Dra. Vivanco-Orellana, Violeta

Nondegenerate optimality conditions for Pareto and weak Pareto optimal solutions to multiobjective optimization problems with inequality and multi-equality constraints determined by Fréchet differentiable functions are established. First, weak and strong regularity conditions are derived, in order to determine weak Karush–Kuhn–Tucker (positivity of at least one Lagrange multiplier associated with objective functions) and strong Karush–Kuhn–Tucker (positivity of all the Lagrange multipliers associated with objective functions) conditions. Subsequently, the class of problems for which every weak (resp. strong) Karush–Kuhn–Tucker point is weak (resp. strong) Pareto solution is characterized. In addition examples that illustrate our results are presented.

We consider an optimal control problem governed by a system of nonlinear difference equations. We obtain the existence of the optimal control as well as first-order optimality conditions of Pontryagin type by using the Dubovitskii–Milyutin formalism. Also, we give the necessary and sufficient conditions for global optimality.

We define a new class of optimal control problems and show that this class is the largest one of control problems where every admissible process that satisfies the Extended Pontryaguin Maximum Principle is an optimal solution of nonregular optimal control problems. In this class of problems the local and global minimum coincide. A dual problem is also proposed, which may be seen as a generalization of the Mond–Weir-type dual problem, and it is shown that the 2-invexity notion is a necessary and sufficient condition to establish weak, strong, and converse duality results between a nonregular optimal control problem and its dual problem. We also present an example to illustrate our results.

In this work sufficient conditions are established to ensure that all feasible points are (properly) efficient solutions in non trivial situations, for a class of non-differentiable, non-convex multiobjective minimization problems. Considering locally Lipschitz functions and some results of non-differentiable analysis introduced by F. H. Clarke.