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A nonmonotone trust-region method for generalized Nash equilibrium and related problems with strong convergence properties. (English) Zbl 1390.90515
Summary: The generalized Nash equilibrium problem (GNEP) is often difficult to solve by Newton-type methods since the problem tends to have locally nonunique solutions. Here we take an existing trust-region method which is known to be locally fast convergent under a relatively mild error bound condition, and modify this method by a nonmonotone strategy in order to obtain a more reliable and efficient solver. The nonmonotone trust-region method inherits the nice local convergence properties of its monotone counterpart and is also shown to have the same global convergence properties. Numerical results indicate that the nonmonotone trust-region method is significantly better than the monotone version, and is at least competitive to an existing software applied to the same reformulation used within our trust-region framework. Additional tests on quasi-variational inequalities (QVI) are also presented to validate efficiency of the proposed extension.

MSC:
90C30 Nonlinear programming
65H10 Numerical computation of solutions to systems of equations
Software:
levmar; QVILIB; STRSCNE
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[1] Bellavia, S; Macconi, M; Morini, B, An affine scaling trust-region approach to bound-constrained nonlinear systems, Appl. Numer. Math., 44, 180-257, (2003) · Zbl 1018.65067
[2] Bellavia, S; Macconi, M; Morini, B, STRSCNE: a scaled trust region solver for constrained nonlinear equations, Comput. Optim. Appl., 28, 31-50, (2004) · Zbl 1056.90128
[3] Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999) · Zbl 1015.90077
[4] Calamai, PH; Moré, JJ, Projected gradient methods for linear constrained problems, Math. Program., 39, 93-116, (1987) · Zbl 0634.90064
[5] Dan, H; Yamashita, N; Fukushima, M, Convergence properties of the inexact Levenberg-Marquardt method under local error bound conditions, Optim. Methods Softw., 17, 605-626, (2002) · Zbl 1030.65049
[6] Dolan, ED; Moré, JJ, Benchmarking optimization software with performance profiles, Math. Program., 91, 201-213, (2002) · Zbl 1049.90004
[7] Dreves, A; Facchinei, F; Fischer, A; Herrich, M, A new error bound result for generalized Nash equilibrium problems and its algorithmic application, Comput. Optim. Appl., 59, 63-84, (2014) · Zbl 1307.91117
[8] Dreves, A; Facchinei, F; Kanzow, C; Sagratella, S, On the solution of the KKT conditions of generalized Nash equilibrium problems, SIAM J. Optim., 21, 1082-1108, (2011) · Zbl 1230.90176
[9] Facchinei, F; Kanzow, C; Sagratella, S, QVILIB: a library of quasi-variational inequality test problems, Pac. J. Optim., 9, 225-250, (2013) · Zbl 1267.65078
[10] Facchinei, F; Kanzow, C; Sagratella, S, Solving quasi-variational inequalities via their KKT conditions, Math. Program., 114, 369-412, (2014) · Zbl 1293.65100
[11] Facchinei, F; Kanzow, C, Generalized Nash equilibrium problems, Ann. Oper. Res., 175, 177-211, (2010) · Zbl 1185.91016
[12] Facchinei, F; Kanzow, C; Karl, S; Sagratella, S, The semismooth Newton method for the solution of quasi-variational inequalities, Comput. Optim. Appl., 62, 85-109, (2015) · Zbl 1331.90083
[13] Fan, J; Yuan, Y, On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption, Computing, 74, 23-39, (2005) · Zbl 1076.65047
[14] Fischer, A; Herrich, M; Schönefeld, K, Generalized Nash equilibrium problems—recent advances and challenges, Pesquisa Operacional, 34, 521-558, (2014)
[15] Fischer, A; Shukla, PK; Wang, M, On the inexactness level of robust Levenberg-Marquardt methods, Optimization, 59, 273-287, (2010) · Zbl 1196.65097
[16] Herrich, M.: Local convergence of Newton-type methods for nonsmooth constrained equations and applications. Ph.D. Thesis, Institute of Mathematics, Technical University of Dresden, Germany (2014) · Zbl 1030.65049
[17] Izmailov, AF; Solodov, MV, On error bounds and Newton-type methods for generalized Nash equilibrium problems, Comput. Optim. Appl., 59, 201-218, (2014) · Zbl 1307.91118
[18] Kanzow, C; Steck, D, Augmented Lagrangian methods for the solution of generalized Nash equilibrium problems, SIAM J. Optim., 26, 2034-2058, (2016) · Zbl 1351.65037
[19] Kanzow, C; Yamashita, N; Fukushima, M, Levenberg-Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints, J. Comput. Appl. Math., 172, 375-397, (2004) · Zbl 1064.65037
[20] Qi, L; Tong, XJ; Li, DH, Active-set projected trust-region algorithm for box-constrained nonsmooth equations, J. Optim. Theory Appl., 120, 601-625, (2004) · Zbl 1140.65331
[21] Toint, PL, Non-monotone trust-region algorithms for nonlinear optimization subject to convex constraints, Math. Program., 77, 69-94, (1997) · Zbl 0891.90153
[22] Tong, XJ; Qi, L, On the convergence of a trust-region method for solving constrained nonlinear equations with degenerate solutions, J. Optim. Theory Appl., 123, 187-211, (2004) · Zbl 1069.65055
[23] Yamashita, N; Fukushima, M, On the rate of convergence of the Levenberg-Marquardt method, Computing, 15, 239-249, (2001) · Zbl 1001.65047
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