Optimality conditions for a bilevel optimization problem in terms of KKT multipliers and convexificators

In this paper we investigate a bilevel optimization problem by using the optimistic approach. Under a non smooth generalized Guignard constraint qualification, due the optimal value reformulation, the necessary optimality conditions in terms of convexificators and Karush-Kuhn-Tucker (KKT) multipliers are given.</p

Optimality conditions with respect to an ordering map using an exact separation principle

In this note, we are concerned with a multiobjective optimization problem. Using a special (nonlinear) scalarization [1], together with an exact separation principle recently introduced by&nbsp;Zheng,Yang and Zou [11], we give necessary optimality conditions for locally weakly nondominated solutions with respect to a given ordering map. To get the results, a nonsmooth sequential Guignard&nbsp;constraint qualication is introduced

A note on the paper “Necessary and sufficient optimality conditions using convexifactors for mathematical programs with equilibrium constraints”

In this work, some counterexamples are given to refute some results in the paper by Kohli [RAIRO:OR 53 (2019) 1617–1632]. We correct the fault in some of his results

Optimality conditions for MPECs in terms of directional upper convexifactors

In this paper, we investigate necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. For this goal, we introduce an appropriate type of MPEC regularity condition and a stationary concept given in terms of directional upper convexificators and directional upper semi-regular convexificators. The appearing functions are not necessarily smooth/locally Lipschitz/convex/continuous, and the continuity directions’ sets are not assumed to be compact or convex. Finally, notions of directional pseudoconvexity and directional quasiconvexity are used to establish sufficient optimality conditions for MPECs

Necessary optimality conditions for robust nonsmooth multiobjective optimization problems

This paper deals with a robust multiobjective optimization problem involving nonsmooth/nonconvex real-valued functions. Under an appropriate constraint qualification, we establish necessary optimality conditions for weakly robust efficient solutions of the considered problem. These optimality conditions are presented in terms of Karush-Kuhn-Tucker multipliers and convexificators of the related functions. Examples illustrating our findings are also given

Optimality conditions in set-valued optimization using approximations as generalized derivatives

The set criterion is an appropriate defining approach regarding the solutions for the set-valued optimization problems. By using approximations as generalized derivatives of set-valued mappings, we establish necessary optimality conditions for a constrained set-valued optimization problem in the sense of set optimization in terms of asymptotical pointwise compact approximations. Sufficient optimality conditions are then obtained through first-order strong approximations of data set-valued mappings

Comments on “Optimality conditions for nonsmooth interval-valued and multiobjective semi-infinite programming”

Necessary optimality conditions for a nonsmooth semi-infinite interval-valued vec- tor programming problem are given in the paper by Jennane et al. (RAIRO:OR (2020). DOI: 10.1051/ro/202006

A note on the paper “Optimality conditions for nonsmooth interval-valued and multiobjective semi-infinite programming”

A nonsmooth semi-infinite interval-valued vector programming problem is solved in the paper by Jennane et al. (RAIRO:OR 55 (2021) 1–11.). The necessary optimality condition obtained by the authors, as well as its proof, is false. Some counterexamples are given to call into question some results on which the main result (Jennane et al. [6] Thm. 4.5) is based. For the convenience of the reader, we correct the faulty in those results, propose a correct formulation of Theorem 4.5, and give also a short proof

On interval-valued bilevel optimization problems using upper convexificators

In this paper, we investigate a bilevel interval valued optimization problem. Reducing the problem into a one-level nonlinear and nonsmooth program, necessary optimality conditions are developed in terms of upper convexificators. Our approach consists of using an Abadie’s constraint qualification together with an appropriate optimal value reformulation. Later on, using an upper estimate for upper convexificators of the optimal value function, we give a more detailed result in terms of the initial data. The appearing functions are not necessarily Lipschitz continuous, and neither the objective function nor the constraint functions of the lower-level optimization problem are assumed to be convex. There are additional examples highlighting both our results and the limitations of certain past studies