Nonlinear separation functions and constrained extremum problems

Xu, Yanhong; Li, S. J.

doi:10.1007/s11590-013-0644-3

Cited by 12 publications

(14 citation statements)

References 19 publications

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“…The motivation behind the term "separating function" comes from a geometric interpretation of many penalty and augmented Lagrangian function as nonlinear functions separating some nonconvex sets. This point of view on penalty and augmented Lagrangian functions is systematically utilized within the image space analysis [56,57,85,69,78,132,133,116]. Remark 2.3.…”

Section: Preliminariesmentioning

confidence: 99%

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A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

Долгополик¹

2018

J Optim Theory Appl

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In this two-part study we develop a unified approach to the analysis of the global exactness of various penalty and augmented Lagrangian functions for constrained optimization problems in finite dimensional spaces. This approach allows one to verify in a simple and straightforward manner whether a given penalty/augmented Lagrangian function is exact, i.e. whether the problem of unconstrained minimization of this function is equivalent (in some sense) to the original constrained problem, provided the penalty parameter is sufficiently large. Our approach is based on the so-called localization principle that reduces the study of global exactness to a local analysis of a chosen merit function near globally optimal solutions. In turn, such local analysis can usually be performed with the use of sufficient optimality conditions and constraint qualifications.In the first paper we introduce the concept of global parametric exactness and derive the localization principle in the parametric form. With the use of this version of the localization principle we recover existing simple necessary and sufficient conditions for the global exactness of linear penalty functions, and for the existence of augmented Lagrange multipliers of Rockafellar-Wets' augmented Lagrangian. Also, we obtain completely new necessary and sufficient conditions for the global exactness of general nonlinear penalty functions, and for the global exactness of a continuously differentiable penalty function for nonlinear second-order cone programming problems. We briefly discuss how one can construct a continuously differentiable exact penalty function for nonlinear semidefinite programming problems, as well.

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Section: Preliminariesmentioning

confidence: 99%

“…A general duality theory for nonlinear Lagrangian and penalty functions was developed in [94,97,90,109]. Another general approach to the study of duality based on the image space analysis was systematically studied in [48,56,57,85,69,132,133,116].…”

Section: Introductionmentioning

confidence: 99%

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

Долгополик¹

2018

J Optim Theory Appl

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“…Note that parametric penalty functions can be viewed as a particular case of separation functions that are studied within the image space analysis (see [21,28,36,49,55] and references therein). However, surprisingly, no results of this paper (even the necessary and sufficient condition for the zero duality gap property) can be derived from the general results on separation functions known to the author (see also Remark 9 below).…”

Section: Introductionmentioning

confidence: 99%

“…Proof. Applying Theorem 3.6 to the penalty function b λ (x) = f (x) + λψ δ (d(0, G(x)) (see (49)) one obtains that b λ is exact. Therefore, as it is easy to see, the penalty function h λ is exact on the set {x ∈ A | d(0, G(x)) < θ} for any θ < δ (cf.…”

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confidence: 99%

A unifying theory of exactness of linear penalty functions II: parametric penalty functions

Долгополик

2017

Optimization

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In this article we develop a general theory of exact parametric penalty functions for constrained optimization problems. The main advantage of the method of parametric penalty functions is the fact that a parametric penalty function can be both smooth and exact unlike the standard (i.e. non-parametric) exact penalty functions that are always nonsmooth. We obtain several necessary and/or sufficient conditions for the exactness of parametric penalty functions, and for the zero duality gap property to hold true for these functions. We also prove some convergence results for the method of parametric penalty functions, and derive necessary and sufficient conditions for a parametric penalty function to not have any stationary points outside the set of feasible points of the constrained optimization problem under consideration. In the second part of the paper, we apply the general theory of exact parametric penalty functions to a class of parametric penalty functions introduced by Huyer and Neumaier, and to smoothing approximations of nonsmooth exact penalty functions. The general approach adopted in this article allowed us to unify and significantly sharpen many existing results on parametric penalty functions.

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“…The separation between the two suitable subsets is proved by showing that they lie in two disjoint level sets of a separating functional. Since then, ISA is extensively applied to establish separation theorems and theorems of alternative of constrained extremum problems by various kinds of separation functions such as linear and nonlinear cases, and then, Kuhn-Tucker type, Fritz-John type and Lagrangian-type optimality conditions of the constrained extremum problems are established by the corresponding theorems of separation and theorems of alternative; see [6,28,33,11,12,13,14,15,20,21,29,16,30]. Compared with the existing other approaches, a prominent advantage of ISA is that it can be applied to deal with nonconvex, nonsmooth and even discontinuous constrained optimization problems from the perspective of geometry.…”

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confidence: 99%

Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points

Chen¹,

Li²,

Yao³

2020

Journal of Industrial &Amp; Management Optimization

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In this paper, we consider a class of constrained vector optimization problems by using image space analysis. A class of vector-valued separation functions and a C-solution notion are proposed for the constrained vector optimization problems, respectively. Moreover, existence of a saddle point for the vector-valued separation function is characterized by the (regular) separation of two suitable subsets of the image space. By employing the separation function, we introduce a class of generalized vector-valued Lagrangian functions without involving any elements of the feasible set of constrained vector optimization problems. The relationships between the type-I(II) saddle points of the generalized Lagrangian functions and that of the function corresponding to the separation function are also established. Finally, optimality conditions for C-solutions of constrained vector optimization problems are derived by the saddle-point conditions.

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Nonlinear separation functions and constrained extremum problems

Cited by 12 publications

References 19 publications

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

A unifying theory of exactness of linear penalty functions II: parametric penalty functions

Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points

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