Improved Convergence Properties of the Relaxation Schemes of Kadrani et al. and Kanzow and Schwartz for MPEC

2018 ◽  
Vol 35 (01) ◽  
pp. 1850008
Author(s):  
Na Xu ◽  
Xide Zhu ◽  
Li-Ping Pang ◽  
Jian Lv
Keyword(s):  
Linear Dependence ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Accumulation Point ◽  
Stationary Points ◽  
Nondegeneracy Condition ◽  
Equilibrium Constraints ◽  
Convergence Properties ◽  
Constant Rank Constraint Qualification ◽  
Relaxation Schemes

This paper concentrates on improving the convergence properties of the relaxation schemes introduced by Kadrani et al. and Kanzow and Schwartz for mathematical program with equilibrium constraints (MPEC) by weakening the original constraint qualifications. It has been known that MPEC relaxed constant positive-linear dependence (MPEC-RCPLD) is a class of extremely weak constraint qualifications for MPEC, which can be strictly implied by MPEC relaxed constant rank constraint qualification (MPEC-RCRCQ) and MPEC relaxed constant positive-linear dependence (MPEC-rCPLD), of course also by the MPEC constant positive-linear dependence (MPEC-CPLD). We show that any accumulation point of stationary points of these two approximation problems is M-stationarity under the MPEC-RCPLD constraint qualification, and further show that the accumulation point can even be S-stationarity coupled with the asymptotically weak nondegeneracy condition.

scholarly journals R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization

2021 ◽  
Author(s):  
Patrick Mehlitz ◽  
Leonid I. Minchenko
Keyword(s):  
Linear Dependence ◽  
Constraint Qualification ◽  
Parametric Optimization ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Bilevel Optimization ◽  
Regularity Property ◽  
Constant Rank Constraint Qualification ◽  
Existence Criterion ◽  
Set Valued Mappings

AbstractThe presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.

scholarly journals A New Sequential Optimality Condition of Cardinality-Constrained Optimization Problem and Application

2021 ◽  
Author(s):  
Liping Pang ◽  
Menglong Xue ◽  
Na Xu
Keyword(s):  
Constrained Optimization ◽  
Optimality Condition ◽  
Optimization Problem ◽  
Constraint Qualification ◽  
Accumulation Point ◽  
Iterative Sequence ◽  
Stationary Points ◽  
Constrained Optimization Problem ◽  
Cardinality Constraint ◽  
Continuous Relaxation

Abstract In this paper, we consider the cardinality-constrained optimization problem and propose a new sequential optimality condition for the continuous relaxation reformulation which is popular recently. It is stronger than the existing results and is still a first-order necessity condition for the cardinality constraint problem without any additional assumptions. Meanwhile, we provide a problem-tailored weaker constraint qualification, which can guarantee that new sequential conditions are Mordukhovich-type stationary points. On the other hand, we improve the theoretical results of the augmented Lagrangian algorithm. Under the same condition as the existing results, we prove that any feasible accumulation point of the iterative sequence generated by the algorithm satisfies the new sequence optimality condition. Furthermore, the algorithm can converge to the Mordukhovich-type (essentially strong) stationary point if the problem-tailored constraint qualification is satisfied.

New Constraint Qualifications for S-Stationarity for MPEC with Nonsmooth Objective

2019 ◽  
Vol 36 (02) ◽  
pp. 1940001
Author(s):  
Peng Zhang ◽  
Jin Zhang ◽  
Gui-Hua Lin ◽  
Xinmin Yang
Keyword(s):  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Mathematical Problem ◽  
Equilibrium Constraints ◽  
Lipschitz Continuous ◽  
Nonlinear Programs ◽  
Locally Lipschitz ◽  
Locally Lipschitz Continuous

This paper considers a mathematical problem with equilibrium constraints (MPEC) in which the objective is locally Lipschitz continuous but not continuously differentiable everywhere. Our focus is on constraint qualifications for the nonsmooth S-stationarity in the sense of the limiting subdifferentials. First, although the MPEC-LICQ is not a constraint qualification for the nonsmooth S-stationarity, we show that the MPEC-LICQ can serve as a constraint qualification for the nonsmooth S-stationarity under some kind of regularity. Then, we extend some new constraint qualifications for nonlinear programs to the considered nonsmooth MPEC and show that all of them can serve as constraint qualifications for the nonsmooth S-stationarity. We further extend these results to the multiobjective case.

scholarly journals Notes on Convergence Properties for a Smoothing-Regularization Approach to Mathematical Programs with Vanishing Constraints

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qingjie Hu ◽  
Yu Chen ◽  
Zhibin Zhu ◽  
Bishan Zhang
Keyword(s):  
Regularization Method ◽  
Accumulation Point ◽  
Stationary Points ◽  
Convergence Properties ◽  
Convergence Behavior ◽  
Mathematical Programs ◽  
Convergence Results ◽  
Vanishing Constraints ◽  
Regularization Problem ◽  
Strongly Stationary

We give some improved convergence results about the smoothing-regularization approach to mathematical programs with vanishing constraints (MPVC for short), which is proposed in Achtziger et al. (2013). We show that the Mangasarian-Fromovitz constraints qualification for the smoothing-regularization problem still holds under the VC-MFCQ (see Definition 5) which is weaker than the VC-LICQ (see Definition 7) and the condition of asymptotic nondegeneracy. We also analyze the convergence behavior of the smoothing-regularization method and prove that any accumulation point of a sequence of stationary points for the smoothing-regularization problem is still strongly-stationary under the VC-MFCQ and the condition of asymptotic nondegeneracy.

scholarly journals Asymptotic stationarity and regularity for nonsmooth optimization problems

2020 ◽  
Author(s):  
Patrick Mehlitz
Keyword(s):  
Nonsmooth Optimization ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Constraint Qualifications ◽  
Variational Calculus ◽  
Stationary Points ◽  
Mathematical Programs ◽  
Disjunctive Constraints ◽  
The Relationship ◽  
Type Constraint

Based on the tools of limiting variational analysis, we derive a sequential necessary optimality condition for nonsmooth mathematical programs which holds without any additional assumptions. In order to ensure that stationary points in this new sense are already Mordukhovich-stationary, the presence of a constraint qualification which we call AM-regularity is necessary. We investigate the relationship between AM-regularity and other constraint qualifications from nonsmooth optimization like metric (sub-)regularity of the underlying feasibility mapping. Our findings are applied to optimization problems with geometric and, particularly, disjunctive constraints. This way, it is shown that AM-regularity recovers recently introduced cone-continuity-type constraint qualifications, sometimes referred to as AKKT-regularity, from standard nonlinear and complementarity-constrained optimization. Finally, we discuss some consequences of AM-regularity for the limiting variational calculus.

scholarly journals CONSTRAINT QUALIFICATIONS IN PARTIAL IDENTIFICATION

2021 ◽  
pp. 1-24
Author(s):  
Hiroaki Kaido ◽  
Francesca Molinari ◽  
Jörg Stoye
Keyword(s):  
Stochastic Programming ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Partial Identification ◽  
Moment Inequalities ◽  
Pure Moment ◽  
High Level

The literature on stochastic programming typically restricts attention to problems that fulfill constraint qualifications. The literature on estimation and inference under partial identification frequently restricts the geometry of identified sets with diverse high-level assumptions. These superficially appear to be different approaches to closely related problems. We extensively analyze their relation. Among other things, we show that for partial identification through pure moment inequalities, numerous assumptions from the literature essentially coincide with the Mangasarian–Fromowitz constraint qualification. This clarifies the relation between well-known contributions, including within econometrics, and elucidates stringency, as well as ease of verification, of some high-level assumptions in seminal papers.

scholarly journals On stability and stationary points in nonlinear optimization

1986 ◽  
Vol 28 (1) ◽  
pp. 36-56 ◽  
Author(s):  
J. Guddat ◽  
H. Th. Jongen ◽  
J. Rueckmann
Keyword(s):  
Stability Condition ◽  
Minimization Problem ◽  
Level Sets ◽  
Constraint Qualification ◽  
Stationary Points ◽  
Feasible Set ◽  
Manifold With Boundary ◽  
Feasible Sets ◽  
Constrained Minimization Problem ◽  
Image Position

This paper presents three theorems concerning stability and stationary points of the constrained minimization problem:In summary, we provethat, given the Mangasarian-Fromovitz constraint qualification (MFCQ), the feasible setM[H, G] is a topological manifold with boundary, with specified dimension; (ℬ) a compact feasible setM[H, G] is stable (perturbations ofHandGproduce homeomorphic feasible sets) if and only if MFCQ holds;under a stability condition, two lower level sets offwith a Kuhn-Tucker point between them are homotopically related by attachment of ak-cell (kbeing the stationary index in the sense of Kojima).

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