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.

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 Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity Constraints

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Yu-xin Li ◽  
Jie Zhang ◽  
Zun-quan Xia
Keyword(s):  
Recent Literature ◽  
Mathematical Program ◽  
Uncertain Data ◽  
Almost Sure Convergence ◽  
Sample Average Approximation ◽  
Accumulation Point ◽  
Complementarity Constraints ◽  
Convergence Properties ◽  
Sample Average ◽  
Average Approximation

To reflect uncertain data in practical problems, stochastic versions of the mathematical program with complementarity constraints (MPCC) have drawn much attention in the recent literature. Our concern is the detailed analysis of convergence properties of a regularization sample average approximation (SAA) method for solving a stochastic mathematical program with complementarity constraints (SMPCC). The analysis of this regularization method is carried out in three steps: First, the almost sure convergence of optimal solutions of the regularized SAA problem to that of the true problem is established by the notion of epiconvergence in variational analysis. Second, under MPCC-MFCQ, which is weaker than MPCC-LICQ, we show that any accumulation point of Karash-Kuhn-Tucker points of the regularized SAA problem is almost surely a kind of stationary point of SMPCC as the sample size tends to infinity. Finally, some numerical results are reported to show the efficiency of the method proposed.

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.

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