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Variational convergence of composed convex functions

POSITIVITY 2005年第1期9卷 115-147页

作者： Laghdir, M Thibault, L Fac Sci Dept Math El Jadida Morocco Univ Montpellier 2 Dept Math F-34095 Montpellier France

In this paper we introduce a concept of variational convergence for mappings taking values in order topological vector spaces. This variational convergence notion is shown to be well adapted to the Gamma(epi)-convergence of composed convex functions, in the sense that it is preserved after composition with nondecreasing functions. It is proved how this stability result can be applied to the continuity of multipliers under perturbations associated with a family of constrained optimization problems. Other applications are also given.

关键词： variational convergence composed convex functions epiconvergence constraint qualifications partial order

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Farkas-type results for inequality systems with composed convex functions via conjugate duality

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2006年第1期322卷 316-328页

作者： Bot, Radu Ioan Hodrea, Ioan Bogdan Wanka, Gert Tech Univ Chemnitz Fac Math D-09107 Chemnitz Germany

We present some Farkas-type results for inequality systems involving finitely many functions. Therefore we use a conjugate duality approach applied to an optimization problem with a composed convex objective function and convex inequality constraints. Some recently obtained results are rediscovered as special cases of our main result. (c) 2005 Elsevier Inc. All rights reserved.

关键词： Farkas-type results composed convex functions conjugate functions conjugate duality

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Generalized Moreau-Rockafellar results for composed convex functions

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OPTIMIZATION 2009年第7期58卷 917-933页

作者： Bot, Radu Ioan Grad, Sorin-Mihai Wanka, Gert Tech Univ Chemnitz Fac Math D-09107 Chemnitz Germany

We give two generalized Moreau-Rockafellar-type results for the sum of a convex function with a composition of convex functions in separated locally convex spaces. Then we equivalently characterize the stable strong duality for composed convex optimization problems through two new regularity conditions, which also guarantee two formulae of the subdifferential of the mentioned sum of functions. We also treat some special cases, rediscovering older results in the literature. A discussion on the topological assumptions for the vector function used in the composition closes this article.

关键词： conjugate functions Moreau-Rockafellar results regularity conditions stable strong duality composed convex functions

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Optimality conditions for weak efficiency to vector optimization problems with composed convex functions

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CENTRAL EUROPEAN JOURNAL OF MATHEMATICS 2008年第3期6卷 453-468页

作者： Bot, Radu Ioan Hodrea, Ioan Bogdan Wanka, Gert Tech Univ Chemnitz Fac Math D-09107 Chemnitz Germany

We consider a convex optimization problem with a vector valued function as objective function and convex cone inequality constraints. We suppose that each entry of the objective function is the composition of some convex functions. Our aim is to provide necessary and sufficient conditions for the weakly efficient solutions of this vector problem. Moreover, a multiobjective dual treatment is given and weak and strong duality assertions are proved.

关键词： multiobjective optimization composed convex functions conjugate duality weak efficiency

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Lower semicontinuous type regularity conditions for subdifferential calculus

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OPTIMIZATION METHODS & SOFTWARE 2010年第1期25卷 37-48页

作者： Bot, Radu Ioan Grad, Sorin-Mihai Tech Univ Chemnitz Fac Math D-09107 Chemnitz Germany

We give a lower semicontinuous type regularity condition and a closedness type one which turn out to be necessary and sufficient for the fulfilment of two different formulae involving the epsilon-subdifferential of a perturbation function, respectively. These regularity conditions prove to be sufficient also for having formulae for the classical subdifferential of a perturbation function. Some recently published results concerning epsilon-subdifferentials are rediscovered as special cases.

关键词： conjugate functions convex subdifferentials regularity conditions stable strong duality composed convex functions

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REGULARITY CONDITIONS FOR FORMULAE OF BICONJUGATE functions

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TAIWANESE JOURNAL OF MATHEMATICS 2008年第8期12卷 1921-1942页

作者： Bot, Radu Ioan Grad, Sorin-Mihai Tech Univ Chemnitz Fac Math D-09107 Chemnitz Germany

When the dual of a normed space X is endowed with the weak* topology, the biconjugates of the proper convex lower semicontinuous functions defined on X coincide with the functions themselves. This is not the case when X* is endowed with the strong topology. Working in the latter framework, we give formulae for the biconjugates of some functions that appear often in convex optimization, which hold provided the validity of some suitable regularity conditions. We also treat some special cases, rediscovering and improving recent results in the literature. Finally, we give a regularity condition that guarantees that the biconjugate of the supremum of a possibly infinite family of proper convex lower semicontinuous functions defined on a separated locally convex space coincides with the supremum of their biconjugates.

关键词： Conjugate functions Regularity conditions Dual problems composed convex functions

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Conjugate duality for multiobjective composed optimization problems

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ACTA MATHEMATICA HUNGARICA 2007年第3期116卷 177-196页

作者： Bot, R. I. Vargyas, E. Wanka, G. Tech Univ Chemnitz Fac Math D-09107 Chemnitz Germany

Given a multiobjective optimization problem with the components of the objective function as well as the constraint functions being composed convex functions, we introduce, by using the Fenchel-Moreau conjugate of the functions involved, a suitable dual problem. Under a standard constraint qualification and some convexity as well as monotonicity conditions we prove the existence of strong duality. Finally, some particular cases of this problem are presented.

关键词： composed convex functions scalar duality multiobjective duality optimality conditions

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Duality for Multiobjective Bilevel Programming Problems with Extremal-Value Function

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Journal of Mathematical Research with Applications 2015年第3期35卷 311-320页

作者： Haijun WANG Ruifang ZHANG Department of Mathematics Taiyuan Normal University

For a multiobjective bilevel programming problem(P) with an extremal-value function,its dual problem is constructed by using the Fenchel-Moreau conjugate of the functions *** some convexity and monotonicity assumption... 详细信息

关键词： multiobjective optimization bilevel programming problems conjugate duality convex programming composed convex functions

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ɛ -Optimality conditions for composed convex optimization problems

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Journal of Approximation Theory 2008年第1期153卷 108-121页

作者： Radu Ioan Boţ Ioan Bogdan Hodrea Gert Wanka Faculty of Mathematics Chemnitz University of Technology D-09107 Chemnitz Germany

The aim of the present paper is to provide a formula for the ɛ -subdifferential of f + g ∘ h different from the ones which can be found in the existent literature. Further we equivalently characterize this formula by using a so-called closedness type regularity condition expressed by means of the epigraphs of the conjugates of the functions involved. Even more, using the ɛ -subdifferential formula we are able to derive necessary and sufficient conditions for the ɛ -optimal solutions of composed convex optimization problems.

关键词： Conjugate functions composed convex functions ɛ -subdifferential ɛ -optimality conditions

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