Journal of Convex Analysis 18 (2011), No. 1, 085--103
Copyright Heldermann Verlag 2011
Higher Order Strong Convexity and Global Strict Minimizers in Multiobjective Optimization
César Gutiérrez
Dep. de Matemática Aplicada, Universidad de Valladolid, Edificio Tec. Inf. Telecomunicaciones, Campus Miguel Delibes, 47011 Valladolid, Spain
cesargv@mat.uva.es
Bienvenido Jiménez
Dep. de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, Ciudad Universitaria, 28040 Madrid, Spain
bjimenez@ind.uned.es
Vicente Novo
Dep. de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, Ciudad Universitaria, 28040 Madrid, Spain
vnovo@ind.uned.es
We extend the scalar concept of strong convex function of order k due to G. H. Lin and M. Fukushima ["Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints", J. Optim. Theory Appl. 118 (2003) 67--80] to a vector-valued function, by considering a partial order given by a convex cone. We analyze some properties of higher order strong convex functions, and we give two characterizations of this kind of strong convexity for locally Lipschitz functions, one of them, through a new property of the Clarke generalized Jacobian, called strong monotonicity of order k. Similar results are obtained for Fréchet differentiable functions. In the second part, we study connections between strong convexity of order k and global strict minimizers of order k, and we establish sufficient optimality conditions for this class of minimizers in multiobjective optimization problems involving strong cone-convex functions.
Keywords: Vector optimization, higher order strong convexity, optimality conditions, strict minimizers, generalized Jacobian, higher order strong monotonicity, partial-quasiconvexity.
MSC: 52A41, 90C29, 90C46, 49K27, 49J52
[ Fulltext-pdf (176 KB)] for subscribers only.