Second-Order M-Composed Tangent Derivative and Its Applications

2018 ◽  
Vol 35 (05) ◽  
pp. 1850029 ◽  
Author(s):  
Yi-Hong Xu ◽  
Zhen-Hua Peng
Keyword(s):  
Optimality Condition ◽  
Optimization Problem ◽  
Convex Sets ◽  
Necessary Optimality Conditions ◽  
Second Order ◽  
Sufficient Optimality Condition ◽  
Point Pair ◽  
Weak Minimizer ◽  
Necessary And Sufficient ◽  
Order Tangent

A new kind of second-order tangent derivative, second-order [Formula: see text]-composed tangent derivative, for a set-valued function is introduced with help of a modified Dubovitskij–Miljutin cone. By using the concept, several generalized convex set-valued functions are introduced. When both the objective function and constrained function are second-order [Formula: see text]-composed derivable, under the assumption of nearly cone-subconvexlikeness, by applying a separation theorem for convex sets, Fritz John and Kuhn–Tucker second-order necessary optimality conditions are obtained for a point pair to be a weak minimizer of set-valued optimization problem. Under the assumption of generalized pseudoconvexity, a Kuhn–Tucker second-order sufficient optimality condition is obtained for a point pair to be a weak minimizer of set-valued optimization problem. A unified second-order necessary and sufficient optimality condition is derived in terms of second-order [Formula: see text]-composed tangent derivatives.

scholarly journals Optimality Conditions and Scalarization of Approximate Quasi Weak Efficient Solutions for Vector Equilibrium Problem

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yameng Zhang ◽  
Guolin Yu ◽  
Wenyan Han
Keyword(s):  
Optimality Conditions ◽  
Equilibrium Problem ◽  
Optimality Condition ◽  
Convex Sets ◽  
Efficient Solutions ◽  
Vector Equilibrium Problem ◽  
Weak Efficiency ◽  
Sufficient Optimality Condition ◽  
Vector Equilibrium ◽  
Quasirelative Interior

This paper is devoted to the investigation of optimality conditions for approximate quasi weak efficient solutions for a class of vector equilibrium problem (VEP). First, a necessary optimality condition for approximate quasi weak efficient solutions to VEP is established by utilizing the separation theorem with respect to the quasirelative interior of convex sets and the properties of the Clarke subdifferential. Second, the concept of approximate pseudoconvex function is introduced and its existence is verified by a concrete example. Under the assumption of introduced convexity, a sufficient optimality condition for VEP in sense of approximate quasi weak efficiency is also presented. Finally, by using Tammer’s function and the directed distance function, the scalarization theorems of the approximate quasi weak efficient solutions of the VEP are proposed.

scholarly journals A Second-Order Sufficient Optimality Condition for Risk-Neutral Bi-level Stochastic Linear Programs

2020 ◽  
Author(s):  
Matthias Claus
Keyword(s):  
Optimality Condition ◽  
Linear Models ◽  
Sufficient Conditions ◽  
Strong Stability ◽  
Second Order ◽  
Linear Programs ◽  
Sufficient Optimality Condition ◽  
Lipschitz Continuous ◽  
Risk Neutral ◽  
Stochastic Linear Programs

Abstract The expectation functionals, which arise in risk-neutral bi-level stochastic linear models with random lower-level right-hand side, are known to be continuously differentiable, if the underlying probability measure has a Lebesgue density. We show that the gradient may fail to be local Lipschitz continuous under this assumption. Our main result provides sufficient conditions for Lipschitz continuity of the gradient of the expectation functional and paves the way for a second-order optimality condition in terms of generalized Hessians. Moreover, we study geometric properties of regions of strong stability and derive representation results, which may facilitate the computation of gradients.

Necessary and sufficient optimality conditions for set-valued optimization problems

2011 ◽  
Vol 18 (1) ◽  
pp. 53-66
Author(s):  
Najia Benkenza ◽  
Nazih Gadhi ◽  
Lahoussine Lafhim
Keyword(s):  
Optimality Conditions ◽  
Vector Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Necessary Optimality Conditions ◽  
Sufficient Optimality Conditions ◽  
Necessary And Sufficient ◽  
First Time ◽  
Funct Anal

Abstract Using a special scalarization employed for the first time for the study of necessary optimality conditions in vector optimization by Ciligot-Travain [Numer. Funct. Anal. Optim. 15: 689–693, 1994], we give necessary optimality conditions for a set-valued optimization problem by establishing the existence of Lagrange–Fritz–John multipliers. Also, sufficient optimality conditions are given without any Lipschitz assumption.

Generalized Lagrange multiplier rule for non-convex vector optimization problems

2016 ◽  
Vol 146 (2) ◽  
pp. 297-308
Author(s):  
Maria Bernadette Donato
Keyword(s):  
Vector Optimization ◽  
Lagrange Multiplier ◽  
Optimization Problem ◽  
Constraint Qualification ◽  
Vector Optimization Problem ◽  
Sufficient Optimality Condition ◽  
Lagrange Multiplier Rule ◽  
Convex Vector Optimization ◽  
Multiplier Rule ◽  
Necessary And Sufficient

In this paper a non-convex vector optimization problem among infinite-dimensional spaces is presented. In particular, a generalized Lagrange multiplier rule is formulated as a necessary and sufficient optimality condition for weakly minimal solutions of a constrained vector optimization problem, without requiring that the ordering cone that defines the inequality constraints has non-empty interior. This paper extends the result of Donato (J. Funct. Analysis261 (2011), 2083–2093) to the general setting of vector optimization by introducing a constraint qualification assumption that involves the Fréchet differentiability of the maps and the tangent cone to the image set. Moreover, the constraint qualification is a necessary and sufficient condition for the Lagrange multiplier rule to hold.

Uniqueness of multipliers in optimal control: the missing piece

2018 ◽  
Vol 36 (4) ◽  
pp. 1395-1411
Author(s):  
Jorge A Becerril ◽  
Karla L Cortez ◽  
Javier F Rosenblueth
Keyword(s):  
Optimal Control ◽  
Nonlinear Programming ◽  
Optimality Conditions ◽  
Lagrange Multipliers ◽  
Constraint Qualification ◽  
Necessary Optimality Conditions ◽  
Second Order ◽  
Simple Condition ◽  
Necessary And Sufficient

Abstract In a well-known paper by Kyparisis it is proved that, in nonlinear programming, the uniqueness of Lagrange multipliers is equivalent to a strict version of the Mangasarian–Fromovitz constraint qualification which, in turn, implies the satisfaction of second-order necessary optimality conditions. This is no longer the case in optimal control where, as shown in a recent paper, the corresponding strict constraint qualification is only sufficient for the uniqueness of multipliers. In this paper we exhibit the missing piece: a new, simple condition, implied by the strict constraint qualification, which is necessary and sufficient for the uniqueness of multipliers in optimal control.

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