Efficiency through variational-like inequalities with Lipschitz functions

In this work, first we introduce several notions of invexity and pseudoinvexity for a locally Lipschitz function by means of the generalized Jacobian. We study relationships between these concepts, in particular the implications between preinvexity and invexity. Next, we obtain necessary and sufficient optimality conditions for efficient and weak efficient solutions of finite-dimensional (non necessarily Pareto) vector optimization problems with locally Lipschitz objective functions through solutions of vector variational-like inequality problems. These conditions are stated via the generalized Jacobian and under pseudoinvexity hypotheses, and they show that a vector optimization problem can be reformulated as a vector variational-like inequality problem. This work extends and improves several previous papers, where the objective function of the vector optimization problem is assumed to be differentiable, or being locally Lipschitz, the authors consider the componentwise subdifferential based on the Clarke’s generalized gradients of the components of the objective function. Throughout the paper some simple examples are given in order to illustrate the main concepts and results.