Strict directional solutions in vectorial problems: necessary optimality conditions

We study directional strict efficiency in vector optimization and equilibrium problems with set-valued map objectives. We devise several possibilities to define a meaningful concept of strict efficiency in a directional sense for these kinds of problems and then we present necessary optimality conditions from several perspectives by means of generalized differentiation calculus. A concept of generalized convexity for multimappings is employed as well and its role in getting equivalence between some classes of solutions is emphasized.

[1]  Boris S. Mordukhovich,et al.  Relative Pareto minimizers for multiobjective problems: existence and optimality conditions , 2009, Math. Program..

[2]  Marius Durea,et al.  On some Fermat rules for set-valued optimization problems , 2011 .

[3]  M. Ferris,et al.  Weak sharp minima in mathematical programming , 1993 .

[4]  Christiane Tammer,et al.  Fuzzy necessary optimality conditions for vector optimization problems , 2009 .

[5]  On strong quasiconvex functions and boundedness of level sets , 1989 .

[6]  N. D. Yen,et al.  Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming , 2006 .

[7]  Bienvenido Jiménez,et al.  Strict Efficiency in Set-Valued Optimization , 2009, SIAM J. Control. Optim..

[8]  Ewa M. Bednarczuk * Weak sharp efficiency and growth condition for vector-valued functions with applications , 2004 .

[9]  Marius Durea,et al.  Necessary optimality conditions for weak sharp minima in set-valued optimization , 2010 .

[10]  B. Mordukhovich Variational analysis and generalized differentiation , 2006 .

[11]  A. Göpfert Variational methods in partially ordered spaces , 2003 .

[12]  Marius Durea,et al.  Directional Pareto Efficiency: Concepts and Optimality Conditions , 2019, J. Optim. Theory Appl..

[14]  S. J. Li,et al.  Strong Fermat Rules for Constrained Set-Valued Optimization Problems on Banach Spaces , 2012 .

[15]  Ewa M. Bednarczuk,et al.  Weak sharp efficiency and growth condition for vector-valued functions with applications , 2004 .

[16]  Boris S. Mordukhovich,et al.  Necessary conditions for super minimizers in constrained multiobjective optimization , 2009, J. Glob. Optim..

[17]  M. Durea,et al.  Generalized penalization and maximization of vectorial nonsmooth functions , 2017 .