A steepest descent method for vector optimization

• Published in: Journal of computational and applied mathematics Vol. 175; no. 2; pp. 395 - 414
• Main Authors: Graña Drummond, L.M., Svaiter, B.F.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.03.2005; Elsevier
• Subjects: [formula omitted]-convexity; Calculus of variations and optimal control; Exact sciences and technology; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in mathematical programming, optimization and calculus of variations; Numerical methods in optimization and calculus of variations; Pareto optimality; Quasi-Féjer convergence; Sciences and techniques of general use; Steepest descent; Vector optimization; Pareto optimality; Quasi-Féjer convergence; K -convexity; Steepest descent; Vector optimization; Quasi Féjer convergence; Necessary condition; Numerical analysis; Unconstrained optimization; Pareto optimum; Applied mathematics; Steepest descent method; Optimization method; Optimality condition; K convexity; Vector method; Mathematical programming
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2004.06.018

In this work we propose a Cauchy-like method for solving smooth unconstrained vector optimization problems. When the partial order under consideration is the one induced by the nonnegative orthant, we regain the steepest descent method for multicriteria optimization recently proposed by Fliege and Svaiter. We prove that every accumulation point of the generated sequence satisfies a certain first-order necessary condition for optimality, which extends to the vector case the well known “gradient equal zero” condition for real-valued minimization. Finally, under some reasonable additional hypotheses, we prove (global) convergence to a weak unconstrained minimizer. As a by-product, we show that the problem of finding a weak constrained minimizer can be viewed as a particular case of the so-called Abstract Equilibrium problem.