New exact penalty function for solving constrained finite min-max problems

Ma, Cheng; Li, Xun; Yiu, Ka Fai Cedric; Zhang, Liansheng

doi:10.1007/s10483-012-1548-6

Cited by 8 publications

(12 citation statements)

References 17 publications

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“…Let us also obtains an interesting property of cluster points of a minimizing sequence constructed with the use of a parametric penalty function (i.e. a sequence satisfying (35)) in the case when this penalty function is not exact. Proposition 3.22.…”

Section: The Following Inequalities Hold Truementioning

confidence: 99%

“…Throughout this article, we refer to the penalty function proposed in [23] as a singular exact penalty function, since one achieves smoothness of this exact penalty function via the introduction of a singular term into the definition of this function. Recently, singular exact penalty functions have attracted a lot of attention of researchers [6,14,15,44], and were successfully applied to various constrained optimization problems [25,27,31,32,35,51,52]. It should be noted that the main feature of both smoothing approximations of exact penalty functions and singular exact penalty functions is the fact that they depend on some additional parameters apart from the penalty parameter.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: The Following Inequalities Hold Truementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Долгополик

2017

Optimization

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“…h GOPF algorithm provides an alternative method to solve (P1). In GOPF algorithm, it does not need to increase the parameter M to 1, which differs from other penalty function methods [17][18][19][20]. Theorem 2.3 is obviously of great theoretical interest, however, if we use standard optimization methods to solve the sequences of problem (PM) we may possibly find local solutions of problem (PM).…”

Section: Proofmentioning

confidence: 99%

“…Recently, for dealing with finite constrained minimax problems, Obasanjo et al [19] presented a primal-dual interior-point method to solve the equivalent problem (P2). Ma et al [20] introduced a new exact and smooth penalty function to tackle minimax problems with equality constraints, in which the penalty parameter needed to be increased gradually to infinity. Based on the equivalent reformulation (P2) and a novel continuously differentiable exact objective penalty function, we propose a penalty function method to solve the minimax problem (P1) by taking a finite penalty parameter.…”

Section: Introductionmentioning

confidence: 99%

A New Objective Penalty Function Approach for Solving Constrained Minimax Problems

Long

2014

J. Oper. Res. Soc. China

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In this paper, a new objective penalty function approach is proposed for solving minimax programming problems with equality and inequality constraints. This new objective penalty function combines the objective penalty and constraint penalty. By the new objective penalty function, a constrained minimax problem is converted to minimizations of a sequence of continuously differentiable functions with a simple box constraint. One can thus apply any efficient gradient minimization methods to solve the minimizations with box constraint at each step of the sequence. Some relationships between the original constrained minimax problem and the corresponding minimization problems with box constraint are established. Based on these results, an algorithm for finding a global solution of the constrained minimax problems is proposed by integrating the particular structure of minimax problems and its global convergence is proved under some conditions. Furthermore, an algorithm is developed for finding a local solution of the constrained minimax problems, with its convergence proved under certain conditions. Preliminary results This research was supported by Natural Science Foundation of Chongqing (Nos. cstc2013jjB00001 and cstc2011jjA00010), by Chongqing Municipal Education Commission (No.KJ120616).

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“…extended or parametric penalty function) was introduced by Huyer and Neumaier [22] in 2003. This penalty function was generalized and, later on, applied to various optimization problems in [1,35,24,30,26,23,25,31,39,13]. In [12], it was shown that Huyer and Neumaier's extended penalty function is exact if and only if the standard nonsmooth penalty function is exact, and some relations between the least exact penalty parameters of these functions were obtained.…”

Section: Introductionmentioning

confidence: 99%

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions II: Extended Exactness

Долгополик

2018

J Optim Theory Appl

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In the second part of our study we introduce the concept of global extended exactness of penalty and augmented Lagrangian functions, and derive the localization principle in the extended form. The main idea behind the extended exactness consists in an extension of the original constrained optimization problem by adding some extra variables, and then construction of a penalty/augmented Lagrangian function for the extended problem. This approach allows one to design extended penalty/augmented Lagrangian functions having some useful properties (such as smoothness), which their counterparts for the original problem might not possess. In turn, the global exactness of such extended merit functions can be easily proved with the use of the localization principle presented in this paper, which reduces the study of global exactness to a local analysis of a merit function based on sufficient optimality conditions and constraint qualifications. We utilize the localization principle in order to obtain simple necessary and sufficient conditions for the global exactness of the extended penalty function introduced by Huyer and Neumaier, and in order to construct a globally exact continuously differentiable augmented Lagrangian function for nonlinear semidefinite programming problems.

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New exact penalty function for solving constrained finite min-max problems

Cited by 8 publications

References 17 publications

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

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

A New Objective Penalty Function Approach for Solving Constrained Minimax Problems

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions II: Extended Exactness

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