scholarly journals A new elementary proof for M-stationarity under MPCC-GCQ for mathematical programs with complementarity constraints

2021 ◽  
Vol Volume 2 (Original research articles) ◽  
Author(s):  
Felix Harder
Keyword(s):  
Constraint Qualification ◽  
Elementary Proof ◽  
Stationary Points ◽  
Complementarity Constraints ◽  
Normal Cones ◽  
Local Minimizers ◽  
Diploma Thesis ◽  
Mathematical Programs ◽  
Technical Theory ◽  
Crucial Ingredient

It is known in the literature that local minimizers of mathematical programs with complementarity constraints (MPCCs) are so-called M-stationary points, if a weak MPCC-tailored Guignard constraint qualification (called MPCC-GCQ) holds. In this paper we present a new elementary proof for this result. Our proof is significantly simpler than existing proofs and does not rely on deeper technical theory such as calculus rules for limiting normal cones. A crucial ingredient is a proof of a (to the best of our knowledge previously open) conjecture, which was formulated in a Diploma thesis by Schinabeck.

A LOCALLY SMOOTHING METHOD FOR MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS

2015 ◽  
Vol 56 (3) ◽  
pp. 299-315 ◽  
Author(s):  
YU CHEN ◽  
ZHONG WAN
Keyword(s):  
Stationary Point ◽  
Sufficient Conditions ◽  
Accumulation Point ◽  
Smoothing Method ◽  
Stationary Points ◽  
Local Perturbation ◽  
Complementarity Constraints ◽  
Approximate Problem ◽  
Mathematical Programs ◽  
Strongly Stationary

We propose a locally smoothing method for some mathematical programs with complementarity constraints, which only incurs a local perturbation on these constraints. For the approximate problem obtained from the smoothing method, we show that the Mangasarian–Fromovitz constraints qualification holds under certain conditions. We also analyse the convergence behaviour of the smoothing method, and present some sufficient conditions such that an accumulation point of a sequence of stationary points for the approximate problems is a C-stationary point, an M-stationary point or a strongly stationary point. Numerical experiments are employed to test the performance of the algorithm developed. The results obtained demonstrate that our algorithm is much more promising than the similar ones in the literature.

scholarly journals A New Smoothing Method for Mathematical Programs with Complementarity Constraints Based on Logarithm-Exponential Function

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Yu Chen ◽  
Zhong Wan
Keyword(s):  
Approximate Solution ◽  
Exponential Function ◽  
Constraint Qualification ◽  
Mathematical Program ◽  
Accumulation Point ◽  
Smoothing Method ◽  
Complementarity Constraints ◽  
Smoothing Methods ◽  
Mathematical Programs ◽  
Solution Sequence

We present a new smoothing method based on a logarithm-exponential function for mathematical program with complementarity constraints (MPCC). Different from the existing smoothing methods available in the literature, we construct an approximate smooth problem of MPCC by partly smoothing the complementarity constraints. With this new method, it is proved that the Mangasarian-Fromovitz constraint qualification holds for the approximate smooth problem. Convergence of the approximate solution sequence, generated by solving a series of smooth perturbed subproblems, is investigated. Under the weaker constraint qualification MPCC-Cone-Continuity Property, it is proved that any accumulation point of the approximate solution sequence is a M-stationary point of the original MPCC. Preliminary numerical results indicate that the developed algorithm based on the partly smoothing method is efficient, particularly in comparison with the other similar ones.

CONVERGENCE ANALYSIS OF A REGULARIZED SAMPLE AVERAGE APPROXIMATION METHOD FOR STOCHASTIC MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS

2011 ◽  
Vol 28 (06) ◽  
pp. 755-771 ◽  
Author(s):  
YONGCHAO LIU ◽  
GUI-HUA LIN
Keyword(s):  
Convergence Analysis ◽  
Approximation Method ◽  
Regularization Parameter ◽  
Sample Average Approximation ◽  
Stationary Points ◽  
Complementarity Constraints ◽  
One Stage ◽  
Mathematical Programs ◽  
Sample Average ◽  
Average Approximation

Regularization method proposed by Scholtes (2011) has been a recognized approach for deterministic mathematical programs with complementarity constraints (MPCC). Meng and Xu (2006) applied the approach coupled with Monte Carlo techniques to solve a class of one stage stochastic MPCC and presented some promising numerical results. However, Meng and Xu have not presented any convergence analysis of the regularized sample approximation method. In this paper, we fill out this gap. Specifically, we consider a general class of one stage stochastic mathematical programs with complementarity constraint where the objective and constraint functions are expected values of random functions. We carry out extensive convergence analysis of the regularized sample average approximation problems including the convergence of statistical estimators of optimal solutions, C-stationary points, M-stationary points and B-stationary points as sample size increases and the regularization parameter tends to zero.

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 MPCC: Strong Stability of M-stationary Points

2021 ◽  
Author(s):  
Harald Günzel ◽  
Daniel Hernández Escobar ◽  
Jan-J. Rückmann
Keyword(s):  
Stationary Point ◽  
Constraint Qualification ◽  
Local Existence ◽  
Strong Stability ◽  
Stationary Points ◽  
Algebraic Characterization ◽  
Describing Functions ◽  
Mathematical Programs ◽  
Local Existence And Uniqueness ◽  
Linear Independence Constraint Qualification

AbstractIn this paper we study the class of mathematical programs with complementarity constraints MPCC. Under the Linear Independence constraint qualification MPCC-LICQ we state a topological as well as an equivalent algebraic characterization for the strong stability (in the sense of Kojima) of an M-stationary point for MPCC. By allowing perturbations of the describing functions up to second order, the concept of strong stability refers here to the local existence and uniqueness of an M-stationary point for any sufficiently small perturbed problem where this unique solution depends continuously on the perturbation. Finally, some relations to S- and C-stationarity are briefly discussed.

scholarly journals Topological Approach to Mathematical Programs with Switching Constraints

2021 ◽  
Author(s):  
Vladimir Shikhman
Keyword(s):  
Stationary Point ◽  
Mountain Pass Theorem ◽  
Strong Stability ◽  
Complete Characterization ◽  
Stationary Points ◽  
Mathematical Programs ◽  
Linear Independence Constraint Qualification ◽  
Point Set ◽  
Stationary Level

AbstractWe study mathematical programs with switching constraints (for short, MPSC) from the topological perspective. Two basic theorems from Morse theory are proved. Outside the W-stationary point set, continuous deformation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension w equals the W-index of the nondegenerate W-stationary point. The W-index depends on both the number of negative eigenvalues of the restricted Lagrangian’s Hessian and the number of bi-active switching constraints. As a consequence, we show the mountain pass theorem for MPSC. Additionally, we address the question if the assumption on the nondegeneracy of W-stationary points is too restrictive in the context of MPSC. It turns out that all W-stationary points are generically nondegenerate. Besides, we examine the gap between nondegeneracy and strong stability of W-stationary points. A complete characterization of strong stability for W-stationary points by means of first and second order information of the MPSC defining functions under linear independence constraint qualification is provided. In particular, no bi-active Lagrange multipliers of a strongly stable W-stationary point can vanish.

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