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Solution of monotone complementarity problems with locally Lipschitzian functions. (English) Zbl 0871.90097

Summary: The paper deals with complementarity problems CP(F), where the underlying function F is assumed to be locally Lipschitzian. Based on a special equivalent reformulation of CP(F) as a system of equations Φ(x)=0 or as the problem of minimizing the merit function Ψ=1 2|Φ| 2 2 , we extend results which hold for sufficiently smooth functions F to the nonsmooth case.

In particular, if F is monotone in a neighbourhood of x, it is proved that 0Ψ(x) is necessary and sufficient for x to be a solution of CP(F). Moreover, for monotone functions F, a simple derivative-free algorithm that reduces Ψ is shown to possess global convergence properties. Finally, the local behaviour of a generalized Newton method is analyzed. To this end, the result by Mifflin that the composition of semismooth functions is again semismooth is extended to p-order semismooth functions. Under a suitable regularity condition and if F is p-order semismooth the generalized Newton method is shown to be locally well defined and superlinearly convergent with the order of 1+p.

MSC:
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
Software:
QPCOMP
References:
[1]S.C. Billups and M.C. Ferris, QPCOMP: A quadratic programming based solver for mixed complementarity problems,Mathematical Programming 76 (1997) 533–562 (this issue).
[2]B. Chen and P.T. Harker, Smooth approximations to nonlinear complementarity problems, Technical Report, Department of Management and Systems, College of Business and Economics, Washington State University (Pullman, WA 1995).
[3]C. Chen and 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 · doi:10.1007/BF00249052
[4]F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley, New York, 1983).
[5]R.W. Cottle, J.-S. Pang and R.E. Stone,The Linear Complementarity Problem (Academic, New York, 1992).
[6]T. De Luca, F. Facchinci and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems,Mathematical Programming 75 (1996) 407–439.
[7]S.P. Dirkse and M.C. Ferris, The PATH solver: A non-monotone stabilization scheme for mixed complementarity problems.Optimization Methods and Software 5 (1995) 123–156. · doi:10.1080/10556789508805606
[8]F. Facchinei, A. Fischer and C. Kanzow, A semismooth Newton method for variational inequalities: Theoretical results and preliminary numerical experience, Preprint MATH-NM-21-1995, Institute for Numerical Mathematics, Technical University of Dresden (Dresden, 1995).
[9]F. Facchinei and J. Soares, A new merit function for nonlinear complementarity problems and a related algorithm,SIAM Journal on Optimization 7 (1997) 225–247. · Zbl 0873.90096 · doi:10.1137/S1052623494279110
[10]M.C. Ferris and D. Ralph, Projected gradient methods for nonlinear complementarity problems via normal maps, in: D.Z. Du, L. Qi and R.S. Womersley, eds.,Recent Advances in Nonsmooth Optimization (World Scientific, Singapore, 1995) 57–87.
[11]M.C. Ferris and S. Lucidi, Nonmonotone stabilization methods for nonlinear equations,Journal of Optimization Theory and Applications 81 (1994) 53–71. · Zbl 0803.65070 · doi:10.1007/BF02190313
[12]A. Fischer, A special Newton-type optimization method,Optimization 24 (1992) 269–284. · Zbl 0814.65063 · doi:10.1080/02331939208843795
[13]A. Fischer, A Newton-type method for positive semidefinite linear complementarity problems,Journal of Optimization Theory and Applications 86 (1995) 585–608. · Zbl 0839.90121 · doi:10.1007/BF02192160
[14]A. Fischer, On the superlinear convergence of a Newton-type method for LCP under weak conditions,Optimization Methods and Software 6 (1995) 83–107. · doi:10.1080/10556789508805627
[15]A. Fischer, An NCP-function and its use for the solution of complementarity problems, in: D.Z. Du, L. Qi and R.S. Womersley, eds.,Recent Advances in Nonsmooth Optimization (World Scientific, Singapore, 1995) 88–105.
[16]A. Friedlander, J.M. Martinez and S.A. Santos, Solution of linear complementarity problems using minimization with simple bounds,Global Optimization 6 (1995) 253–267. · Zbl 0835.90101 · doi:10.1007/BF01099464
[17]C. Geiger and C. Kanzow, On the resolution of monotone complementarity problems,Computational Optimization and Applications 5 (1996) 155–173. · Zbl 0859.90113 · doi:10.1007/BF00249054
[18]P.T. Harker and B. Xian, Newton’s method for the nonlinear complementarity problem: a B-differentiable equation approach,Mathematical Programming 48 (1990) 339–357. · Zbl 0724.90071 · doi:10.1007/BF01582262
[19]J.-B. Hiriart-Urruty and C. Lemaréchal,Convex Analysis and Minimization Algorithms (Springer, Heidelberg, 1993).
[20]H. Jiang, Unconstrained minimization approaches to nonlinear complementarity problems,Journal of Global Optimization 9 (1996) 169–181. · Zbl 0868.90122 · doi:10.1007/BF00121662
[21]H. Jiang and L. Qi, Local uniqueness and convergence of iterative methods for nonsmooth variational inequalities,Journal of Mathematical Analysis and Applications 196 (1995) 314–331. · Zbl 0845.65028 · doi:10.1006/jmaa.1995.1412
[22]H. Jiang and L. Qi, Globally and superlinearly convergent trust region algorithm for convex SC1 minimization problems and its application to stochastic programs,Journal of Optimization Theory and Applications 90 (1996) 653–673. · Zbl 0866.90093 · doi:10.1007/BF02189800
[23]H. Jiang and L. Qi, A new nonsmooth equations approach to nonlinear complementarity problems,SIAM Journal on Control and Optimization 35 (1997) 178–193. · Zbl 0872.90097 · doi:10.1137/S0363012994276494
[24]S.-P. Han, J.-S. Pang and N. Rangaraj, Globally convergent Newton methods for nonsmooth equations,Mathematics of Operations Research 17 (1992) 586–607. · Zbl 0777.90057 · doi:10.1287/moor.17.3.586
[25]C. Kanzow, Some equation-based methods for the nonlinear complementarity problem,Optimization Methods and Saftware 3 (1994) 327–340. · doi:10.1080/10556789408805573
[26]C. Kanzow, An unconstrained optimization technique for large-scale linearly constrained convex minimization problems,Computing 53 (1994) 101–117. · Zbl 0820.90099 · doi:10.1007/BF02252984
[27]C. Kanzow, Global convergence properties of some iterative methods for linear complementarity problems,SIAM Journal on Optimization, to appear.
[28]M. Kojima and S. Shindo, Extensions of Newton and quasi-Newton methods to systems ofPC 1 equations,Journal of Operations Research Society of Japan 29 (1986) 352–374.
[29]B. Kummer, Newton’s method for non-differentiable functions, in: J. Guddat et al., eds.,Mathematical Research, Advances in Mathematical Optimization (Akademie, Berlin, 1988) pp. 114–125.
[30]B. Kummer, Newton’s method based on generalized derivatives for nonsmooth functions: Convergence analysis, in: W. Oettli and D. Pallaschke, eds.,Advances in Optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 382 (Springer, Heidelberg, 1992) pp. 171–194.
[31]O.L. Mangasarian, Equivalence of the complementarity problem to a system of nonlinear equations,SIAM Journal on Applied Mathematics 31 (1976) 89–92. · Zbl 0339.90051 · doi:10.1137/0131009
[32]O.L. Mangasarian and M.V. Solodov, Nonlinear complementarity as unconstrained and constrained minimization,Mathematical Programming 62 (1993) 277–297. · Zbl 0813.90117 · doi:10.1007/BF01585171
[33]R. Mifflin, Semismooth and semiconvex functions in constrained optimization,SIAM Journal on Control and Optimization 15 (1977) 959–972. · Zbl 0376.90081 · doi:10.1137/0315061
[34]J.J. Moré, Global methods for nonlinear complementarity problems,Mathematics of Operations Research, to appear.
[35]J.-S. Pang, Newton’s method for B-differentiable equations,Mathematics of Operations Research 15 (1990) 311–341. · Zbl 0716.90090 · doi:10.1287/moor.15.2.311
[36]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 · doi:10.1007/BF01586928
[37]J.-S. Pang and S.A. Gabriel, NE/SQP: A robust algorithm for the nonlinear complementarity problem,Mathematical Programming 60 (1993) 295–337. · Zbl 0808.90123 · doi:10.1007/BF01580617
[38]J.-S. Pang and L. Qi, Nonsmooth equations: Motivation and algorithms,SIAM Journal on Optimization 3 (1993) 443–465. · Zbl 0784.90082 · doi:10.1137/0803021
[39]J.-S. Pang and L. Qi, A globally convergent Newton method for convex SC1 minimization problems,Journal of Optimization Theory and Applications 85 (1995) 633–648. · Zbl 0831.90095 · doi:10.1007/BF02193060
[40]L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations,Mathematics of Operations Research 18 (1993) 227–244. · Zbl 0776.65037 · doi:10.1287/moor.18.1.227
[41]L. Qi and H. Jiang, Karush-Kuhn-Tucker equations and convergence analysis of Newton methods and quasi-Newton methods for solving these equations,Mathematics of Operations Research, to appear.
[42]L. Qi and J. Sun, A nonsmooth version of Newton’s method,Mathematical Programming 58 (1993) 353–367. · Zbl 0780.90090 · doi:10.1007/BF01581275
[43]D. Ralph, Global convergence of damped Newton’s method for nonsmooth equations via the path search,Mathematics of Operations Research 19 (1994) 352–389. · Zbl 0819.90102 · doi:10.1287/moor.19.2.352
[44]S.M. Robinson, Mathematical foundations of nonsmooth embedding methods,Mathematical Programming 48 (1990) 221–229. · Zbl 0728.90084 · doi:10.1007/BF01582256
[45]S.M. Robinson, Newton’s method for a class of nonsmooth functions,Set Valued Analysis 2 (1994) 291–305. · Zbl 0804.65062 · doi:10.1007/BF01027107
[46]R.T. Rockafellar, Computational schemes for large-scale problems in extended linear quadratic programming,Mathematical Programming 48 (1990) 447–474. · Zbl 0735.90050 · doi:10.1007/BF01582268
[47]R.T. Rockafellar and R.J.-B. Wets, A Lagrangian finite-generation technique for solving linear-quadratic problems in stochastic programming,Mathematical Programming Study 28 (1986) 63–93. · Zbl 0599.90090 · doi:10.1007/BFb0121126
[48]P.K. Subramanian, Gauss-Newton methods for the complementarity problem,Journal of Optimization Theory and Applications 77 (1993) 467–482. · Zbl 0792.90082 · doi:10.1007/BF00940445
[49]P. Tseng, Growth behaviour of a class of merit functions for the nonlinear complementarity problem,Journal of Optimization Theory and Applications 89 (1996) 17–37. · Zbl 0866.90127 · doi:10.1007/BF02192639
[50]P. Tseng, An infeasible path-following method for monotone complementarity problem,SIAM Journal on Optimization, to appear.
[51]P. Tseng, N. Yamashita and M. Fukushima, Equivalence of complementarity problems to differentiable minimization: A unified approach,SIAM Journal on Optimization, to appear.
[52]N. Yamashita and 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 · doi:10.1007/BF02191990