A Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity Constraints
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...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2014/321010 |
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| author | Yu-xin Li Jie Zhang Zun-quan Xia |
| author_facet | Yu-xin Li Jie Zhang Zun-quan Xia |
| author_sort | Yu-xin Li |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-0de80fffdb7d49bea3b83a72e2d768c8 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-0de80fffdb7d49bea3b83a72e2d768c82025-02-03T06:12:03ZengWileyJournal of Applied Mathematics1110-757X1687-00422014-01-01201410.1155/2014/321010321010A Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity ConstraintsYu-xin Li0Jie Zhang1Zun-quan Xia2Institute of ORCT, School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, ChinaSchool of Mathematics, Liaoning Normal University, Dalian 116029, ChinaInstitute of ORCT, School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, ChinaTo 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.http://dx.doi.org/10.1155/2014/321010 |
| spellingShingle | Yu-xin Li Jie Zhang Zun-quan Xia A Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity Constraints Journal of Applied Mathematics |
| title | A Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity Constraints |
| title_full | A Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity Constraints |
| title_fullStr | A Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity Constraints |
| title_full_unstemmed | A Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity Constraints |
| title_short | A Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity Constraints |
| title_sort | regularization saa scheme for a stochastic mathematical program with complementarity constraints |
| url | http://dx.doi.org/10.1155/2014/321010 |
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