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Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities. (English) Zbl 0920.90134
Summary: The D-gap function, recently introduced by J.-M. Peng, allows a smooth unconstrained minimization reformulation of the general variational inequality problem. This paper is concerned with the D-gap function for variational inequality problems over a box or, equivalently, mixed complementarity problems. The purpose of this paper is twofold. First we investigate theoretical properties in depth of the D-gap function, such as the optimality of stationary points, bounded level sets, global error bounds and generalized Hessians. Next, we present a nonsmooth Gauss-Newton type algorithm for minimizing the D-gap function, and report extensive numerical results for the whole set of problems in the MCPLIB test problem collection.

MSC:
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
49J40 Variational inequalities
Software:
MCPLIB
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[1] P.T. Harker, J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathematical Programming 48 (1990) 161–220. · Zbl 0734.90098
[2] M.C. Ferris, J.-S. Pang, Engineering and economic applications of complementarity problems, SIAM Review (to appear). · Zbl 0891.90158
[3] S.P. Dirkse, M.C. Ferris, MCPLIB: A collection of nonlinear mixed complementarity problems, Optimisation: Methods and Software 5 (1995) 319–345.
[4] S.C. Billups, S.P. Dirkse, M.C. Ferris, A comparison of algorithms for large scale mixed complementarity problems, Computational Optimization and Applications 7 (1997) 3–25. · Zbl 0883.90116
[5] C. Chen, O.L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optimization and Applications 5 (1996) 97–138. · Zbl 0859.90112
[6] F. Facchinei, A. Fischer, C. Kanzow, A semismooth Newton method for variational inequalities: The case of box constraints, in: M.C. Ferris, J.-S. Pang (Eds.), Complementarity and Variational Problems: State of the Art, SIAM, Philadelphia, PA, 1997, pp. 76–90. · Zbl 0886.90152
[7] S.A. Gabriel, J.J. Moré, Smoothing of mixed complementarity problems, in: M.C. Ferris, J.-S. Pang (Eds.), Complementarity and Variational Problems: State of the Art, SIAM, Philadelphia, PA, 1997, pp. 105–116. · Zbl 0886.90154
[8] S.P. Dirkse, M.C. Ferris, P.V. Preckel, T. Rutherford, The GAMS callable program library for variational and complementarity solvers Technical Report 94-07, Computer Sciences Department, University of Wisconsin, Madison, WI, 1994.
[9] M. Fukushima, Merit functions for variational inequality and complementarity problems, in: G. Di Pillo, F. Giannessi (Eds.), Nonlinear Optimization and Applications, Plenum Press, New York, 1996, pp. 155–170. · Zbl 0996.90082
[10] J.-M. Peng, Equivalence of variational inequality problems to unconstrained minimization, Mathematical Programming (to appear). · Zbl 0887.90171
[11] N. Yamashita, K. Taji, M. Fukushima, Unconstrained optimization reformulations of variational inequality problems, Journal of Optimization Theory and Applications 92 (1997) 439–456. · Zbl 0879.90180
[12] M. Fukushima, J.-S. Pang. Minimizing and stationary sequences of merit functions for complementarity problems and variational inequalities, in: M.C. Ferris, J.-S. Pang (Eds.), Complementarity and Variational Problems: State of the Art, SIAM, Philadelphia, PA, 1997, pp. 91–104. · Zbl 0886.90153
[13] D. Sun, M. Fukushima, L. Qi, A computable generalized Hessian of the D-gap function and Newton-type methods for variational inequality problems in: M.C. Ferris, J.-S. Pang (Eds.), Complementarity and Variational Problems: State of the Art, SIAM, Philadelphia, PA, 1997, pp. 452–473. · Zbl 0886.90165
[14] A. Auslender, Optimisation: Méthodes Numériques, Masson, Paris, 1976.
[15] P. Marcotte, J.-P. Dussault, A note on a globally convergent Newton method for solving monotone variational inequalities. Operations Research Letters 6 (1987) 35–42. · Zbl 0623.65073
[16] G. Auchmuty, Variational principles for variational inequalities, Numerical Functional Analysis and Optimization 10 (1989) 863–874. · Zbl 0678.49010
[17] M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Mathematical Programming 53 (1992) 99–110. · Zbl 0756.90081
[18] J.-M. Peng, Y.-X. Yuan, Unconstrained methods for generalized complementarity problems, Technical Report, Institute of Computational Mathematics and Scientific Computing, Academia Sinica, Beijing, China, 1994. · Zbl 1006.65069
[19] O. L. Mangasarian, M.V. Solodov, Nonlinear complementarity as unconstrained and constrained minimization, Mathematical Programming 62 (1993) 277–297. · Zbl 0813.90117
[20] N. Yamashita, M. Fukushima, On stationary points of the implicit Lagrangian for nonlinear complementarity problems, Journal of Optimization Theory and Applications 84 (1995) 653–663. · Zbl 0824.90131
[21] Z.-Q. Luo, O.L. Mangasarian, J. Ren, M.V. Solodov, New error bounds for the linear complementarity problem, Mathematics of Operations Research 19 (1994) 880–892. · Zbl 0833.90113
[22] C. Kanzow, Nonlinear complementarity as unconstrained optimization, Journal of Optimization Theory and Applications 88 (1996) 139–155. · Zbl 0845.90120
[23] H. Jiang, Unconstrained minimization approaches to nonlinear complementarity problems, Journal of Global Optimization 9 (1996) 169–181. · Zbl 0868.90122
[24] F. Facchinei, C. Kanzow, On unconstrained and constrained stationary points of the implicit Lagrangian, Journal of Optimization Theory and Applications 92 (1997) 99–115. · Zbl 0914.90249
[25] J.-M. Peng, The convexity of the implicit Lagrangian, Journal of Optimization Theory and Applications 92 (1997) 331–341. · Zbl 0886.90147
[26] N. Yamashita, M. Fukushima, Equivalent unconstrained minimization and global error bounds for variational inequality problems, SIAM Journal on Control and Optimization 35 (1997) 273–284. · Zbl 0873.49006
[27] R. Andreani, A. Friedlander, J.M. Martínez, On the solution of finite-dimensional variational inequalities using smooth optimization with simple bounds, Technical Report, Department of Applied Mathematics, University of Campinas, Campinas, Brazil, September 1995.
[28] F. Facchinei, A. Fischer, C. Kanzow, Inexact Newton methods for semismooth equations with applications to variational inequality problems, in: G. Di Pillo, F. Giannessi (Eds.), Nonlinear Optimization and Applications, Plenum Press, New York, 1996, pp. 125–139. · Zbl 0980.90101
[29] F. Facchinei, A. Fischer, C. Kanzow, Regularity properties of a semismooth reformulation of variational inequalities. SIAM Journal of Optimization (to appear). · Zbl 0913.90249
[30] J.-S. Pang, Newton’s method for B-differentiable equations, Mathematics of Operations Research 15 (1990) 311–341. · Zbl 0716.90090
[31] J.-S. Pang, A B-differentiable equation-based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems, Mathematical Programming 51 (1991) 101–131. · Zbl 0733.90063
[32] B. Xiao, P.T. Harker, A nonsmooth Newton method for variational inequalities, I: Theory, Mathematical Programming 65 (1994) 151–194. · Zbl 0812.65048
[33] B. Xiao, P.T. Harker, A nonsmooth Newton method for variational inequalities, II: Numerical results, Mathematical Programming 65 (1994) 195–216. · Zbl 0812.65049
[34] D.P. Bertsekas, J.N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ, 1989.
[35] L. Qi, J. Sun, A nonsmooth version of Newton’s method, Mathematical Programming 58 (1993) 353–367. · Zbl 0780.90090
[36] L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research 18 (1993) 227–244. · Zbl 0776.65037
[37] H. Jiang, L. Qi, X. Chen, D. Sun, Semismoothness and superlinear convergence in nonsmooth optimization and nonsmooth equations, in: G. Di Pillo, F. Giannessi (Eds.), Nonlinear Optimization and Applications, Plenum Press, New York, 1996, pp. 197–212. · Zbl 0991.90123
[38] F. Facchinei, Minimization ofSC 1 functions and the Maratos effect, Operations Research Letters 17 (1995) 131–137. · Zbl 0843.90108
[39] C. Geiger, C. Kanzow, On the resolution of monotone complementarity problems, Computational Optimization Applications 5 (1996) 155–173. · Zbl 0859.90113
[40] F. Facchinei, J. Soares, A new merit function for nonlinear complementarity problems and a related algorithm, SIAM Journal of Optimization 7 (1997) 225–247. · Zbl 0873.90096
[41] A. Fischer, A special Newton-type optimization method, Optimization 24 (1992) 269–284. · Zbl 0814.65063
[42] C. Kanzow, N. Yamashita, M. Fukushima, New NCP-functions and their properties, Journal of Optimization Theory and Applications (to appear). · Zbl 0886.90146
[43] J.H. Wu, M. Florian, P. Marcotte, A general descent framework for the monotone variational inequality problem, Mathematical Programming 61 (1993) 281–300. · Zbl 0813.90111
[44] J.E. Dennis, Jr., R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, NJ, 1983.
[45] H. Jiang, L. Qi, Local uniqueness and iterative methods for nonsmooth variational inequalities, Journal of Mathematical Analysis and Applications 196 (1995) 314–331. · Zbl 0845.65028
[46] J.-S. Pang, L. Qi, Nonsmooth equations: Motivation and algorithms, SIAM Journal of Optimization 3 (1993) 443–465. · Zbl 0784.90082
[47] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983 (reprint by SIAM, Philadelphia, PA, 1990). · Zbl 0582.49001
[48] T. De Luca, F. Facchinei, C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Mathematical Programming 75 (1996) 407–439. · Zbl 0874.90185
[49] R. Barrett, M. Berry, T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, PA, 1994. · Zbl 0814.65030
[50] L. Grippo, F. Lampariello, S. Lucidi, A nonmonotone linesearch technique for Newton’s method, SIAM Journal of Numerical Analysis 23 (1986) 707–716. · Zbl 0616.65067
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