A Proof of the Necessity of Linear Independence Condition and Strong Second-Order Sufficient Optimality Condition for Lipschitzian Stability in Nonlinear Programming

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.

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On a sufficient optimality condition over convex feasible regions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700023194 ◽

1977 ◽

Vol 16 (2) ◽

pp. 199-202

Author(s):

C.D. Alders ◽

V.A. Sposito

Keyword(s):

Nonlinear Programming ◽

Objective Function ◽

Programming Problem ◽

Optimality Condition ◽

Feasible Region ◽

Sufficient Condition ◽

Sufficient Optimality Condition ◽

Arbitrary Cone

In this note a sufficient optimality condition is established for nonlinear programming problems over arbitrary cone domains. A Kuhn-Tucker type sufficient condition is established if the programming problem has a pseudoconvex objective function and a convex feasible region.

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Second Order Sufficient Optimality Condition for a Class of Major Constraints Vector Optimization Problem

Proceedings of the 3rd International Conference on Advances in Management Science and Engineering (IC-AMSE 2020) ◽

10.2991/aebmr.k.200402.014 ◽

2020 ◽

Author(s):

Xuanwei Zhou

Keyword(s):

Vector Optimization ◽

Optimality Condition ◽

Optimization Problem ◽

Vector Optimization Problem ◽

Second Order ◽

Sufficient Optimality Condition

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Local Convergence of Exact and Inexact Augmented Lagrangian Methods under the Second-Order Sufficient Optimality Condition

SIAM Journal on Optimization ◽

10.1137/10081085x ◽

2012 ◽

Vol 22 (2) ◽

pp. 384-407 ◽

Cited By ~ 33

Author(s):

D. Fernández ◽

M. V. Solodov

Keyword(s):

Optimality Condition ◽

Local Convergence ◽

Augmented Lagrangian ◽

Second Order ◽

Sufficient Optimality Condition ◽

Augmented Lagrangian Methods ◽

Lagrangian Methods

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Second-Order M-Composed Tangent Derivative and Its Applications

Asia Pacific Journal of Operational Research ◽

10.1142/s021759591850029x ◽

2018 ◽

Vol 35 (05) ◽

pp. 1850029 ◽

Cited By ~ 1

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.

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Optimality of Approximate Quasi-Weakly Efficient Solutions for Vector Equilibrium Problems via Convexificators

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595921500470 ◽

2021 ◽

Author(s):

Guolin Yu ◽

Siqi Li ◽

Xiao Pan ◽

Wenyan Han

Keyword(s):

Optimality Condition ◽

Generalized Convexity ◽

Equilibrium Problems ◽

Efficient Solutions ◽

Sufficient Optimality Condition ◽

Necessary Optimality Condition ◽

Weakly Efficient Solutions ◽

Convexity Assumption ◽

Pseudoconvex Function ◽

Vector Equilibrium

This paper is devoted to the investigation of optimality conditions for approximate quasi-weakly efficient solutions to a class of nonsmooth Vector Equilibrium Problem (VEP) via convexificators. First, a necessary optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is presented by making use of the properties of convexificators. Second, the notion of approximate pseudoconvex function in the form of convexificators is introduced, and its existence is verified by a concrete example. Under the introduced generalized convexity assumption, a sufficient optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is also established. Finally, a scalar characterization for approximate quasi-weakly efficient solutions to problem (VEP) is obtained by taking advantage of Tammer’s function.

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Characterizations of Lipschitzian Stability in Nonlinear Programming

mathematical programming with data perturbations ◽

10.1201/9781003072119-4 ◽

2020 ◽

pp. 65-82 ◽

Cited By ~ 1

Author(s):

A. L. Dontchev ◽

R. T. Rockafellar

Keyword(s):

Nonlinear Programming ◽

Lipschitzian Stability

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Distributed-Order Non-Local Optimal Control

Axioms ◽

10.3390/axioms9040124 ◽

2020 ◽

Vol 9 (4) ◽

pp. 124

Author(s):

Faïçal Ndaïrou ◽

Delfim F. M. Torres

Keyword(s):

Optimal Control ◽

Maximum Principle ◽

Control Theory ◽

Optimality Condition ◽

Optimal Control Theory ◽

Sufficient Optimality Condition ◽

Weak Version ◽

Fractional Optimal Control ◽

Non Local ◽

Distributed Order

Distributed-order fractional non-local operators were introduced and studied by Caputo at the end of the 20th century. They generalize fractional order derivatives/integrals in the sense that such operators are defined by a weighted integral of different orders of differentiation over a certain range. The subject of distributed-order non-local derivatives is currently under strong development due to its applications in modeling some complex real world phenomena. Fractional optimal control theory deals with the optimization of a performance index functional, subject to a fractional control system. One of the most important results in classical and fractional optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. In our work, we extend the fractional optimal control theory by considering dynamical system constraints depending on distributed-order fractional derivatives. Precisely, we prove a weak version of Pontryagin’s maximum principle and a sufficient optimality condition under appropriate convexity assumptions.

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