scholarly journals Optimality Conditions, Qualifications and Approximation Method for a Class of Non-Lipschitz Mathematical Programs with Switching Constraints

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2915
Author(s):  
Jinman Lv ◽  
Zhenhua Peng ◽  
Zhongping Wan
Keyword(s):  
Optimality Conditions ◽  
Approximation Method ◽  
Constraint Qualifications ◽  
Necessary Optimality Conditions ◽  
Accumulation Point ◽  
Local Minimizer ◽  
Necessary Condition ◽  
Local Minimizers ◽  
Mathematical Programs ◽  
The Given

In this paper, we consider a class of mathematical programs with switching constraints (MPSCs) where the objective involves a non-Lipschitz term. Due to the non-Lipschitz continuity of the objective function, the existing constraint qualifications for local Lipschitz MPSCs are invalid to ensure that necessary conditions hold at the local minimizer. Therefore, we propose some MPSC-tailored qualifications which are related to the constraints and the non-Lipschitz term to ensure that local minimizers satisfy the necessary optimality conditions. Moreover, we study the weak, Mordukhovich, Bouligand, strongly (W-, M-, B-, S-) stationay, analyze what qualifications making local minimizers satisfy the (M-, B-, S-) stationay, and discuss the relationship between the given MPSC-tailored qualifications. Finally, an approximation method for solving the non-Lipschitz MPSCs is given, and we show that the accumulation point of the sequence generated by the approximation method satisfies S-stationary under the second-order necessary condition and MPSC Mangasarian-Fromovitz (MF) qualification.

Necessary and sufficient optimality conditions using convexifactors for mathematical programs with equilibrium constraints

2019 ◽  
Vol 53 (5) ◽  
pp. 1617-1632 ◽  
Author(s):  
Bhawna Kohli
Keyword(s):  
Optimality Conditions ◽  
Constraint Qualification ◽  
Necessary Optimality Conditions ◽  
Sufficient Optimality Conditions ◽  
Equilibrium Constraints ◽  
Necessary Optimality Condition ◽  
First Order ◽  
Mathematical Programs ◽  
Strong Stationarity ◽  
Necessary And Sufficient

The main aim of this paper is to develop necessary Optimality conditions using Convexifactors for mathematical programs with equilibrium constraints (MPEC). For this purpose a nonsmooth version of the standard Guignard constraint qualification (GCQ) and strong stationarity are introduced in terms of convexifactors for MPEC. It is shown that Strong stationarity is the first order necessary optimality condition under nonsmooth version of the standard GCQ. Finally, notions of asymptotic pseudoconvexity and asymptotic quasiconvexity are used to establish the sufficient optimality conditions for MPEC.

scholarly journals Sequential optimality conditions for cardinality-constrained optimization problems with applications

2021 ◽  
Author(s):  
Christian Kanzow ◽  
Andreas B. Raharja ◽  
Alexandra Schwartz
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Optimality Condition ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Local Minimizer ◽  
Constrained Optimization Problems ◽  
New Approach ◽  
Mathematical Programs ◽  
Sequential Optimality Conditions

AbstractRecently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed. Following this approach, we derive a problem-tailored sequential optimality condition, which is satisfied at every local minimizer without requiring any constraint qualification. We relate this condition to an existing M-type stationary concept by introducing a weak sequential constraint qualification based on a cone-continuity property. Finally, we present two algorithmic applications: We improve existing results for a known regularization method by proving that it generates limit points satisfying the aforementioned optimality conditions even if the subproblems are only solved inexactly. And we show that, under a suitable Kurdyka–Łojasiewicz-type assumption, any limit point of a standard (safeguarded) multiplier penalty method applied directly to the reformulated problem also satisfies the optimality condition. These results are stronger than corresponding ones known for the related class of mathematical programs with complementarity constraints.

scholarly journals Sufficient optimality conditions and Mond-Weir duality results for a fractional multiobjective optimization problem

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nazih Abderrazzak Gadhi ◽  
Fatima Zahra Rahou
Keyword(s):  
Multiobjective Optimization ◽  
Optimality Conditions ◽  
Optimization Problem ◽  
Necessary Optimality Conditions ◽  
Sufficient Optimality Conditions ◽  
Multiobjective Optimization Problem ◽  
Support Functions ◽  
Duality Results ◽  
The Given ◽  
Generalized Convexities

<p style='text-indent:20px;'>In this work, we are concerned with a fractional multiobjective optimization problem <inline-formula><tex-math id="M1">\begin{document}$ (P) $\end{document}</tex-math></inline-formula> involving set-valued maps. Based on necessary optimality conditions given by Gadhi et al. [<xref ref-type="bibr" rid="b14">14</xref>], using support functions, we derive sufficient optimality conditions for <inline-formula><tex-math id="M2">\begin{document}$ \left( P\right) , $\end{document}</tex-math></inline-formula> and we establish various duality results by associating the given problem with its Mond-Weir dual problem <inline-formula><tex-math id="M3">\begin{document}$ \left( D\right) . $\end{document}</tex-math></inline-formula> The main tools we exploit are convexificators and generalized convexities. Examples that illustrates our findings are also given.</p>

scholarly journals Constraint qualifications in nonsmooth multiobjective optimization problem

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 781-797 ◽  
Author(s):  
Rekha Gupta ◽  
Manjari Srivastava
Keyword(s):  
Multiobjective Optimization ◽  
Optimality Conditions ◽  
Optimization Problem ◽  
Constraint Qualifications ◽  
Necessary Optimality Conditions ◽  
Inequality Constraint ◽  
Efficient Solutions ◽  
Equality Constraints ◽  
Regularity Conditions ◽  
Multiobjective Optimization Problem

A multiobjective optimization problem (MOP) with inequality and equality constraints is considered where the objective and inequality constraint functions are locally Lipschitz and equality constraint functions are differentiable. Burachik and Rizvi [J. Optim. Theory Appl. 155, 477-491 (2012)] gave Guignard and generalized Abadie regularity conditions for a differentiable programming problem and derived Karush-Kuhn-Tucker (KKT) type necessary optimality conditions. In this paper, we have defined the nonsmooth versions of Guignard and generalized Abadie regularity conditions given by Burachik and Rizvi and obtained KKT necessary optimality conditions for efficient and weak efficient solutions of (MOP). Further several constraint qualifications sufficient for the above newly defined constraint qualifications are introduced for (MOP) with no equality constraints. Relationships between them are presented and examples are constructed to support the results.

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