Finding Second-Order Stationary Points in Constrained Minimization: A Feasible Direction Approach

This paper introduces a method for computing points satisfying the second-order necessary optimality conditions for nonconvex minimization problems subject to a closed and convex constraint set. The method comprises two independent steps corresponding to the first- and second-order conditions. The f...

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Published in:Journal of optimization theory and applications Vol. 186; no. 2; pp. 480 - 503
Main Authors: Hallak, Nadav, Teboulle, Marc
Format: Journal Article
Language:English
Published: New York Springer US 01.08.2020
Springer Nature B.V
Subjects:
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Constraints
Engineering
First order algorithms
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Theory of Computation
90C31
90C30
65K05
90C46
Feasible direction methods
Second-order methods
Second-order necessary optimality conditions
90C26
Constrained optimization
ISSN:0022-3239, 1573-2878
Online Access:Get full text
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Abstract This paper introduces a method for computing points satisfying the second-order necessary optimality conditions for nonconvex minimization problems subject to a closed and convex constraint set. The method comprises two independent steps corresponding to the first- and second-order conditions. The first-order step is a generic closed map algorithm, which can be chosen from a variety of first-order algorithms, making it adjustable to the given problem. The second-order step can be viewed as a second-order feasible direction step for nonconvex minimization subject to a convex set. We prove that any limit point of the resulting scheme satisfies the second-order necessary optimality condition, and establish the scheme’s convergence rate and complexity, under standard and mild assumptions. Numerical tests illustrate the proposed scheme.
AbstractList This paper introduces a method for computing points satisfying the second-order necessary optimality conditions for nonconvex minimization problems subject to a closed and convex constraint set. The method comprises two independent steps corresponding to the first- and second-order conditions. The first-order step is a generic closed map algorithm, which can be chosen from a variety of first-order algorithms, making it adjustable to the given problem. The second-order step can be viewed as a second-order feasible direction step for nonconvex minimization subject to a convex set. We prove that any limit point of the resulting scheme satisfies the second-order necessary optimality condition, and establish the scheme’s convergence rate and complexity, under standard and mild assumptions. Numerical tests illustrate the proposed scheme.
Author Teboulle, Marc
Hallak, Nadav
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  givenname: Marc
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  surname: Teboulle
  fullname: Teboulle, Marc
  email: teboulle@tauex.tau.ac.il
  organization: School of Mathematical Sciences, Tel-Aviv University
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crossref_primary_10_1007_s10589_023_00501_3
crossref_primary_10_1137_21M1430868
crossref_primary_10_1080_10556788_2023_2296982
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SubjectTerms Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Constraints
Engineering
First order algorithms
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Theory of Computation
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Title Finding Second-Order Stationary Points in Constrained Minimization: A Feasible Direction Approach
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