of the members of this family is the inexact spectral projected gradient (ISPG) method for minimization with convex constraints. At each iteration of these methods a strictly convex quadratic function with convex constraints must be (inexactly) minimized. In the case of the ISPG method it was shown that, in some important applications, iterative projection methods can be used for this minimization. In this paper the particular case in which the convex domain is a polytope described by a finite set of linear inequalities is considered. For solving the linearly constrained convex quadratic subproblem a dual approach is adopted, by means of which subproblems become (not necessarily strictly) convex quadratic minimization problems with box constraints. These subproblems are solved by means of an active-set box-constraint quadratic optimizer with a proximal-point type unconstrained algorithm for minimization within the current faces. Convergence results and numerical experiments are presented.