Higher Order Strong Convexity and Global Strict Minimizers in Multiobjective Optimization

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.