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Summary: Spectral projected gradient and variable metric methods for
optimization with linear inequalities
Roberto Andreani # Ernesto G. Birgin + Jos’e Mario Mart’nez #
Jinyun Yuan §
July 6, 2004, 10.52 hs
Abstract
A family of variable metric methods for convex constrained optimization was intro
duced recently by Birgin, Mart’nez 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 ISPG 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 min
imization problems with box constraints. These subproblems are solved by means of
an an activeset boxconstraint quadratic optimizer with a proximalpoint type un
constrained algorithm for minimization within the current faces. Convergence results
and numerical experiments are presented.
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