scholarly journals Sufficient Optimality Conditions in Bilevel Programming

2021 ◽  
Author(s):  
Patrick Mehlitz ◽  
Alain B. Zemkoho
Keyword(s):  
Optimality Conditions ◽  
Optimization Problems ◽  
Bilevel Optimization ◽  
Second Order ◽  
Initial Problem ◽  
Directional Derivatives ◽  
Sufficient Optimality Conditions ◽  
Smooth Functions ◽  
Level Problem ◽  
Problem Data

This paper is concerned with the derivation of first- and second-order sufficient optimality conditions for optimistic bilevel optimization problems involving smooth functions. First-order sufficient optimality conditions are obtained by estimating the tangent cone to the feasible set of the bilevel program in terms of initial problem data. This is done by exploiting several different reformulations of the hierarchical model as a single-level problem. To obtain second-order sufficient optimality conditions, we exploit the so-called value function reformulation of the bilevel optimization problem, which is then tackled with the aid of second-order directional derivatives. The resulting conditions can be stated in terms of initial problem data in several interesting situations comprising the settings where the lower level is linear or possesses strongly stable solutions.

scholarly journals Necessary and Sufficient Second-Order Optimality Conditions on Hadamard Manifolds

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1152
Author(s):  
Gabriel Ruiz-Garzón ◽  
Jaime Ruiz-Zapatero ◽  
Rafaela Osuna-Gómez ◽  
Antonio Rufián-Lizana
Keyword(s):  
Optimality Conditions ◽  
Optimization Problems ◽  
Second Order ◽  
Stationary Points ◽  
Directional Derivatives ◽  
Sufficient Optimality Conditions ◽  
General Notion ◽  
Hadamard Manifolds ◽  
Scalar Optimization ◽  
Necessary And Sufficient

This work is intended to lead a study of necessary and sufficient optimality conditions for scalar optimization problems on Hadamard manifolds. In the context of this geometry, we obtain and present new function types characterized by the property of having all their second-order stationary points be global minimums. In order to do so, we extend the concept convexity in Euclidean space to a more general notion of invexity on Hadamard manifolds. This is done employing notions of second-order directional derivatives, second-order pseudoinvexity functions, and the second-order Karush–Kuhn–Tucker-pseudoinvexity problem. Thus, we prove that every second-order stationary point is a global minimum if and only if the problem is either second-order pseudoinvex or second-order KKT-pseudoinvex depending on whether the problem regards unconstrained or constrained scalar optimization, respectively. This result has not been presented in the literature before. Finally, examples of these new characterizations are provided in the context of “Higgs Boson like” potentials, among others.

scholarly journals On generalized derivatives forC1,1vector optimization problems

2003 ◽  
Vol 2003 (7) ◽  
pp. 365-376 ◽  
Author(s):  
Davide La Torre
Keyword(s):  
Optimality Conditions ◽  
Vector Optimization ◽  
Optimization Problems ◽  
Second Order ◽  
Clarke Subdifferential ◽  
Vector Optimization Problems ◽  
Directional Derivatives ◽  
Generalized Derivatives ◽  
Second Order Optimality Conditions

We introduce generalized definitions of Peano and Riemann directional derivatives in order to obtain second-order optimality conditions for vector optimization problems involvingC1,1data. We show that these conditions are stronger than those in literature obtained by means of second-order Clarke subdifferential.

scholarly journals Second-Order Optimality Conditions: An extension to Hadamard Manifolds

2020 ◽  
Author(s):  
Gabriel Ruiz-Garzón ◽  
Jaime Ruiz-Zapatero ◽  
Rafaela Osuna-Gómez ◽  
Antonio Rufián-Lizana
Keyword(s):  
Optimality Conditions ◽  
Optimization Problems ◽  
Directional Derivative ◽  
Second Order ◽  
Stationary Points ◽  
Sufficient Optimality Conditions ◽  
General Notion ◽  
Hadamard Manifolds ◽  
Scalar Optimization ◽  
Second Order Directional Derivative

This work is intended to lead a study of necessary and sufficient optimality conditions for scalar optimization problems on Hadamard manifolds. In the context of this geometry, we obtain and present new function types characterized by the property of having all their second-order stationary points to be global minimums. In order to do so, we extend the concept convexity in Euclidean space to a more general notion of invexity on Hadamard manifolds. This is done employing notions of second-order directional derivative, second-order pseudoinvexity functions and the second-order Karush-Kuhn-Tucker-pseudoinvexity problem. Thus, we prove that every second-order stationary point is a global minimum if and only if the problem is either second-order pseudoinvex or second-order KKT-pseudoinvex depending on whether the problem regards unconstrained or constrained scalar optimization respectively. This result has not been presented in the literature before. Finally, examples of these new characterizations are provided in the context of \textit{"Higgs Boson like"} potentials among others.

New second-order radial epiderivatives and applications to optimality conditions

2020 ◽  
Vol 54 (4) ◽  
pp. 949-959
Author(s):  
Xiaoyan Zhang ◽  
Qilin Wang
Keyword(s):  
Optimality Conditions ◽  
Recent Literature ◽  
Optimization Problems ◽  
Necessary Optimality Conditions ◽  
Efficient Solutions ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Contingent Epiderivative ◽  
Proper Efficient Solutions

In this paper, we introduce the second-order weakly composed radial epiderivative of set-valued maps, discuss its relationship to the second-order weakly composed contingent epiderivative, and obtain some of its properties. Then we establish the necessary optimality conditions and sufficient optimality conditions of Benson proper efficient solutions of constrained set-valued optimization problems by means of the second-order epiderivative. Some of our results improve and imply the corresponding ones in recent literature.

scholarly journals Constructive generalization of classical sufficient second-order optimality conditions

2019 ◽  
Vol 487 (5) ◽  
pp. 493-495
Author(s):  
Yu. G. Evtushenko ◽  
A. A. Tret’yakov
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Constrained Optimization Problems ◽  
Constrained Problems ◽  
Slack Variables ◽  
Equality Constrained Optimization

In this paper, we consider new sufficient conditions of optimality of the second-order for equality constrained optimization problems, which essentially enhance and complement the classical ones and are constructive. For example, they establish equivalence between sufficient conditions in the equality constrained optimization problems and sufficient conditions for optimality in equality constrained problems by reducing the latter to equalities with the help of introducing slack variables. Previously, when using the classical sufficient optimality conditions, this fact was not considered to be true, that is, the existing classical sufficient conditions were not complete, so the proposed optimality conditions complement the classical ones and close the question of the equivalence of the problems with inequalities and the problems with equalities when reducing the first to the second by introducing slack variables.

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

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