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, Martínez 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.

Spectral Projected Gradient Methods: Review and Perspectives

“…The case in which the convex feasible set of the problem to be solved by SPG is defined by linear equality and inequality constraints has been considered in Birgin et al (2003b), Andreani et al (2005), Martínez, Pilotta, and Raydan (2005), and . A crucial observation is that this type of set is not necessarily one in which it is easy to project.…”

Spectral Projected Gradient Methods

“…Application and implementation of the spectral methods to particular optimization problems: Linear inequality constraints were considered in [1]. In [38] the SPG method was used to solve Augmented Lagrangian subproblems.…”

Partial spectral projected gradient method with active-set strategy for linearly constrained optimization

“…This method combines the basic spectral-step ideas [8,53,54] with projected gradient strategies [10,37,45]. The extension of the spectral gradient method to smooth convex programming problems [5,15,14,17,18,19] was motivated by the surprisingly good performance of the spectral gradient for large-scale unconstrained minimization [54]. Nonmonotone strategies, like the one proposed by Grippo, Lampariello and Lucidi [40], turned out to be an important ingredient for the success of the spectral idea for unconstrained minimization and other extensions.…”

“…Bound constrained minimization was addressed in [6,14,15]. Linearly constrained optimization and nonlinear systems were considered in [5,48] and [42,67], respectively. In [48] a penalty approach was used and in [5] problems were solved by means of sequential quadratic programming.…”

“…Linearly constrained optimization and nonlinear systems were considered in [5,48] and [42,67], respectively. In [48] a penalty approach was used and in [5] problems were solved by means of sequential quadratic programming. For general nonlinear programming, a method combining SPG with the augmented Lagrangian was proposed in [30], and augmented Lagrangian methods that use the methods introduced in [6,15] to solve bound constrained subproblems were developed in [3,4,12].…”