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Inexact projected gradient method for vector optimization

Ellen H. Fukuda(ellen***at***ime.usp.br)
L. M. Graņa Drummond(lmgd***at***facc.ufrj.br)

Abstract: In this work, we propose an inexact projected gradient-like method for solving smooth constrained vector optimization problems. In the unconstrained case, we retrieve the steepest descent method introduced by Graņa Drummond and Svaiter. In the constrained setting, the method we present extends the exact one proposed by Graņa Drummond and Iusem, since it admits relative errors on the search directions. At each iteration, a decrease of the objective value is obtained by means of an Armijo-like rule. The convergence results of this new method extend those obtained by Fukuda and Graņa Drummond for the exact version. Basically, for antisymmetric and non-antisymmetric partial orders, under some reasonable hypotheses, global convergence to weakly efficient points of all sequences produced by the inexact projected gradient method is established for convex (respect to the ordering cone) objective functions. In the convergence analysis we also establish a connection between the so-called weighting method and the one we propose.

Keywords: Weak efficiency, multiobjective optimization, projected gradient method, vector optimization

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 2: Other Topics (Multi-Criteria Optimization )

Citation: State University of Campinas and Federal University of Rio de Janeiro, Brazil, June/2011.

Download: [PDF]

Entry Submitted: 06/15/2011
Entry Accepted: 06/15/2011
Entry Last Modified: 06/15/2011

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