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image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Journal of Applied M...arrow_drop_down
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Journal of Applied Mathematics and Computing
Article . 2021 . Peer-reviewed
License: Springer TDM
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image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
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Article . 2022
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On sufficiency and duality theorems for nonsmooth semi-infinite mathematical programming problem with equilibrium constraints

Authors: Tran Van Su; Dinh Dieu Hang;

On sufficiency and duality theorems for nonsmooth semi-infinite mathematical programming problem with equilibrium constraints

Abstract

The semi-infinite MPEC \(f(x)\rightarrow \min_{x\in C}\) subject to the semi-infinite constraint \(g(x,t)\le 0 \;\forall t\in \Delta\), the equality constraint \(h(x)=0\) and the MPEC constraint \(G(x)\ge 0\), \(H(x)\ge 0\), \(G(x)^TH(x)=0\) is considered. Using the \(\Psi\)-invexity of \(C\) and the \(\Psi\)-preinvexity of the problem functions \(f,g(\cdot,t), \pm h_j, -G_i, -H_i\) sufficient conditions for global optimality are derived. The main tool is the contingent epi-derivative \(D_\uparrow f(\bar x)v\) in direction \(v\) of the contingend/Bouligand cone \(T(C,\bar x)\) [\textit{J.-P. Aubin} and \textit{H. Frankowska}, Set-valued analysis. Boston etc.: Birkhäuser (1990; Zbl 0713.49021)]. (GA) stationary points together with above \(\Psi\) pre-invexity (Theorem 1) or \(\Psi\)-quasi-convexity (Theorem 2) are global optimal whenever the last sum in the (GA) definition (see below) vanishes (\(\Leftarrow L_u=\emptyset\)). Since the inequality of differentials in Definition 2 ((GA)-stationary vector) of the paper is incomplete, the paper is badly to understand. The inequality for (GA)-stationary \(\bar x\) (generalized alternatively stationary point) should be written (see, e.g., [\textit{Y. Pandey} and \textit{S. K. Mishra}, J. Optim. Theory Appl. 171, No. 2, 694--707 (2016; Zbl 1349.90837), Definition 2.8]) \begin{multline*} D_{\uparrow} f(\bar x)v+\sum_{i\in I_k}\lambda^g_iD_{\uparrow}g(\bar x, t_i)v+\sum_{j\in I_n}\left[\lambda^h_jD_{\uparrow}h_j(\bar x)v+\mu^h_jD_{\uparrow}(-h_j)(\bar x)v\right] \\ +\sum_{i\in I_p}\left[\lambda^G_iD_{\uparrow}(-G_i)(\bar x)v+\lambda^H_iD_{\uparrow}(-H_i)(\bar x)v\right]+\sum_{i\in I_p}\left[\mu^G_iD_{\uparrow}(G_i)(\bar x)v+\mu^H_iD_{\uparrow}(H_i)(\bar x)v\right]\ge 0 \end{multline*} for all \(v\in T(C,\bar x)\). In Section 4, weak (Theorem 3, Theorem 5) and strong Mond-Weir/Wolfe duality (for local solutions in Theorem 4, Theorem 6) results are shown under the above invexity/quasi-convexity conditions and \(L_\mu=\emptyset\). For strong duality results, some generalized Slater condition is additionally necessary. Also, in Definition 4 of the Mond-Weir/Wolfe duality, the above last sum with the multipliers \(\mu_i^G,\mu_i^H\) must be added for understanding the subsequent formulations. However, there is no discussion whenever for a given problem the index set \(L_\mu\) is empty being a crucial condition for the validity of Theorem 1--Theorem 6.

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Keywords

\(\Psi\)-invex set, contingent epiderivative, Applications of functional analysis in optimization, convex analysis, mathematical programming, economics, GA-stationary vectors, Nonconvex programming, global optimization, generalized Slater constraint qualifications, Mond-Weir and Wolfe types dual model, nonsmooth semi-infinite mathematical programming problem with equilibrium constraints, \(\Psi\)-preinvex function, sufficiency and duality theorems, Optimality conditions and duality in mathematical programming, \(\Psi\)-quasi-convex function, Semi-infinite programming, Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming), Duality theory (optimization), Convexity of real functions of several variables, generalizations

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
6
Top 10%
Average
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