Two Error Bounds for Constrained Optimization Problems and Their Applications

Wang, Changyu; Zhang, Jianzhong; Zhao, Wenling

doi:10.1007/s00245-007-9023-8

Cited by 5 publications

(4 citation statements)

References 25 publications

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“…By (37) and (38), we obtain that the condition (a) in Definition 4 holds. Now we will prove the condition (b) in Definition 4 also holds.…”

Section: Some Examplesmentioning

confidence: 94%

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The Augmented Weak Sharpness of Solution Sets in Equilibrium Problems

Wang,

Zhao,

Song

et al. 2024

Mathematics

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This study considers equilibrium problems, focusing on identifying finite solutions for feasible solution sequences. We introduce an innovative extension of the weak sharp minimum concept from convex programming to equilibrium problems, coining this as weak sharpness for solution sets. Recognizing situations where the solution set may not exhibit weak sharpness, we propose an augmented mapping approach to mitigate this limitation. The core of our research is the formulation of augmented weak sharpness for the solution set. This comprehensive concept encapsulates both weak sharpness and strong non-degeneracy within feasible solution sequences. Crucially, we identify a necessary and sufficient condition for the finite termination of these sequences under the premise of augmented weak sharpness for the solution set in equilibrium problems. This condition significantly broadens the scope of the existing literature, which often assumes the solution set to be weakly sharp or strongly non-degenerate, especially in mathematical programming and variational inequality problems. Our findings not only shed light on the termination conditions in equilibrium problems but also introduce a less stringent sufficient condition for the finite termination of various optimization algorithms. This research, therefore, makes a substantial contribution to the field by enhancing our understanding of termination conditions in equilibrium problems and expanding the applicability of established theories to a wider range of optimization scenarios.

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“…By (37) and (38), we obtain that the condition (a) in Definition 4 holds. Now we will prove the condition (b) in Definition 4 also holds.…”

Section: Some Examplesmentioning

confidence: 94%

“…By the assumptions of the strong non-degeneracy of x and the continuity of ∇ y ϕ(x, x), and (Proposition 5.1 [37]), we know that x is an isolated point of S, Thus, we have…”

Section: The Smooth Casementioning

confidence: 99%

The Augmented Weak Sharpness of Solution Sets in Equilibrium Problems

Wang,

Zhao,

Song

et al. 2024

Mathematics

View full text Add to dashboard Cite

show abstract

“…Then, we give the definition of the augmented weak sharpness of solution sets relative to feasible solution sequence for VIP, and clarify, by examples, the new notion is an extension of the weak sharpness and strong non-degeneracy of the solution set of monotone variational inequality. Now we give the notions of weak sharpness and strong non-degeneracy of solution sets for V IP (see [16]).…”

Section: The Augmented Weak Sharpness Of Solution Setsmentioning

confidence: 99%

Finite Convergence for Feasible Solution Sequence of Variational Inequality Problems

Zhao

Wang

Zhang

2017

MCA

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We establish the notion of augmented weak sharpness of solution sets for the variational inequality problems which can be abbreviated to VIPs. This notion of augmented weak sharpness is an extension of the weak sharpness of the solution set of monotone variational inequality, and it overcomes the defect of the solution set not satisfying the weak sharpness in many cases. Under the condition of the solution set being augmented weak sharp, we present a necessary and sufficient condition for finite convergence for feasible solution sequence of VIP. The result is an extension of published results, and the augmented weak sharpness also provides weaker sufficient conditions for the finite convergence of many optimization algorithms.

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“…Another objective of this paper is to address the finite termination of the algorithm; that is, the optimal solution can be obtained after finite iterations. This issue has drawn the attention of many authors [1,[9][10][11]17,18]. Particularly, Burke and Ferris [5] developed the necessary Downloaded by [University of Auckland Library] at 17:47 18 November 2014 and sufficient conditions for the finite termination of a class of iterative algorithms for solving convex programming problems, provided that the solution set is weakly sharp.…”

Section: Introductionmentioning

confidence: 99%

Global convergence and finite termination of a class of smooth penalty function algorithms

Wang

Zhao

Zhou

et al. 2013

Optimization Methods and Software

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In this paper, based on a new class of asymptotic l 1 exact penalty functions, we propose a smooth penalty function for solving nonlinear programming problems. One of the main features of our algorithm is that at each iteration, we do not need to solve the global minimum of penalty functions. Furthermore, global convergence property is established without requiring any constraint qualifications. By addressing perturbation functions, we obtain that the lower semi-continuity of the perturbation function at zero is a necessary and sufficient condition to ensure the convergence of the objective function values generated by the algorithm to the optimal value of the primal problem. Since the perturbation function is only dependent on the data of the primal problem, these results enable us to check the convergence property of the algorithm in advance. Finally, we discuss the finite termination of the algorithm when the solution set is non-degenerate or weakly sharp.

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Two Error Bounds for Constrained Optimization Problems and Their Applications

Cited by 5 publications

References 25 publications

The Augmented Weak Sharpness of Solution Sets in Equilibrium Problems

The Augmented Weak Sharpness of Solution Sets in Equilibrium Problems

Finite Convergence for Feasible Solution Sequence of Variational Inequality Problems

Global convergence and finite termination of a class of smooth penalty function algorithms

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