Constrained Extremum Problems and Image Space Analysis—Part II: Duality and Penalization

Li, Shengjie; Xu, Yanhong; You, Manxue; Zhu, Shengkun

doi:10.1007/s10957-018-1248-y

Cited by 15 publications

(3 citation statements)

References 6 publications

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“…Some deep connections among aspects, such as duality, gap functions, error bounds, that might be not evident from other perspective can be revealed by this approach; see [40,41,42]. For more details, we refer to [43,44,45] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Weak separation functions constructed by Gerstewitz and topical functions with applications in conjugate duality

2023

J. Nonlinear Var. Anal.

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This paper aims to construct some nonlinear weak separation functions in image space analysis by virtue of the Gerstewitz and topical functions. Then, applying these separation functions, a framework of conjugate type duality for constrained vector optimization problems is introduced. The primal problem is scalarized and then the separation functions are applied to give a scalar dual problem. Meanwhile, equivalent characterizations of the zero duality gap as well as the strong duality are established via subdifferential calculus, separation properties, and saddle point assertions.

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Section: Introductionmentioning

confidence: 99%

Weak separation functions constructed by Gerstewitz and topical functions with applications in conjugate duality

2023

J. Nonlinear Var. Anal.

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“…The ISA has shown to be a useful tool in unifying several fields of the mathematical optimization theory and to allow one to find new results. After 2005, there have been lots of theoretical results based on the ISA, including optimality conditions, duality, penalty methods, regularity and gap functions, developed; see, e.g., [2,10,16,17,[33][34][35][36][40][41][42][43]45,50,53,55,56].…”

Section: Introductionmentioning

confidence: 99%

Image Space Analysis for Set Optimization Problems with Applications

Zhou

Zhu

2021

J Optim Theory Appl

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In this paper, we consider a set optimization problem with a partial order relation, which is defined by Minkowski difference. By using the image space analysis, we establish the relationships among the set optimization problem, a vector optimization problem and a set-valued optimization with vector criterion related to the image of the set optimization problem. In addition, two nonlinear regular weak separation functions are proposed for the set optimization problem. Based on the two nonlinear regular weak separation functions, saddle point sufficient optimality conditions, gap functions and error bounds for the set optimization problem, are obtained. Finally, we explore some applications of the obtained results to investigate robust multi-objective optimization problems and verify the validity of the results in shortest path problems with data uncertainty and multi-criteria traffic network equilibrium problems with interval-valued cost functions.

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“…Image space analysis (shortly, ISA) was initiated in [13] and has been extensively used as a powerful tool for studying optimality conditions, duality, variational principles, penalty methods, gap functions and error bounds for various mathematical topics such as constrained extremum problems, variational inequalities, vector equilibrium problems, vector optimization and set optimization, see, e.g., [6,7,8,20,21,28,29] and the references therein More generally, it can be used for any problem, which can be expressed under the form of the infeasibility of a parametric system. The infeasibility of a parametric system is characterized by the disjunction of two suitable subsets in the image space.…”

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confidence: 99%

Image space analysis for uncertain multiobjective optimization problems: Robust optimality conditions

Al-Homidan

Ansari

et al. 2023

JIMO

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<p style='text-indent:20px;'>We introduce the <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{C} $\end{document}</tex-math></inline-formula>-robust efficient solution and optimistic <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{C} $\end{document}</tex-math></inline-formula>-robust efficient solution of uncertain multiobjective optimization problems (UMOP). By using image space analysis, robust optimality conditions as well as saddle point sufficient optimality conditions for uncertain multiobjective optimization problems are established based on real-valued linear (regular) weak separation function and real-valued (vector-valued) nonlinear (regular) weak separation functions. We also introduce two inclusion problems by using the image sets of robust counterpart of (UMOP) and establish the relations between the solution of the inclusion problems and the <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{C} $\end{document}</tex-math></inline-formula>-robust efficient solution (respectively, optimistic <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{C} $\end{document}</tex-math></inline-formula>-robust efficient solution) of (UMOP).</p>

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Constrained Extremum Problems and Image Space Analysis—Part II: Duality and Penalization

Cited by 15 publications

References 6 publications

Weak separation functions constructed by Gerstewitz and topical functions with applications in conjugate duality

Weak separation functions constructed by Gerstewitz and topical functions with applications in conjugate duality

Image Space Analysis for Set Optimization Problems with Applications

Image space analysis for uncertain multiobjective optimization problems: Robust optimality conditions

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