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A descent algorithm for generalized complementarity problems based on generalized Fischer-Burmeister functions. (English) Zbl 1393.90122
Summary: We study an unconstrained minimization approach to the generalized complementarity problem $$\mathrm{GCP}(f,g)$$ based on the generalized Fischer-Burmeister function and its generalizations when the underlying functions are $$C^1$$. Also, we show how, under appropriate regularity conditions, minimizing the merit function corresponding to $$f$$ and $$g$$ leads to a solution of the generalized complementarity problem. Moreover, we propose a descent algorithm for $$\mathrm{GCP}(f,g)$$ and show a result on the global convergence of a descent algorithm for solving generalized complementarity problem. Finally, we present some preliminary numerical results. Our results further give a unified/generalization treatment of such results for the nonlinear complementarity problem based on generalized Fischer-Burmeister function and its generalizations.
##### MSC:
 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 90C56 Derivative-free methods and methods using generalized derivatives
MCPLIB
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##### References:
 [1] Aashtiani, H; Magnanti, T, Equilibrium on a congested transportation network, SIAM J Algebr Discrete Methods, 42, 213-226, (1981) · Zbl 0501.90033 [2] Agdeppa, RP; Yamashita, N; Fukushima, M, The traffic equilibrium problem with nonadditive costs and its monotone mixed complementarity problem formulation, Transp Res Part B, 41, 862874, (2007) [3] Andreani, R; Friedlander, A; Santos, SA, On the resolution of the generalized nonlinear complementarity problem, SIAM J Optim, 12, 303-321, (2002) · Zbl 1006.65068 [4] Chen J-S, Huang Z-H, She C-Y (2011) A new class of penalized NCP-functions and its properties. Comput Optim Appl 50:49-73 · Zbl 1254.90253 [5] Chen, M; Bernstein, DH; Chien, SIJ; Mouskos, K, Simplified formulation of the toll design problem, Transp Res Rec, 1667, 8895, (1999) [6] Chen, J-S, The semismooth-related properties of a merit function and a descent method for the nonlinear complementarity problem, J Glob Optim, 36, 565580, (2006) · Zbl 1144.90493 [7] Chen, Jein-shan, On some NCP-functions based on the generalized fischer-burmeister function, Asia Pac J Oper Res, 24, 401-420, (2007) · Zbl 1141.90557 [8] Chen, J-S; Gao, H-T; Pan, S-H, An R-linearly convergent derivative-free algorithm for nonlinear complementarity problems based on the generalized fischer-burmeister merit function, J Comput Appl Math, 232, 455-471, (2009) · Zbl 1175.65070 [9] Chen, J-S; Pan, S-H, A family of NCP functions and a descent method for the nonlinear complementarity problems, Comput Optim Appl, 40, 389-404, (2008) · Zbl 1153.90542 [10] Company R, Egorova VN, Jdar L (2014) Solving American option pricing models by the front fixing method: numerical analysis and computing. Abstr Appl Anal, Article ID 146745 [11] Cottle RW, Giannessi F, Lions J-L (eds) (1980) Variational inequalities and complementarity problems: theory and applications. Wiley, New York · Zbl 0484.90088 [12] Cottle RW, Pang J-S, Stone RE (1992) The linear complementarity problem. Academic Press, Boston · Zbl 0757.90078 [13] Dirkse, SP; Ferris, M, MCPLIB: a collection of nonlinear mixed complementarity problems, Optim Methods Softw, 5, 319-345, (1994) [14] Di Pillo G, Giannessi F (eds) (1996) Nonlinear optimization and applications. Plenum Press, New York · Zbl 0941.00047 [15] Facchinei, F; Kanzow, C, On unconstrained and constrained stationary points of the implicit Lagrangian, J Optim Theory Appl, 92, 99-115, (1997) · Zbl 0914.90249 [16] Facchinei F, Pang J-S (2003) Finite dimensional variational inequalities and complementarity problems. Springer-Verlag, New York · Zbl 1062.90002 [17] Facchinei, F; Soares, J, A new merit function for nonlinear complementarity problems and related algorithm, SIAM J Optim, 7, 225-247, (1997) · Zbl 0873.90096 [18] Feng L, Linetsky V, Morales JL, Nocedal J (2011) On the solution of complementarity problems arising in American options pricing. Optim Methods Softw 26(4-5):813-825 · Zbl 1229.90230 [19] Ferris MC, Pang J-S (1997b) Engineering and economic applications of complementarity problems. SIAM Rev 39:669-713 · Zbl 0891.90158 [20] Ferris MC, Pang J-S (eds) (1997a) Complementarity and variational problems: state of the art. SIAM, Philadelphia · Zbl 0828.90127 [21] Ferris MC, Ralph D (1995) Projected gradient methods for nonlinear complementarity problems via normal maps. In: Du DZ, Qi L, Womersley RS (eds) Recent advances in nonsmooth optimization. World Scientific Publishers, Singapore, pp 57-87 · Zbl 0946.90090 [22] Fischer, A, Solution of monotone complementarity problems with locally Lipschitzian functions, Math Program, 76, 513-532, (1997) · Zbl 0871.90097 [23] Fischer, A, A new constrained optimization reformulation for complementarity problems, J Optim Theory Appl, 97, 105-117, (1998) · Zbl 0907.90260 [24] Gabriel, SA; Bernstein, D, The traffic equilibrium problem with nonadditive path costs, Transp Sci, 31, 337-348, (1997) · Zbl 0920.90058 [25] Galántai, A, Properties and construction of NCP functions, Comput Optim Appl, 52, 805-824, (2012) · Zbl 1282.90194 [26] Geiger, C; Kanzow, C, On the resolution of monotone complementarity problems, Comput Optim, 5, 155-173, (1996) · Zbl 0859.90113 [27] Gu W-Z, Tawhid MA (2014) Generalized complementarity problems based on generalized Fischer-Burmeister functions as unconstrained optimization. Adv Model Optim 16(2):269-284 · Zbl 1413.90285 [28] Harker, PT; Pang, J-S, Finite dimension variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math Program, 48, 161-220, (1990) · Zbl 0734.90098 [29] Hu S-L, Huang Z-H, Chen J-S (2009) Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems. J Comput Appl Math 230:68-82 · Zbl 1172.65029 [30] Huang J, Pang J-S (1998) Option pricing and linear complementarity. J Comput Finance 2:31-60 · Zbl 0813.90117 [31] Hyer DH, Isac G, Rassias TM (1997) Topics in nonlinear analysis and applications. World scientific Publishing Company, Singapore [32] Isac G (1992) Complementarity problems, Lecture Notes in Mathematics, vol 1528. Springer Verlag, Berlin · Zbl 0795.90072 [33] Jiang, H, Unconstrained minimization approaches to nonlinear complementarity problems, J Glob Optim, 9, 169-181, (1996) · Zbl 0868.90122 [34] Jiang, H; Fukushima, M; Qi, L; Sun, D, A trust region method for solving generalized complementarity problems, SIAM J Optim, 8, 140-157, (1998) · Zbl 0911.90324 [35] Kanzow, C, Nonlinear complementarity as unconstrained optimization, J Optim Theory Appl, 88, 139-155, (1996) · Zbl 0845.90120 [36] Lo, HK; Chen, A, Traffic equilibrium problem with route-specific costs: formulation and algorithms, Transp Res Part B, 34, 493-513, (2000) [37] Luca, TD; Facchinei, F; Kanzow, C, A semismooth equation approach to the solution of nonlinear complementarity problems, Math Program, 75, 407-439, (1996) · Zbl 0874.90185 [38] Mangasarian, OL; Solodov, MV, Nonlinear complementarity as unconstrained and constrained minimization, Math Program, 62, 277-297, (1993) · Zbl 0813.90117 [39] McKean, HP, Appendix: a free boundary problem for the heat equation arising from a problem in mathematical economics, Ind Manag Rev, 6, 3239, (1965) [40] Noor MA (1993) General nonlinear complementarity problems. In: Srivastava HM, Rassias TM (eds) Analysis, geometry, and groups: a riemann legacy volume. Hadronic Press, Palm Harbor, pp 337-371 [41] Noor, MA, Quasi-complementarity problem, J Math Anal Appl, 130, 344-353, (1988) · Zbl 0645.90086 [42] Ortega JM, Rheinboldt WC (1970) Iterative solution of nonlinear equations in several variables. Academic Press, New York, San Francisco, London · Zbl 0241.65046 [43] Outrata, JV; Zowe, J, A Newton method for a class of quasi-variational inequalities, Comput Optim Appl, 4, 5-21, (1995) · Zbl 0827.49007 [44] Pang JS (1981) The implicit complementarity problem. In: Mangasarian OL, Meyer RR, Robinson SM (eds) Nonlinear programming. Academic Press, New York, pp 487-518 · Zbl 1006.65068 [45] Patriksson M (1994) The traffic assignment problem: models and methods. VSP, Utrecht · Zbl 0828.90127 [46] Patriksson, M, Algorithms for computing traffic equilibria, Netw Spatial Econ, 4, 2338, (2004) · Zbl 1079.90141 [47] Sheffi Y (1985) Urban transportation networks: equilibrium analysis with mathematical programming methods. Prentice Hall, New Jersey [48] Tawhid, MA, An application of $$H$$-differentiability to nonnegative and unrestricted generalized complementarity, Comput Optim Appl, 39, 51-74, (2008) · Zbl 1147.90405 [49] Tseng, P, Growth behavior of a class of merit functions for the nonlinear complementarity problem, J Optim Theory Appl, 89, 17-37, (1996) · Zbl 0866.90127 [50] Moerbeke, P, On optimal stopping and free boundary problems, Arch Ration Mech Anal, 60, 101148, (1976) · Zbl 0336.35047 [51] Wilmott P, Howison S, Dewynne J (1995) The mathematics of financial derivatives. Cambridge University Press, Cambridge · Zbl 0842.90008 [52] Xu M, Gao ZY (2011) A complementary formulation for traffic equilibrium problem with a new nonadditive route cost. Sci China Technol Sci 54(9):2525-2530 · Zbl 1237.90059 [53] Yamada K, Yamashita N, Fukushima M (2000) A new derivative-free descent method for the nonlinear complementarity problems. In: Pillo GD, Giannessi F (eds) Nonlinear optimization and related topics. Kluwer Academic Publishers, Netherlands, pp 463-487 · Zbl 0996.90085 [54] Yamashita, N; Fukushima, M, On stationary points of the implicit Lagrangian for nonlinear complementarity problems, J Optim Theory Appl, 84, 653-663, (1995) · Zbl 0824.90131
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