A Projected Gradient Method for Vector Optimization Problems

Vector optimization problems are a significant extension of multiobjective optimization, which has a large number of real life applications. In vector optimization the preference order is related to an arbitrary closed and convex cone, rather than the nonnegative orthant. We consider extensions of the projected gradient gradient method to vector optimization, which work directly with vector-valued functions, without using scalar-valued objectives. We provide a direction which adequately substitutes for the projected gradient, and establish results which mirror those available for the scalar-valued case, namely stationarity of the cluster points (if any) without convexity assumptions, and convergence of the full sequence generated by the algorithm to a weakly efficient optimum in the convex case, under mild assumptions. We also prove that our results still hold when the search direction is only approximately computed.

[1]  A. M. Geoffrion.  Proper efficiency and the theory of vector maximization , 1968 .

[2]  Roger M. Y. Ho,et al.  Goal programming and extensions , 1976 .

[3]  D. J. White,et al.  A Bibliography on the Applications of Mathematical Programming Multiple-objective Methods , 1990 .

[4]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

[5]  J. Jahn.  Mathematical vector optimization in partially ordered linear spaces , 1986 .

[6]  K. G. Murty,et al.  Convergence of the steepest descent method for minimizing quasiconvex functions , 1996 .

[7]  Johannes Jahn,et al.  Optimality conditions for set-valued optimization problems , 1998, Math. Methods Oper. Res..

[8]  A. Iusem,et al.  Full convergence of the steepest descent method with inexact line searches , 1995 .

[9]  C. Tammer,et al.  Theory of Vector Optimization , 2003 .

[10]  Jörg Fliege,et al.  Steepest descent methods for multicriteria optimization , 2000, Math. Methods Oper. Res..

[11]  E. M. L. Beale,et al.  Nonlinear Programming: A Unified Approach. , 1970 .

[12]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[13]  Charalambos D. Aliprantis,et al.  GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS , 2002 .

[14]  V. G. Karmanov,et al.  The stabilizing properties of the gradient method , 1987 .

[15]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[16]  Charalambos D. Aliprantis,et al.  General equilibrium analysis in ordered topological vector spaces , 2004 .

[17]  Alfredo N. Iusem,et al.  On the projected subgradient method for nonsmooth convex optimization in a Hilbert space , 1998, Math. Program..

[18]  Alfredo N. Iusem,et al.  A proximal regularization of the steepest descent method , 1995 .

[19]  E. Polak,et al.  Constrained minimization under vector-valued criteria in finite dimensional spaces☆ , 1967 .

[20]  Claude Lemaréchal,et al.  Convergence of some algorithms for convex minimization , 1993, Math. Program..

[21]  B. Svaiter,et al.  A steepest descent method for vector optimization , 2005 .

[22]  D. Bertsekas,et al.  CONVERGENCE OF A GRADIENT PROJECTION METHOD , 1982 .

[23]  Thomas Hanne,et al.  On the Development and Future Aspects of Vector Optimization and MCDM , 1997 .