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NE/SQP: A robust algorithm for the nonlinear complementarity problem

Mathematical Programming volume 60, pages 295–337 (1993)Cite this article

Abstract

In this paper, we present a new iterative method for solving the nonlinear complementarity problem. This method, which we call NE/SQP (for Nonsmooth Equations/Successive Quadratic Programming), is a damped Gauss—Newton algorithm applied to solve a certain nonsmooth-equation formulation of the complementarity problem; it is intended to overcome a major deficiency of several previous methods of this type. Unlike these earlier algorithms whose convergence critically depends on a solvability assumption on the subproblems, the NE/SQP method involves solving a sequence of nonnegatively constrained convex quadratic programs of the least-squares type; the latter programs are always solvable and their solution can be obtained by a host of efficient quadratic programming subroutines. Hence, the new method is a robust procedure which, not only is very easy to describe and simple to implement, but also has the potential advantage of being capable of solving problems of very large size. Besides the desirable feature of robustness and ease of implementation, the NE/SQP method retains two fundamental attractions of a typical member in the Gauss—Newton family of algorithms; namely, it is globally and locally quadratically convergent. Besides presenting the detailed description of the NE/SQP method and the associated convergence theory, we also report the numerical results of an extensive computational study which is aimed at demonstrating the practical efficiency of the method for solving a wide variety of realistic nonlinear complementarity problems.

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References

  1. H.Z. Aashtiani, “The multi-modal traffic assignment problem,” Ph.D. thesis, Sloan School of Management, Massachusetts Institute of Technology (Cambridge, MA, 1979).

    Google Scholar 

  2. H.Z. Aashtiani and T.L. Magnanti, “Equilibria on a congested transportation network,”SIAM Journal on Algebraic and Discrete Methods 2 (1981) 213–226.

    Google Scholar 

  3. B.H. Ahn,Comutation of Market Equilibria for Policy Analysis: The Project Independence Evaluation System (PIES)Approach (Garland, New York, 1979).

  4. D.P. Bertsekas and E.M. Gafni, “Projection methods for variational inequalities with application to the traffic assignment problem,”Mathematical Programming Study 17 (1982) 139–159.

    Google Scholar 

  5. J.V. Burke, “Methods for solving generalized inequalities with applications to nonlinear programming,” Ph.D. thesis, Department of Mathematics, University of Illinois (Urbana-Champaign, IL, 1984).

    Google Scholar 

  6. J.V. Burke and S.P. Han, “A Gauss—Newton approach to solving generalized inequalities,”Mathematics of Operations Research 11 (1986) 632–643.

    Google Scholar 

  7. J.V. Burke and S.P. Han, “A robust sequential quadratic programming method,”Mathematical Programming 43 (1989) 277–303.

    Google Scholar 

  8. S.C. Choi, W. DeSarbo and P.T. Harker, “Product positioning under price competition,”Management Science 36 (1990) 175–199.

    Google Scholar 

  9. F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley-Interscience, New York, 1983).

    Google Scholar 

  10. R.C. Cottle, J.S. Pang and R.E. Stone,The Linear Complementarity Problem (Academic Press, Boston, MA, 1992).

    Google Scholar 

  11. G.B. Dantzig and A.S. Manne, “A complementarity algorithm for an optimal capital path with invariant proportions,”Journal of Economic Theory 9 (1974) 312–323.

    Google Scholar 

  12. B.C. Eaves, “Thoughts on computing market equilibrium with SLCP,” in: D. Talman and G. van der Laan, eds.,The Computation and Modelling of Economic Equilibria (North-Holland, Amsterdam, 1987) pp. 1–18.

    Google Scholar 

  13. M. Ferris and S. Lucidi, “Globally convergent methods for nonlinear equations,” manuscript, Computer Sciences Department, University of Wisconsin (Madison, WI, 1991).

    Google Scholar 

  14. R. Fletcher, “Numerical experiments with an exactL 1 penalty function method,” in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear Programming 4 (Academic Press, New York, 1981) pp. 99–129.

    Google Scholar 

  15. M. Frank and P. Wolfe, “An algorithm for quadratic programming,”Naval Research Logistics Quarterly 3 (1956) 95–110.

    Google Scholar 

  16. T.L. Friesz, R.L. Tobin, T.E. Smith and P.T. Harker, “A nonlinear complementarity formulation and solution procedure for the general derived demand network equilibrium problem,”Journal of Regional Sciences 23 (1983) 337–359.

    Google Scholar 

  17. M. Fukushima, “Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,”Mathematical Programming 53 (1992) 99–110.

    Google Scholar 

  18. S.A. Gabriel, “Algorithms for the nonlinear complementarity problem: NE/SQP and extensions,” Ph.D. thesis, Department of Mathematical Sciences, The Johns Hopkins University (Baltimore, MD, 1992).

    Google Scholar 

  19. S.A. Gabriel and J.S. Pang, “An inexact NE/SQP method for solving the nonlinear complementarity problem,” manuscript, Department of Mathematical Sciences, The Johns Hopkins University (Baltimore, MD, 1992).

    Google Scholar 

  20. P.E. Gill, W. Murray, M.S. Saunders and M.H. Wright, “User's guide for QPSOL (Version 3.2): A FORTRAN package for quadratic programming, “Technical Report SOL 84-6, Systems Optimization Laboratory, Department of Operations Research, Stanford University (Stanford, CA, 1984).

    Google Scholar 

  21. M. El Hallabi and R.A. Tapia, “A global convergence theory for arbitrary norm trust-region methods for nonlinear equations,” Technical Report TR87-25, Department of Mathematical Sciences, Rice University (Houston, TX, 1989).

    Google Scholar 

  22. S.P. Han, J.S. Pang and N. Rangaraj, “Globally convergent Newton methods for nonsmooth equations,” to appear in:Mathematics of Operations Research (1992).

  23. T. Hansen and T.C. Koopmans, “On the definition and computation of a capital stock invariant under optimization,”Journal of Economic Theory 5 (1972) 487–523.

    Google Scholar 

  24. P.T. Harker, “Dispersed spatial price equilibrium,”Environment and Planning A 20 (1988) 353–368.

    Google Scholar 

  25. P.T. Harker, “Accelerating the convergence of the diagonalization and projection algorithms for finite-dimensional variational inequalities,”Mathematical Programming 41 (1988) 25–59.

    Google Scholar 

  26. P.T. Harker and J.S. Pang, “Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications,”Mathematical Programming 48 (1990) 161–220.

    Google Scholar 

  27. P.T. Harker and B. Xiao, “Newton's method for the nonlinear complementarity problem: a B-differentiable equation approach,”Mathematical Programming 48 (1990) 339–357.

    Google Scholar 

  28. R.H.W. Hoppe and H.D. Mittelmann, “A multi-grid continuation method for parameter-dependent variational inequalities,”Journal of Computational and Applied Mathematics 26 (1989) 35–46.

    Google Scholar 

  29. W.W. Hogan, “Energy policy models for project independence,”Computers and Operations Research 2 (1975) 251–271.

    Google Scholar 

  30. C.M. Ip and J. Kyparisis, “Local convergence of quasi-Newton methods for B-differentiable equations,”Mathematical Programming 56 (1992) 71–89.

    Google Scholar 

  31. P.C. Jones, “Computing an optimal invariant capital stock,”SIAM Journal on Algebraic and Discrete Methods 3 (1982) 145–150.

    Google Scholar 

  32. N.H. Josephy, “A Newton method for the PIES energy model,” MRC Technical Summary Report #1971, Mathematics Research Center, University of Wisconsin (Madison, WI, 1979).

    Google Scholar 

  33. M. Kojima, “Computational methods for solving the nonlinear complementarity problem”Keio Engineering Reports 27 (1974) 1–41.

    Google Scholar 

  34. M. Kojima and S. Shindo, “Extensions of Newton and quasi-Newton methods to systems of PC1 equations,”Journal of Operations Research Society of Japan 29 (1986) 352–374.

    Google Scholar 

  35. M.M. Kostreva, “Elasto-hydrodynamic lubrication: a nonlinear complementarity problem,”International Journal for Numerical Methods in Fluids 4 (1984) 377–397.

    Google Scholar 

  36. Y.Y. Lin and J.S. Pang, “Iterative methods for large convex quadratic programs: a survey,”SIAM Journal on Control and Optimization 25 (1987) 383–411.

    Google Scholar 

  37. L. Mathiesen, “Computational experience in solving equilibrium models by a sequence of linear complementarity problems,”Operations Research 33 (1985) 1225–1250.

    Google Scholar 

  38. L. Mathiesen, “An algorithm based on a sequence of linear complementarity problems applied to a Walrasian equilibrium model: an example,”Mathematical Programming 37 (1987) 1–18.

    Google Scholar 

  39. E. Miersemann and H.D. Mittelmann, “Continuation for parameterized variational inequalities,”Journal of Computational and Applied Mathematics 26 (1989) 23–34.

    Google Scholar 

  40. F.H. Murphy, H.D. Sherali and A.L. Soyster, “A mathematical programming approach for determining oligopolistic market equilibrium,”Mathematical Programming 24 (1982) 92–106.

    Google Scholar 

  41. K.P. Oh, “The numerical solution of dynamically loaded elastohydrodynamic contact as a nonlinear complementarity problem,”Transactions of the ASME 106 (1984) 88–95.

    Google Scholar 

  42. K.P. Oh, “The formulation of the mixed lubrication problem as a generalized nonlinear complementarity problem,”Transactions of the ASME 108 (1986) 598–604.

    Google Scholar 

  43. J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Euqations in Several Variables (Academic Press, New York, 1970).

    Google Scholar 

  44. J.S. Pang, “Newton's method for B-differentiable equations,”Mathematics of Operations Research 15 (1990) 311–341.

    Google Scholar 

  45. 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.

    Google Scholar 

  46. J.S. Pang, “Convergence of splitting and Newton methods for complementarity problems: An application of some sensitivity results,”Mathematical Programming 58 (1993) 149–160.

    Google Scholar 

  47. J.S. Pang, S.P. Han and N. Rangaraj, “Minimization of locally Lipschitzian functions,”SIAM Journal on Optimization 1 (1991) 57–82.

    Google Scholar 

  48. J.S. Pang and L. Qi, “Nonsmooth equations: motivation and algorithms,” to appear in:SIAM Journal on Optimization.

  49. J.S. Pang and Z.P. Wang, “Embedding methods for variational inequality and nonlinear complementarity problems,” manuscript, Department of Mathematical Sciences, The Johns Hopkins University (Baltimore, MD, 1990).

    Google Scholar 

  50. J.S. Pang and C.S. Yu, “Linearized simplicial decomposition methods for computing traffic equilibria on networks,”Networks 14 (1984) 427–438.

    Google Scholar 

  51. K. Park, “Newton's method for nonsmooth equations and constrained optimization,” manuscript, Department of Industrial Engineering, University of Wisconsin (Madison, WI, 1989).

    Google Scholar 

  52. M.J.D. Powell, “A hybrid method for nonlinear equations,” in: P. Rabinowitz, ed.,Numerical Methods for Nonlinear Algebraic Equations (Gordon and Breach, London, 1970) pp. 90–93.

    Google Scholar 

  53. P.V. Preckel, “Alternative algorithms for computing economic equilibria,”Mathematical Programming Study 23 (1985) 163–172.

    Google Scholar 

  54. L. Qi, “Convergence analysis of some algorithms for solving nonsmooth equations,” manuscript, School of Mathematics, The University of New South Wales (Kensington, NSW, 1991).

    Google Scholar 

  55. L. Qi and J. Sun, “A nonsmooth version of Newton's method and an interior point algorithm for convex programming,” manuscript, School of Mathematics, The University of New South Wales (Kensington, NSW, 1991).

    Google Scholar 

  56. D. Ralph, “Global convergence of damped Newton's method for nonsmooth equations, via the path search,” manuscript, Computer Sciences Department, Cornell University (Ithaca, New York, 1991).

    Google Scholar 

  57. A. Reinoza, “Solving generalized equations via homotopies,”Mathematical Programming 31 (1985) 307–320.

    Google Scholar 

  58. S.M. Robinson, “Strongly regular generalized equations,”Mathematics of Operations Research 5 (1980) 43–62.

    Google Scholar 

  59. S.M. Robinson, “Newton's method for a class of nonsmooth functions,” manuscript, Department of Industrial Engineering, University of Wisconson (Madison, WI, 1988).

    Google Scholar 

  60. S.M. Robinson, “Mathematical foundation of embedding methods for nonsmooth equations,”Mathematical Programing 48 (1990) 221–230.

    Google Scholar 

  61. T.F. Rutherford, “A modelling system for applied general equilibrium analysis,” Discussion Paper No. 836, Cowles Foundation for Research in Economics, Yale University (New Haven, CT, 1987).

    Google Scholar 

  62. T.F. Rutherford, “Implementational issues and computational performance solving applied general equilibrium models with SLCP,” Discussion Paper No. 837, Cowles Foundation for Research in Economics, Yale University (New Haven, CT, 1987).

    Google Scholar 

  63. P.A. Samuelson, “Spatial price equilibrium and linear programming,”American Economic Review 42 (1952) 283–303.

    Google Scholar 

  64. H. Scarf,The Computation of Economic Equilibria (Yale University Press, New Haven, CT, 1973).

    Google Scholar 

  65. M.J. Smith, “The existence, uniqueness and stability of traffic equilibria,”Transportation Research 13B (1979) 295–304.

    Google Scholar 

  66. J.C. Stone, “Sequential optimization and complementarity techniques for computing economic equilibria,”Mathematical Programming Study 23 (1985) 173–191.

    Google Scholar 

  67. J.C. Stone, “Formulation and solution of economic equilibrium problems,” Technical Report SOL 88-7, Systems Optimization Laboratory, Department of Operations Research, Stanford University (Stanford, CA, 1987).

    Google Scholar 

  68. K. Taji, M. Fukushima, and T. Ibaraki, “A globally convergent Newton method for solving monotone variational inequalities,” Technical Report #91006, Department of Applied Mathematics and Physics, Kyoto University (Kyoto, Japan, 1991).

    Google Scholar 

  69. R.L. Tobin, “Variable dimension spatial price equilibrium algorithm,”Mathematical Programming 40 (1988) 33–51.

    Google Scholar 

  70. R.L. Tobin, “A method for the analysis of equilibrium multipart prices in oligopolistic markets,” manuscript, GTE Laboratories Incorporated (Waltham, MA, 1990).

    Google Scholar 

  71. B. Xiao, “Global Newton methods for nonlinear programs and variational inequalities: a B-differentiable equation approach,” Ph.D. thesis, Department of Decision Sciences, The Wharton School, University of Pennsylvania (Philadelphia, PA, 1990).

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, The Johns Hopkins University, 21218, Baltimore, MD, USA

    Jong-Shi Pang & Steven A. Gabriel

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  1. Jong-Shi Pang
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  2. Steven A. Gabriel
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This work was based on research supported by the National Science Foundation under grants ECS-8717968 and DDM-9104078.

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Pang, JS., Gabriel, S.A. NE/SQP: A robust algorithm for the nonlinear complementarity problem. Mathematical Programming 60, 295–337 (1993). https://doi.org/10.1007/BF01580617

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Key words

  • Nonlinear complementarity problem
  • nonsmooth equations
  • sequential quadratic programming
  • equilibrium programming