×

zbMATH — the first resource for mathematics

On a relaxation method for mathematical programs with vanishing constraints. (English) Zbl 1256.49032
Summary: The paper considers a numerical approach for the solution of Mathematical Problems with Vanishing Constraints (MPVC). It is well known that direct numerical approaches for the treatment of MPVCs typically fail because standard constraint qualifications usually are not satisfied at a local minimizer. This parallels the situation in the related class of Mathematical Programs with Equilibrium Constraints (MPEC). For MPVC several concepts of stationarity have been proposed in the last few years generalizing the usual KKT-conditions. The paper considers a direct relaxation approach where the original problem is replaced by a perturbed problem. This approach has also been investigated by A. Izmailov and M. Solodov [J. Optim. Theory Appl. 142, No. 3, 501–532 (2009; Zbl 1180.90312)]. Under standard regularity conditions, we prove the convergence of KKT-points of the perturbed problem to suitable stationary points of the MPVC. Moreover, conditions are provided showing convergence to a strongly stationary point (i.e., a KKT-point) of MPVC. The paper closes with a numerical test of this approach for a problem arising in the field of topology optimization of mechanical structures with vanishing stress constraints.

MSC:
49M20 Numerical methods of relaxation type
65K10 Numerical optimization and variational techniques
90C31 Sensitivity, stability, parametric optimization
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C90 Applications of mathematical programming
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Izmailov, Journal of Optimization Theory and Applications 142 pp 501– (2009) · Zbl 1180.90312 · doi:10.1007/s10957-009-9517-4
[2] Achtziger, Computational Optimization and Applications 15 pp 69– (2000) · Zbl 0942.90036 · doi:10.1023/A:1008775015126
[3] Achtziger, Mathematical Programming 114 pp 69– (2008) · Zbl 1151.90046 · doi:10.1007/s10107-006-0083-3
[4] C. Kirches A. Potschka H. Bock S. Sager A parametric active set method for quadratic programs with vanishing constraints, Tech. rep., Optimization Online, 2011, (to appear in Pacific Journal of Optimization). · Zbl 1270.90042
[5] Z. Q. Luo J. S. Pang D. Ralph Mathematical Programs with Equilibrium Constraints (Cambridge University Press, Cambridge, New York, Melbourne, 1996).
[6] J. Outrata M. Kočvara J. Zowe Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Nonconvex Optimization and its Applications (Kluwer Academic Publishers, 1998).
[7] Hoheisel, Optimization 58 pp 431– (2009) · Zbl 1162.90560 · doi:10.1080/02331930701763405
[8] Hoheisel, Journal of Mathematical Analysis and Applications 337 pp 292– (2008) · Zbl 1141.90572 · doi:10.1016/j.jmaa.2007.03.087
[9] Hoheisel, Applications of Mathematics 52 pp 495– (2007) · Zbl 1164.90407 · doi:10.1007/s10492-007-0029-y
[10] Izmailov, Computational Mathematics and Mathematical Physics 49 pp 1128– (2009) · Zbl 1224.90165 · doi:10.1134/S0965542509070069
[11] Dorsch, Journal of Global Optimization 52 pp 591– (2012) · Zbl 1254.90224 · doi:10.1007/s10898-011-9805-z
[12] Hoheisel, Nonlinear Analysis: Theory, Methods, and Applications 72 pp 2514– (2010) · Zbl 1185.90187 · doi:10.1016/j.na.2009.10.047
[13] Hoheisel, Optimization Methods and Software 27 pp 483– (2012) · Zbl 1266.90170 · doi:10.1080/10556788.2010.535170
[14] Scholtes, SIAM Journal on Optimization 11 pp 918– (2001) · Zbl 1010.90086 · doi:10.1137/S1052623499361233
[15] Robinson, SIAM Journal on Numerical Analysis 13 pp 497– (1976) · Zbl 0347.90050 · doi:10.1137/0713043
[16] Robinson, Mathematics of Operations Research 5 pp 43– (1980) · Zbl 0437.90094 · doi:10.1287/moor.5.1.43
[17] M. Kojima Strongly stable stationary solutions in nonlinear programming, in: Analysis and Computations of Fixed Points, edited by S. Robinson (Academic Press Ney York, 1980), pp. 93-138. · doi:10.1016/B978-0-12-590240-3.50009-4
[18] S. Scholtes Introduction to piecewise smooth equations, 1994, Habilitation Thesis, University of Karlsruhe, Karlsruhe, Germany.
[19] M. Bendsøe O. Sigmund Topology Optimization - Theory, Methods and Applications (Springer-Verlag, Heidelberg, Berlin, Germany, 2003), 2nd ed.
[20] W. Achtziger T. Hoheisel C. Kanzow A smoothing-regularization approach to mathematical programs with vanishing constraints, Tech. rep., Institute of Mathematics, University of Würzburg, Würzburg, Germany, 2007, (submitted). · Zbl 1291.90234
[21] G. Allaire Shape Optimization by the Homogenization Method, Applied Mathematical Sciences (Springer, Heidelberg, Berlin, Germany, 2010).
[22] Wächter, Mathematical Programming 106 pp 25– (2006) · Zbl 1134.90542 · doi:10.1007/s10107-004-0559-y
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.