×
Hu, Qingjie; Chen, Yu; Zhu, Zhibin; Zhang, Bishan

Notes on convergence properties for a smoothing-regularization approach to mathematical programs with vanishing constraints. (English) Zbl 1474.90460

Abstr. Appl. Anal. 2014, Article ID 715015, 7 p. (2014).
Summary: We give some improved convergence results about the smoothing-regularization approach to mathematical programs with vanishing constraints (MPVC for short), which is proposed in [W. Achtziger et al., Comput. Optim. Appl. 55, No. 3, 733–767 (2013; Zbl 1291.90234)]. 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.

Cited in 1 Document

MSC:

90C31 Sensitivity, stability, parametric optimization
90C30 Nonlinear programming

Citations:

Zbl 1291.90234
PDF BibTeX XML Cite
\textit{Q. Hu} et al., Abstr. Appl. Anal. 2014, Article ID 715015, 7 p. (2014; Zbl 1474.90460)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[1] Achtziger, W.; Kanzow, C., Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications, Mathematical Programming, 114, 1, 69-99, (2008) · Zbl 1151.90046
[2] Kirches, C.; Potschka, A.; Bock, H. G.; Sager, S., A parametric active set method for quadratic programs with vanishing constraints, Technical Report, (2012), Heidelberg, Germany: Interdisciplinary Center for Scientic Computing, University of Heidelberg, Heidelberg, Germany
[3] Achtziger, W.; Hoheisel, T.; Kanzow, C., A smoothing-regularization approach to mathematical programs with vanishing constraints, Computational Optimization and Applications, 55, 3, 733-767, (2013) · Zbl 1291.90234
[4] Achtziger, W.; Kanzow, C.; Hoheisel, T., On a relaxation method for mathematical programs with vanishing constraints, GAMM-Mitteilungen, 35, 2, 110-130, (2012) · Zbl 1256.49032
[5] Dorsch, D.; Shikhman, V.; Stein, O., Mathematical programs with vanishing constraints: critical point theory, Journal of Global Optimization, 52, 3, 591-605, (2012) · Zbl 1254.90224
[6] Hoheisel, T.; Kanzow, C., On the Abadie and Guignard constraint qualifications for mathematical programmes with vanishing constraints, Optimization, 58, 4, 431-448, (2009) · Zbl 1162.90560
[7] Hoheisel, T.; Kanzow, C., Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications, Journal of Mathematical Analysis and Applications, 337, 1, 292-310, (2008) · Zbl 1141.90572
[8] Hoheisel, T.; Kanzow, C., First- and second-order optimality conditions for mathematical programs with vanishing constraints, Applications of Mathematics, 52, 6, 495-514, (2007) · Zbl 1164.90407
[9] Hoheisel, T.; Kanzow, C.; Outrata, J. V., Exact penalty results for mathematical programs with vanishing constraints, Nonlinear Analysis: Theory, Methods & Applications, 72, 5, 2514-2526, (2010) · Zbl 1185.90187
[10] Hoheisel, T.; Kanzow, C.; Schwartz, A., Convergence of a local regularization approach for mathematical programmes with complementarity or vanishing constraints, Optimization Methods & Software, 27, 3, 483-512, (2012) · Zbl 1266.90170
[11] Izmailov, A. F.; Pogosyan, A. L., Optimality conditions and Newton-type methods for mathematical programs with vanishing constraints, Computational Mathematics and Mathematical Physics, 49, 7, 1128-1140, (2009) · Zbl 1224.90165
[12] Izmailov, A. F.; Solodov, M. V., Mathematical programs with vanishing constraints: optimality conditions, sensitivity, and a relaxation method, Journal of Optimization Theory and Applications, 142, 3, 501-532, (2009) · Zbl 1180.90312
[13] Davis, C., Theory of positive linear dependence, The American Journal of Mathematics, 76, 4, 733-746, (1954) · Zbl 0058.25201
[14] Mangasarian, O. L., Nonliear Programming, (1969), New York, NY, USA: McGraw-Hill, New York, NY, USA
[15] Qi, L.; Wei, Z., On the constant positive linear dependence condition and its application to SQP methods, SIAM Journal on Optimization, 10, 4, 963-981, (2000) · Zbl 0999.90037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.