scholarly journals Optimality Conditions for Group Sparse Constrained Optimization Problems

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 84
Author(s):  
Wenying Wu ◽  
Dingtao Peng
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Stationary Point ◽  
Tangent Cone ◽  
Optimization Problems ◽  
Stationary Points ◽  
Tangent Cones ◽  
Sufficient Optimality Conditions ◽  
Normal Cones ◽  
The Relationship

In this paper, optimality conditions for the group sparse constrained optimization (GSCO) problems are studied. Firstly, the equivalent characterizations of Bouligand tangent cone, Clarke tangent cone and their corresponding normal cones of the group sparse set are derived. Secondly, by using tangent cones and normal cones, four types of stationary points for GSCO problems are given: TB-stationary point, NB-stationary point, TC-stationary point and NC-stationary point, which are used to characterize first-order optimality conditions for GSCO problems. Furthermore, both the relationship among the four types of stationary points and the relationship between stationary points and local minimizers are discussed. Finally, second-order necessary and sufficient optimality conditions for GSCO problems are provided.

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.

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 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.

scholarly journals Robust Approximate Optimality Conditions for Uncertain Nonsmooth Optimization with Infinite Number of Constraints

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 12 ◽  
Author(s):  
Xiangkai Sun ◽  
Hongyong Fu ◽  
Jing Zeng
Keyword(s):  
Optimality Conditions ◽  
Optimization Problem ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Optimization Technique ◽  
Optimal Solution ◽  
Sufficient Optimality Conditions ◽  
Convex Constraint ◽  
Approximate Optimality Conditions ◽  
Necessary And Sufficient

This paper deals with robust quasi approximate optimal solutions for a nonsmooth semi-infinite optimization problems with uncertainty data. By virtue of the epigraphs of the conjugates of the constraint functions, we first introduce a robust type closed convex constraint qualification. Then, by using the robust type closed convex constraint qualification and robust optimization technique, we obtain some necessary and sufficient optimality conditions for robust quasi approximate optimal solution and exact optimal solution of this nonsmooth uncertain semi-infinite optimization problem. Moreover, the obtained results in this paper are applied to a nonsmooth uncertain optimization problem with cone constraints.

On second-order sufficient optimality conditions for c 1,1-optimization problems

Optimization ◽  
1988 ◽  
Vol 19 (2) ◽  
pp. 169-179 ◽  
Author(s):  
D. Klatte ◽  
K. Tammek
Keyword(s):  
Optimality Conditions ◽  
Optimization Problems ◽  
Second Order ◽  
Sufficient Optimality Conditions

Tangent Cones and Convexity

1976 ◽  
Vol 19 (3) ◽  
pp. 257-261 ◽  
Author(s):  
J. Borwein ◽  
R. O’Brien
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Tangent Cones ◽  
Constrained Optimization Problems ◽  
Multiplier Theorems ◽  
Tangent Vectors

The study of general multiplier theorems (Kuhn-Tucker Conditions) for constrained optimization problems has led to extensions of the notion of a differentiable arc. Abadie [1], Varaiya [10], Guignard [5], Zlobec [11] and Massam [12] investigated the so called cone of tangent vectors to a point in a set for optimization purposes.

scholarly journals Sufficient optimality conditions for semi-infinite multiobjective fractional programming under (Ф,ρ)-V-invexity and generalized (Ф,ρ)-V-invexity

Filomat ◽  
2016 ◽  
Vol 30 (14) ◽  
pp. 3649-3665 ◽  
Author(s):  
Tadeusz Antczak
Keyword(s):  
Optimality Conditions ◽  
Fractional Programming ◽  
Generalized Convexity ◽  
Optimization Problems ◽  
Fundamental Property ◽  
Efficient Solutions ◽  
Sufficient Optimality Conditions ◽  
Multiobjective Fractional Programming ◽  
New Class ◽  
Multiobjective Programming Problems

A new class of nonconvex smooth semi-infinite multiobjective fractional programming problems with both inequality and equality constraints is considered. We formulate and establish several parametric sufficient optimality conditions for efficient solutions in such nonconvex vector optimization problems under (?,?)-V-invexity and/or generalized (?,?)-V-invexity hypotheses. With the reference to the said functions, we extend some results of efficiency for a larger class of nonconvex smooth semi-infinite multiobjective programming problems in comparison to those ones previously established in the literature under other generalized convexity notions. Namely, we prove the sufficient optimality conditions for such nonconvex semi-infinite multiobjective fractional programming problems in which not all functions constituting them have the fundamental property of convexity, invexity and most generalized convexity notions.

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