Do Van Luu

Higher-order necessary and sufficient conditions for strict local pareto minima in terms of Studniarski's derivatives

In this article, we establish higher-order necessary and sufficient conditions for strict local Pareto minima of nonsmooth multiobjective programing problems in terms of Studniarskis derivatives of higher order.In this article, we establish higher-order necessary and sufficient conditions for strict local Pareto minima of nonsmooth multiobjective programing problems in terms of Studniarskis derivatives of higher order.

Necessary and Sufficient Conditions for Efficiency Via Convexificators

Based on the extended Ljusternik Theorem by Jiménez-Novo, necessary conditions for weak Pareto minimum of multiobjective programming problems involving inequality, equality and set constraints in terms of convexificators are established. Under assumptions on generalized convexity, necessary conditions for weak Pareto minimum become sufficient conditions.

Convexificators and necessary conditions for efficiency

In this article we establish necessary conditions for local Pareto and weak minima of multiobjective programming problems involving inequality, equality and set constraints in Banach spaces in terms of convexificators.

Efficient solutions and optimality conditions for vector equilibrium problems

Necessary optimality conditions for efficient solutions of unconstrained and vector equilibrium problems with equality and inequality constraints are derived. Under assumptions on generalized convexity, necessary optimality conditions for efficient solutions become sufficient optimality conditions. Note that it is not required here that the ordering cone in the objective space has a nonempty interior.

Optimality Condition for Local Efficient Solutions of Vector Equilibrium Problems via Convexificators and Applications

Fritz John and Karush–Kuhn–Tucker necessary conditions for local efficient solutions of constrained vector equilibrium problems in Banach spaces in which those solutions are regular in the sense of Ioffe via convexificators are established. Under suitable assumptions on generalized convexity, sufficient conditions are derived. Some applications to constrained vector variational inequalities and constrained vector optimization problems are also given.

Constrained minimax for a vector-valued function

Necessary Kuhn-Tucker conditions are established for a weak local minimax of a differentiable vector-valued functions with differentiable constraints, in finite dimensions. These are related to strong optimal conditions with suitably larger order cones. Sufficient conditions are given, under suitable(invex) hypotheses

Necessary conditions for efficiency in terms of the Michel–Penot subdifferentials

In this article, we study a multiobjective optimization problem involving inequality and equality cone constraints and a set constraint in which the functions are either locally Lipschitz or Frechet differentiable (not necessarily C 1-functions). Under various constraint qualifications, Kuhn–Tucker necessary conditions for efficiency in terms of the Michel–Penot subdifferentials are established.In this article, we study a multiobjective optimization problem involving inequality and equality cone constraints and a set constraint in which the functions are either locally Lipschitz or Fréchet differentiable (not necessarily C 1-functions). Under various constraint qualifications, Kuhn–Tucker necessary conditions for efficiency in terms of the Michel–Penot subdifferentials are established.

Strengthened invex and perturbations

If the strengthened invex property holds for a constrained minimization problem, then a Karush-Kuhn-Tucker point is a strict minimum. The strict minimum property is preserved under small perturbations of the problem. This allows sufficient conditions to be given for a minimax, starting from Karush-Kuhn-Tucker conditions. They extend to vector-valued minimax and to nonsmooth (Lipschitz) problems. An example is provided to illustrate the strengthened invex property, also a discussion of quadratic-linear (nonconvex) programming implications.

Higher-order optimality conditions for a minimax

Higher-order necessary and sufficient optimality conditions for a nonsmooth minimax problem with infinitely many constraints of inequality type are established under suitable basic assumptions and regularity conditions.

On Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Equilibrium Constraints

In this article, necessary conditions of Fritz John type for weak efficient solutions of a nonsmooth vector equilibrium problem involving equilibrium constraints (VEPEC) in terms of the Clarke subdifferentials are established. Under constraint qualifications which are suitable for (VEPEC), necessary conditions of Kuhn-Tucker type for efficiency are derived. Under assumptions on generalized convexity of data, sufficient conditions for efficiency are developed. Some applications to vector variational inequalities and vector optimization problems with equilibrium constraints are also given.