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Title: An SQP method for mathematical programs with complementarity constraints with strong convergence properties (English)
Author: Benko, Matus
Author: Gfrerer, Helmut
Language: English
Journal: Kybernetika
ISSN: 0023-5954 (print)
ISSN: 1805-949X (online)
Volume: 52
Issue: 2
Year: 2016
Pages: 169-208
Summary lang: English
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Category: math
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Summary: We propose an SQP algorithm for mathematical programs with complementarity constraints which solves at each iteration a quadratic program with linear complementarity constraints. We demonstrate how strongly M-stationary solutions of this quadratic program can be obtained by an active set method without using enumeration techniques. We show that all limit points of the sequence of iterates generated by our SQP method are at least M-stationary. (English)
Keyword: SQP method
Keyword: active set method
Keyword: mathematical program with complementarity constraints
Keyword: strong M-stationarity
MSC: 49M37
MSC: 90C26
MSC: 90C33
MSC: 90C55
idZBL: Zbl 1357.49124
idMR: MR3501157
DOI: 10.14736/kyb-2016-2-0169
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Date available: 2016-07-17T11:59:41Z
Last updated: 2018-01-10
Stable URL: http://hdl.handle.net/10338.dmlcz/145769
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Reference: [1] Anitescu, M.: Global convergence of an elastic mode approach for a class of mathematical programs with complementarity constraints..SIAM J. Optim. 16 (2005), 120-145. Zbl 1099.65050, MR 2177772, 10.1137/040606855
Reference: [2] Anitescu, M.: On using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints..SIAM J. Optim. 15 (2005), 1203-1236. Zbl 1097.90050, MR 2178496, 10.1137/s1052623402401221
Reference: [3] Biegler, L. T., Raghunathan, A. U.: An interior point method for mathematical programs with complementarity constraints (MPCCs)..SIAM J. Optim. 15 (2005), 720-750. Zbl 1077.90079, MR 2142858, 10.1137/s1052623403429081
Reference: [4] DeMiguel, V., Friedlander, M. P., Nogales, F. J., Scholtes, S.: A two-sided relaxation scheme for mathematical programs with equilibrium constraints..SIAM J. Optim. 16 (2005), 587-609. Zbl 1122.90060, MR 2197997, 10.1137/04060754x
Reference: [5] Facchinei, F., Jiang, H., Qi, L.: A smoothing method for mathematical programs with equilibrium constraints..Math. Programming 85 (1999), 107-134. Zbl 0959.65079, MR 1689366, 10.1007/s101070050048
Reference: [6] Fletcher, R.: Practical Methods of Optimization 2: Constrained Optimization..John Wiley and Sons, Chichester 1981. MR 0633058
Reference: [7] Fletcher, R., Leyffer, S.: Solving mathematical programs with complementary constraints as nonlinear programs.. Zbl 1074.90044
Reference: [8] Fletcher, R., Leyffer, S., Ralph, D., Scholtes, S.: Local convergence of SQP methods for mathematical programs with equilibrium constraints..SIAM J. Optim. 17 (2006), 259-286. Zbl 1112.90098, MR 2219153, 10.1137/s1052623402407382
Reference: [9] Fukushima, M., Tseng, P.: An implementable active-set algorithm for computing a b-stationary point of a mathematical program with linear complementarity constraints..SIAM J. Optim. 12 (2002), 724-739. Zbl 1127.65034, MR 1884914, 10.1137/s1052623499363232
Reference: [10] Gfrerer, H.: Optimality conditions for disjunctive programs based on generalized differentiation with application to mathematical programs with equilibrium constraints..SIAM J. Optim. 24 (2014), 898-931. Zbl 1298.49021, MR 3217222, 10.1137/130914449
Reference: [11] Giallombardo, G., Ralph, D.: Multiplier convergence in trust region methods with application to convergence of decomposition methods for MPECs..Math. Programming 112 (2008), 335-369. Zbl 1145.90073, MR 2361928, 10.1007/s10107-006-0020-5
Reference: [12] Hu, X. M., Ralph, D.: Convergence of a penalty method for mathematical programming with complementarity constraints..J. Optim. Theory Appl. 123 (2004), 365-390. MR 2101411, 10.1007/s10957-004-5154-0
Reference: [13] Izmailov, A. F., Pogosyan, A. L., Solodov, M. V.: Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints..Computational Optim. Appl. 51 (2012), 199-221. Zbl 1245.90124, MR 2872496, 10.1007/s10589-010-9341-7
Reference: [14] Jiang, H., Ralph, D.: QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints..Comput. Optim. Appl. 13 (1999), 25-59. MR 1704113, 10.1023/A:1008696504163
Reference: [15] Jiang, H., Ralph, D.: Extension of quasi-newton methods to mathematical programs with complementarity constraints..Comput. Optim. Appl. 25 (2002), 123-150. Zbl 1038.90100, MR 1996665, 10.1023/A:1022945316191
Reference: [16] Kadrani, A., Dussault, J. P., Benchakroun, A.: A new regularization scheme for mathematical programs with complementarity constraints..SIAM J. Optim. 20 (2009), 78-103. Zbl 1187.65064, MR 2496894, 10.1137/070705490
Reference: [17] Kanzow, C., Schwartz, A.: A new regularization method for mathematical programs with complementarity constraints with strong convergence properties..SIAM J. Optim. 23 (2013), 770-798. Zbl 1282.65069, MR 3045664, 10.1137/100802487
Reference: [18] Kanzow, C., Schwartz, A.: The price of inexactness: convergence properties of relaxation methods for mathematical programs with equilibrium constraints revisited..Math. Oper. Res. 40 (2015), 2, 253-275. MR 3320430, 10.1287/moor.2014.0667
Reference: [19] Leyffer, S.: MacMPEC: AMPL collection of MPECs, 2000..
Reference: [20] Leyffer, S., López-Calva, G., Nocedal, J.: Interior methods for mathematical programs with complementarity constraints..SIAM J. Optim. 17 (2006), 52-77. Zbl 1112.90095, MR 2219144, 10.1137/040621065
Reference: [21] Lin, G. H., Fukushima, M.: A modified relaxation scheme for mathematical programs with complementarity constraints..Ann. Oper. Res. 133 (2005), 63-84. Zbl 1119.90058, MR 2119313, 10.1007/s10479-004-5024-z
Reference: [22] Luo, Z. Q., Pang, J. S., Ralph, D.: Mathematical Programs with Equilibrium Constraints..Cambridge University Press, Cambridge, New York, Melbourne 1996. Zbl 1139.90003, MR 1419501, 10.1017/cbo9780511983658
Reference: [23] Luo, Z. Q., Pang, J. S., Ralph, D.: Piecewise sequential quadratic programming for mathematical programs with nonlinear complementarity constraints..In: Multilevel Optimization: Algorithms, Complexity, and Applications (A. Migdalas, P. Pardalos, and P. Värbrand, eds.), Kluwer Academic Publishers, Dordrecht 1998, pp. 209-229. Zbl 0897.90184, MR 1605239, 10.1007/978-1-4613-0307-7_9
Reference: [24] Luo, Z. Q., Pang, J. S., Ralph, D., Wu, S. Q.: Exact penalization and stationarity conditions of mathematical programs with equilibrium constraints..Math. Programming 75 (1996), 19-76. Zbl 0870.90092, MR 1415093, 10.1007/bf02592205
Reference: [25] Outrata, J. V., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Nonconvex Optimization and its Applications..Kluwer Academic Publishers, Dordrecht 1998. MR 1641213, 10.1007/978-1-4757-2825-5
Reference: [26] Powell, M. J. D.: A fast algorithm for nonlinearly constrained optimization calculations..In: Numerical Analysis Dundee 1977 (G. A. Watson, ed.), Lecture Notes in Mathematics 630, Springer, Berlin, 1978, pp. 144-157. Zbl 0374.65032, MR 0483447, 10.1007/bfb0067703
Reference: [27] Ralph, D., Wright, S. J.: Some properties of regularization and penalization schemes for MPECs..Optim. Methods Software 19 (2004), 527-556. Zbl 1097.90054, MR 2095351, 10.1080/10556780410001709439
Reference: [28] Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity..Math. Oper. Res. 25 (2000), 1-22. Zbl 1073.90557, MR 1854317, 10.1287/moor.25.1.1.15213
Reference: [29] Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints..SIAM J. Optim. 11 (2001), 918-936. Zbl 1010.90086, MR 1855214, 10.1137/s1052623499361233
Reference: [30] Scholtes, S., Stöhr, M.: Exact penalization of mathematical programs with equilibrium constraints..SIAM J. Control Optim. 37 (1999), 617-652. Zbl 0922.90128, MR 1670641, 10.1137/s0363012996306121
Reference: [31] Steffensen, S., Ulbrich, M.: A new regularization scheme for mathematical programs with equilibrium constraints..SIAM J. Optim. 20 (2010), 2504-2539. MR 2678403, 10.1137/090748883
Reference: [32] Stein, O.: Lifting mathematical programs with complementarity constraints..Math. Programming 131 (2012), 71-94. Zbl 1250.90094, MR 2886141, 10.1007/s10107-010-0345-y
Reference: [33] Stöhr, M.: Nonsmooth Trust Region Methods and their Applications to Mathematical Programs with Equilibrium Constraints..Shaker-Verlag, Aachen 1999.
Reference: [34] Zhang, J., Liu, G.: A new extreme point algorithm and its application in psqp algorithms for solving mathematical programs with linear complementarity constraints..J. Glob. Optim. 19 (2001), 335-361. Zbl 1049.90125, MR 1824769, 10.1023/A:1011226232107
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