Inexact variable metric method for convex-constrained optimization problems

• Published in: Optimization Vol. 71; no. 1; pp. 145 - 163
• Main Authors: Gonçalves, Douglas S., Gonçalves, Max L. N., Menezes, Tiago C.
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 02.01.2022; Taylor & Francis LLC
• Subjects: 65Kxx; 90Cxx; approximate solution; Convex-constrained optimization problem; inexact variable metric method; Iterative methods; Optimization; projected gradient method; Quadratic equations; spectral gradient method
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2021.1887181

This paper is concerned with the inexact variable metric method for solving convex-constrained optimization problems. At each iteration of this method, the search direction is obtained by inexactly minimizing a strictly convex quadratic function over the closed convex feasible set. Here, we propose a new inexactness criterion for the search direction subproblems. Under mild assumptions, we prove that any accumulation point of the sequence generated by the new method is a stationary point of the problem under consideration. In order to illustrate the practical advantages of the new approach, we report some numerical experiments. In particular, we present an application where our concept of the inexact solutions is quite appealing.