First-Order Optimality Conditions for Lipschitz Optimization Problems with Vanishing Constraints
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RESEARCH PAPER
First-Order Optimality Conditions for Lipschitz Optimization Problems with Vanishing Constraints Hadis Mokhtavayi1 • Aghileh Heydari1 • Nader Kanzi1 Received: 20 April 2020 / Accepted: 15 September 2020 Ó Shiraz University 2020
Abstract This article deals with a class of non-smooth mathematical programming with vanishing constraints. We introduce several kinds of constraint qualifications for these problems, and we study the relations between them. Then, we apply these constraint qualifications to obtain several stationary conditions. Finally, a sufficient condition for optimality of the problem is presented. Keywords Constraint qualification Stationary conditions Optimality conditions Vanishing constraints
1 Introduction In this paper, motivated by Hoheisel et al. (2010) and Izmailov and Solodov (2009), we consider the following mathematical programming with vanishing constraints (MPVC in short): ðPÞ
min s:t:
f ðxÞ Hi ðxÞ 0;
i 2 I :¼ f1; . . .; mg;
Gi ðxÞHi ðxÞ 0;
i 2 I;
where f ; Hi ; Gi : Rn ! R (for i 2 I) are locally Lipschitz functions. It should be noted that the general form of a MPVC which has been considered in Achtziger and Kanzow (2007), Achtziger et al. (2013), and Hoheisel and Kanzow (2007, 2008, 2009) includes inequality constraints gj ðxÞ 0; j 2 J and equality constraints ht ðxÞ ¼ 0; t 2 T for some finite index sets J and T. Since adding these constraints to problem (P) does not increase the technical
& Nader Kanzi [email protected] Hadis Mokhtavayi [email protected] Aghileh Heydari [email protected] 1
Department of Mathematics, Payame Noor University (PNU), P. O. Box 19395-3697 Tehran, Iran
problems of the issue and just prolongs the formulas, we ignore them and just deal with problem (P) . MPVCs are only studied since 2007 by Achtziger and Kanzow (2007). As in classic form of mathematical programming problems, to get Karush–Kuhn–Tucker (KKT)type necessary optimality conditions, we need some additional assumptions which are called constraint qualifications (CQ). These CQs are divided to two groups: algebraic CQs and geometric CQs. The most important algebraic CQs are linear independent CQ (LICQ), Mangasarian–Fromovitz CQ (MFCQ), Cottle CQ (CCQ), and Slater CQ (SCQ). In Achtziger and Kanzow (2007), it has been shown that LICQ and MFCQ never happen for MPVCs unless under some limited assumptions. On the other hand, since in general, convexity of u1 and u2 functions does not lead to convexity of u1 u2 , and SCQ is also just helpful for convex problems; therefore, the classic definition of SCQ does not seem useful for MPVCs. From geometric CQs, in references Achtziger and Kanzow (2007) and Hoheisel and Kanzow (2009), the Abadie CQ (ACQ) and the Guignard CQ (GCQ) for problem (P) with smooth data have been studied. We will study here the generalizing of these algebraic and geometric CQs in case of non-smooth MPVCs. We will also present some stationary conditions commensurate with defined CQs. Recently, Kazemi an
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