Skip to main content
SpringerLink
Log in

Merit Functions for Complementarity and Related Problems: A Survey

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

Merit functions have become important tools for solving various mathematical problems arising from engineering sciences and economic systems. In this paper, we are surveying basic principles and properties of merit functions and some of their applications. As a particular case we will consider the nonlinear complementarity problem (NCP) and present a collection of different merit functions. We will also introduce and study a class of smooth merit functions for the NCP.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. AganagiĆ, “Newton's method for linear complementarity problems,” Mathematical Programming, vol. 28, pp. 349-362, 1984.

    Google Scholar 

  2. G. Auchmuty, “Variational principles for variational inequalities,” Numerical Functional Analysis and Optimization, vol. 10, pp. 863-874, 1989.

    Google Scholar 

  3. R. Andreani, A. Friedlander, and J.M. Martínez, “Solution of finite-dimensional variational inequalities using smooth optimization with simple bounds,” Journal of Optimization Theory and Applications, vol. 94, pp. 635-657, 1997.

    Google Scholar 

  4. R. Andreani and J.M. Martínez, “On the reformulation of nonlinear complementarity problems using the Fischer-Burmeister function,” Applied Mathematics Letters, vol. 12, pp. 7-12, 1999.

    Google Scholar 

  5. A. Auslender, Optimization Methods Numeriques, Masson: Paris, 1976.

    Google Scholar 

  6. S.C. Billups, Algorithms for complementarity problems and generalized equations, Ph.D. Thesis, Computer Science Department, University of Wisconsin, Madison, 1995.

    Google Scholar 

  7. B. Chen, X. Chen, and C. Kanzow, A penalized “Fischer-Burmeister NCP-function,” Mathematical Programming, vol. 88, pp. 211-216, 2000.

    Google Scholar 

  8. B. Chen and P.T. Harker, “Smooth approximations to nonlinear complementarity problems,” SIAM Journal on Optimization, vol. 7, pp. 403-420, 1997.

    Google Scholar 

  9. C. Chen and O.L. Mangasarian, “Smoothing methods for convex inequalities and linear complementarity problems,” Mathematical Programming, vol. 71, pp. 51-69, 1995.

    Google Scholar 

  10. G.Y. Chen, C.J. Goh, and X.Q. Yang, “On gap function of vector variational inequality,” in Vector Variational Inequalities and Vector Equilibria, F. Giannessi (Ed.), Kluwer Academic Publishers: Dordrecht, pp. 55-72, 1999.

    Google Scholar 

  11. R.W. Cottle, J.-S. Pang, and R.E. Stone, The Linear Complementarity Problem, Academic Press: New York, 1992.

    Google Scholar 

  12. B.C. Eaves, “On the basic theorem of complementarity,” Mathematical Programming, vol. 1, pp. 68-75, 1971.

    Google Scholar 

  13. T. De Luca, F. Facchinei, and C. Kanzow, “A semismooth equation approach to the solution of nonlinear complementarity problems,” Mathematical Programming, vol. 75, pp. 407-439, 1996.

    Google Scholar 

  14. H. Dietrich, “An equivalent formulation of a variational inequality,” Report No. 44, Department of Mathematics, University of Halle, Halle, 1996.

    Google Scholar 

  15. F. Facchinei, “Minimization of SC1 functions and the Maratos effect,” Operations Research Letters, vol. 17, pp. 131-137, 1995.

    Google Scholar 

  16. F. Facchinei, “Structural and stability properties of P0 nonlinear complementarity problems,” Mathematics of Operations Research, vol. 23, pp. 735-745, 1998.

    Google Scholar 

  17. F. Facchinei, A. Fischer, and C. Kanzow, “Inexact Newton methods for semismooth equations with applications to variational inequalities,” in Nonlinear Optimization and Applications, G. Di Pillo and F. Giannessi (Eds.), Plenum Press: New York, 1996, pp. 125-139.

    Google Scholar 

  18. F. Facchinei, A. Fischer, and C. Kanzow, “A semismooth Newton method for variational inequalities: The case of box constraints,” in Complementarity and Variational Problems-State of the Art, M.C. Ferris and J.-S. Pang (Eds.), SIAM: Philadelphia, 1997, pp. 76-90.

    Google Scholar 

  19. F. Facchinei, A. Fischer, and C. Kanzow, “Regularity properties of a semismooth reformulation of variational inequalities,” SIAM Journal on Optimization, vol. 8, pp. 850-869, 1998.

    Google Scholar 

  20. F. Facchinei, A. Fischer, and C. Kanzow, “On the accurate identification of active constraints,” SIAM Journal on Optimization, vol. 9, pp. 14-32, 1998.

    Google Scholar 

  21. F. Facchinei, A. Fischer, and C. Kanzow, “On the identification of zero-variables in an interior-point framework,” Applied Mathematics Report 159, Department of Mathematics, University of Dortmund, Dortmund, 1998, to appear in SIAM Journal on Optimization.

    Google Scholar 

  22. F. Facchinei, A. Fischer, C. Kanzow, and J. Peng, “A simply constrained optimization reformulation of KKT systems arising from variational inequalities,” Applied Mathematics and Optimization, vol. 40, pp. 19-37, 1999.

    Google Scholar 

  23. F. Facchinei, H. Jiang, and L. Qi, “A smoothing method for mathematical programs with equilibrium constraints,” Mathematical Programming, vol. 85, pp. 81-106, 1999.

    Google Scholar 

  24. F. Facchinei and C. Kanzow, “A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems,” Mathematical Programming, vol. 76, pp. 493-512, 1997.

    Google Scholar 

  25. F. Facchinei and C. Kanzow, “On unconstrained and constrained stationary points of the Implicit Lagrangian,” Journal of Optimization Theory and Applications, vol. 92, pp. 99-115, 1997.

    Google Scholar 

  26. F. Facchinei and C. Kanzow, “Beyond monotonicity in regularization methods for complementarity problems,” SIAM Journal on Control and Optimization, vol. 37, pp. 1150-1161, 1999.

    Google Scholar 

  27. F. Facchinei and J.-S. Pang, “Total stability of variational inequalities,” DIS Working Paper 09.98, Computer and Systems Science Department, University of Rome I “La Sapienza”, Rome, 1998.

    Google Scholar 

  28. F. Facchinei and J. Soares, “A new merit function for nonlinear complementarity problems and a related algorithm,” SIAM Journal on Optimization, vol. 7, pp. 225-247, 1997.

    Google Scholar 

  29. M.C. Ferris and D. Ralph, “Projected gradient methods for nonlinear complementarity problems via normal maps,” in Recent Advances in Nonsmooth Optimization, D.Z. Du, L. Qi, and R.S.Womersley (Eds.), World Scientific Publishers: Singapore, 1995, pp. 57-87.

    Google Scholar 

  30. A. Fischer, “A special Newton-type optimization method,” Optimization, vol. 24, pp. 269-284, 1992.

    Google Scholar 

  31. A. Fischer, “A Newton-type method for positive semidefinite linear complementarity problems,” Journal of Optimization Theory and Applications, vol. 86, pp. 585-608, 1995.

    Google Scholar 

  32. A. Fischer, “An NCP-function and its use for the solution of complementarity problems,” in Recent Advances in Nonsmooth Optimization, D. Du, L. Qi, and R.Womersley (Eds.), World Scientific Publishers: Singapore, 1995, pp. 88-105.

    Google Scholar 

  33. A. Fischer, “Solution of monotone complementarity problems with locally Lipschitzian functions,” Mathematical Programming, vol. 76, pp. 513-532, 1997.

    Google Scholar 

  34. A. Fischer, “A new constrained optimization reformulation for complementarity problems,” Journal of Optimization Theory and Applications, vol. 97, pp. 105-117, 1998.

    Google Scholar 

  35. A. Fischer, “Merit Functions and Stability for Complementarity Problems,” in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi (Eds.), Kluwer Academic Publishers: Dordrecht, 1999, pp. 149-159.

    Google Scholar 

  36. A. Fischer and U. Schnabel, “On a newimbedding of positive semidefinite linear complementarity problems” (in German), Technical Report MATH-NM-08-1993, Department of Mathematics, Technical University of Dresden, Dresden, 1993.

    Google Scholar 

  37. A. Friedlander, J.M. Martínez, and S.A. Santos, “Resolution of linear complementarity problems using minimization with simple bounds,” Journal of Global Optimization, vol. 6, pp. 253-267, 1995.

    Google Scholar 

  38. M. Fukushima, “Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,” Mathematical Programming, vol. 53, pp. 99-110, 1992.

    Google Scholar 

  39. M. Fukushima, “Merit functions for variational inequality and complementarity problems,” in Nonlinear Optimization and Applications, G. Di Pillo and F. Giannessi (Eds.), Plenum Press: New York, 1996, pp. 155-170.

    Google Scholar 

  40. S.A. Gabriel and J.J. Moré, “Smoothing of mixed complementarity problems,” in Complementarity and Variational Problems-State of the Art, M.C. Ferris and J.-S. Pang (Eds.), SIAM: Philadelphia, 1997, pp. 105-116.

    Google Scholar 

  41. C. Geiger and C. Kanzow, “On the resolution of monotone complementarity problems,” Computational Optimization and Applications, vol. 5, pp. 155-173, 1996.

    Google Scholar 

  42. F. Giannessi, “Separation of sets and gap functions for quasi-variational inequalities,” in Variational Inequality and Network Equilibrium Problems, F. Giannessi and A. Maugeri (Eds.), Plenum Press: New York, 1995, pp. 101-121.

    Google Scholar 

  43. M.S. Gowda and G. Ravindran, “Algebraic univalence theorems for nonsmooth functions,” Technical Report TR98-01, Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, 1998.

    Google Scholar 

  44. M.S. Gowda and M. Tawhid, “Existence and limiting behavior of trajectories associated with P0-equations,” Computational Optimization and Applications, vol. 12, pp. 229-251, 1999.

    Google Scholar 

  45. W.W. Hager, “Stabilized sequential quadratic programming,” Computational Optimization and Applications, vol. 12, pp. 253-273, 1999.

    Google Scholar 

  46. P.T. Harker and J.-S. Pang, “Finite-dimensional variational inequality and nonlinear complementarity problem: A survey of theory, algorithms and applications,” Mathematical Programming, vol. 48, pp. 161-220, 1990.

    Google Scholar 

  47. P.T. Harker and B. Xiao, “Newton method for the nonlinear complementarity problem: A B-differentiable equation approach,” Mathematical Programming, vol. 48, pp. 339-357, 1990.

    Google Scholar 

  48. H. Jiang, “Unconstrained minimization approaches to nonlinear complementarity problems,” Journal of Global Optimization, vol. 9, pp. 169-181, 1996.

    Google Scholar 

  49. H. Jiang, “Smoothed Fischer-Burmeister equation methods for the complementarity problem,” Technical Report, Department of Mathematics, The University of Melbourne, Melbourne, 1997.

    Google Scholar 

  50. H. Jiang and L. Qi, “A new nonsmooth equations approach to nonlinear complementarity problems,” SIAM Journal on Control and Optimization, vol. 35, pp. 178-193, 1997.

    Google Scholar 

  51. H. Jiang and D. Ralph, “Smooth SQP methods for mathematical programs with nonlinear complementarity constraints,” Technical Report, Department of Mathematics, The University of Melbourne, Melbourne, 1997, to appear in SIAM Journal on Optimization.

    Google Scholar 

  52. H. Jiang and D. Ralph, “Global and local superlinear convergence analysis of Newton-type methods for semismooth equations with smooth least squares,” in Reformulation-Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi (Eds.), Kluwer Academic Publishers: Dordrecht, 1999, pp. 181-209.

    Google Scholar 

  53. C. Kanzow, “Some equation-based methods for the nonlinear complementarity problem,” Optimization Methods and Software, vol. 3, pp. 327-340, 1994.

    Google Scholar 

  54. C. Kanzow and M. Fukushima, “Equivalence of the generalized complementarity problem to differentiable unconstrained minimization,” Journal of Optimization Theory and Applications, vol. 90, pp. 581-603, 1996.

    Google Scholar 

  55. C. Kanzow and M. Fukushima, “Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities,” Mathematical Programming, vol. 83, pp. 55-87, 1998.

    Google Scholar 

  56. C. Kanzow and H. Kleinmichel, “A new class of semismooth Newton-type methods for nonlinear complementarity problems,” Computational Optimization and Applications, vol. 11, pp. 227-251, 1998.

    Google Scholar 

  57. C. Kanzow and H.-D. Qi, “A QP-free constrained Newton-type method for variational inequality problems,” Mathematical Programming, vol. 85, pp. 81-106, 1999.

    Google Scholar 

  58. C. Kanzow, N. Yamashita, and M. Fukushima, “New NCP-functions and their properties,” Journal of Optimization Theory and Applications, vol. 94, pp. 115-135, 1997.

    Google Scholar 

  59. M. Kojima, “Strongly stable stationary solutions in nonlinear programs,” in Analysis and Computation of Fixed Points, S.M. Robinson (Ed.), Academic Press: New York, 1980, pp. 93-138.

    Google Scholar 

  60. B. Kummer, B. Bank, H. Hollatz, P. Kall, D. Klatte, B. Kummer, K. Lommatzsch, K. Tammer, M. Vlach, and K. Zimmermann, “Newton's method for non-differentiable functions,” in Mathematical Research, Advances in Mathematical Optimization, J. Guddat et al. (Eds.), Akademie-Verlag: Berlin, 1988, pp. 114-125.

    Google Scholar 

  61. T. Larsson and M. Patriksson, “A class of gap functions for variational inequalities,” Mathematical Programming, vol. 64, pp. 53-79, 1994.

    Google Scholar 

  62. Z.-Q. Luo and J.-S. Pang, “Error bounds for analytic systems and their applications,” Mathematical Programming, vol. 67, pp. 1-28, 1994.

    Google Scholar 

  63. Z.-Q. Luo and P. Tseng, “Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem,” SIAM Journal on Optimization, vol. 2, pp. 43-54, 1992.

    Google Scholar 

  64. Z.-Q. Luo and P. Tseng, “A new class of merit functions for the nonlinear complementarity problem,” in Complementarity and Variational Problems: State of the Art, M.C. Ferris and J.-S. Pang (Eds.), SIAM: Philadelphia, 1997, pp. 204-225.

    Google Scholar 

  65. O.L. Mangasarian, “Equivalence of the complementarity problem to a system of nonlinear equations,” SIAM Journal on Applied Mathematics, vol. 31, pp. 89-92, 1976.

    Google Scholar 

  66. O.L. Mangasarian and M.V. Solodov, “Nonlinear complementarity as unconstrained and constrained minimization,” Mathematical Programming, vol. 62, pp. 277-297, 1993.

    Google Scholar 

  67. O.L. Mangasarian and M.V. Solodov, “A linearly convergent derivative-free descent method for strongly monotone complementarity problems,” Computational Optimization and Applications, vol. 14, pp. 5-16, 1999.

    Google Scholar 

  68. J.M. Martínez and L. Qi, “Inexact Newton methods for solving nonsmooth equations,” Journal of Computational and Applied Mathematics, vol. 60, pp. 127-145, 1995.

    Google Scholar 

  69. R. Mifflin, “Semismooth and semiconvex functions in constrained optimization,” SIAM Journal on Control and Optimization, vol. 15, pp. 957-972, 1997.

    Google Scholar 

  70. R.D.C. Monteiro, J.-S. Pang, and T. Wang, “Apositive algorithm for the nonlinear complementarity problem,” SIAM Journal on Optimization, vol. 5, pp. 129-148, 1995.

    Google Scholar 

  71. J.J. Moré, “Global methods for nonlinear complementarity problems,” Mathematics of Operations Research, vol. 21, pp. 589-614, 1996.

    Google Scholar 

  72. J. V. Outrata and J. Zowe, “A Newton method for a class of quasi-variational inequalities,” Computational Optimization and Applications, vol. 4, pp. 5-21, 1995.

    Google Scholar 

  73. J.-S. Pang, “Inexact Newton methods for the nonlinear complementarity problem,” Mathematical Programming, vol. 36, pp. 54-71, 1986.

    Google Scholar 

  74. J.-S. Pang, “Newton's methods for B-differentiable equations,” Mathematics of Operations Research, vol. 15, pp. 311-341, 1990.

    Google Scholar 

  75. J.-S. Pang, “A B-differentiable equation based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity, and variational inequality problems,” Mathematical Programming, vol. 51, pp. 101-131, 1991.

    Google Scholar 

  76. J.-S. Pang, “Complementarity problems,” in Handbook of Global Optimization, R. Horst and P. Pardalos (Eds.), Kluwer Academic Publishers: Boston, 1994, pp. 271-338.

    Google Scholar 

  77. J.-S. Pang and L. Qi, “Nonsmooth equations: Motivation and algorithms,” SIAM Journal on Optimization, vol. 3, pp. 443-465, 1993.

    Google Scholar 

  78. M. Patriksson, “Merit functions and descent algorithms for a class of variational inequality problems,” Optimization, vol. 41, pp. 37-55, 1997.

    Google Scholar 

  79. J.M. Peng, “Convexity of the implicit Lagrangian,” Journal of Optimization Theory and Applications, vol. 92, pp. 331-341, 1997.

    Google Scholar 

  80. J.M. Peng, “Equivalence of variational inequality problems to unconstrained optimization,” Mathematical Programming, vol. 78, pp. 347-355, 1997.

    Google Scholar 

  81. J.M. Peng and Y. Yuan, “Unconstrained methods for generalized complementarity problems,” Journal of Computational Mathematics, vol. 15, pp. 253-264, 1997.

    Google Scholar 

  82. L. Qi, “Regular pseudo-smooth NCP and BVIP functions and globally and quadratically convergent generalized Newton methods for complementarity and variational inequality problems,” Mathematics of Operations Research, vol. 24, pp. 440-471, 1999.

    Google Scholar 

  83. L. Qi and J. Sun, “A nonsmooth version of Newton's method,” Mathematical Programming, vol. 58, pp. 353-367, 1993.

    Google Scholar 

  84. L. Qi, D. Sun, and G. Zhou, “A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities,” Mathematical Programming, vol. 87, pp. 1-35, 2000.

    Google Scholar 

  85. G. Ravindran and M.S. Gowda, “Regularization of P0-functions in box variational inequality problems,” Technical Report TR97-07, Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, 1997.

    Google Scholar 

  86. S.M. Robinson, “Homeomorphism conditions for normal maps of polyhedra,” in Optimization and Nonlinear Analysis, A. Ioffe, M. Marcus, and S. Reich (Eds.), Pitman Research Notes in Mathematics Series 244, 1992, pp. 240-248.

  87. S.M. Robinson, “Sensitivity analysis of variational inequalities by normal-map techniques,” in Nonlinear Variational Inequalities and Network Equilibrium Problems, F. Giannessi and A. Maugeri (Eds.), Plenum Press: New York, 1995, pp. 257-269.

    Google Scholar 

  88. M.V. Solodov, “Stationary points of bound constrained minimization reformulations of complementarity problems,” Journal of Optimization Theory and Applications, vol. 94, pp. 449-467, 1997.

    Google Scholar 

  89. D. Sun and L. Qi, “On NCP-functions,” Computational Optimization and Applications, vol. 13, pp. 201-220, 1999.

    Google Scholar 

  90. D. Sun and R.S. Womersley, “A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Gauss-Newton method,” SIAM Journal on Optimization, vol. 9, pp. 388-413, 1999.

    Google Scholar 

  91. K. Taji and M. Fukushima, “A new merit function and a successive quadratic programming algorithm for variational inequality problems,” SIAM Journal on Optimization, vol. 6, pp. 703-713, 1996.

    Google Scholar 

  92. M. Tawhid and M.S. Gowda, “On two applications of H-differentiability to optimization and complementarity problems,” Technical Report TR98-12, Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, 1998.

    Google Scholar 

  93. P. Tseng, “Growth behavior of a class of merit functions for the nonlinear complementarity problem,” Journal of Optimization Theory and Applications, vol. 89, pp. 17-38, 1996.

    Google Scholar 

  94. P. Tseng, “Merit functions for semi-definite complementarity problems,” Mathematical Programming, vol. 83, pp. 159-185, 1998.

    Google Scholar 

  95. P. Tseng, N. Yamashita, and M. Fukushima, “Equivalence of complementarity problems to differentiable minimization: A unified approach,” SIAM Journal on Optimization, vol. 6, pp. 446-460, 1996.

    Google Scholar 

  96. A.P. Wierzbicki, “Note on the equivalence of Kuhn-Tucker complementarity conditions to an equation,” Journal of Optimization Theory and Applications, vol. 37, pp. 401-405, 1982.

    Google Scholar 

  97. S. Wright, “Superlinear convergence of a stabilized SQP method to a degenerate solution,” Preprint ANL/MCS-P643-0297, Mathematics and Computer Science Department, Argonne National Laboratory, Argonne, 1997.

    Google Scholar 

  98. J.H. Wu, M. Florian, and P. Marcotte, “A general descent framework for the monotone variational inequality problem,” Mathematical Programming, vol. 61, pp. 281-300, 1993.

    Google Scholar 

  99. B. Xiao and P.T. Harker, “Anonsmooth Newton method for variational inequalities, I: Theory,” Mathematical Programming, vol. 65, pp. 151-194, 1994.

    Google Scholar 

  100. K. Yamada, N. Yamashita, and M. Fukushima, “A new derivative-free descent method for the nonlinear complementarity problem,” in Nonlinear Optimization and Related Topics, G. Di Pillo and F. Giannessi (Eds.), Kluwer Academic Publishers, Dordrecht, pp. 463-487, 2000.

    Google Scholar 

  101. N. Yamashita and M. Fukushima, “On stationary points of the implicit Lagrangian for nonlinear complementarity problems,” Journal of Optimization Theory and Applications, vol. 84, pp. 653-663, 1995.

    Google Scholar 

  102. N. Yamashita and M. Fukushima, “A new merit function and a descent method for semidefinite complementarity problems,” in Reformulation-Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi (Eds.), Kluwer Academic Publishers: Dordrecht, 1999, pp. 405-420.

    Google Scholar 

  103. N. Yamashita, K. Taji, and M. Fukushima, “Unconstrained optimization reformulations of variational inequality problems,” Journal of Optimization Theory and Applications, vol. 92, pp. 439-456, 1997.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Dortmund, D-44221, Dortmund, Germany

    Andreas Fischer

  2. CSIRO Mathematical and Information Sciences, GPO Box 664, Canberra, ACT, 2601, Australia

    Houyuan Jiang

Authors
  1. Andreas Fischer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Houyuan Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fischer, A., Jiang, H. Merit Functions for Complementarity and Related Problems: A Survey. Computational Optimization and Applications 17, 159–182 (2000). https://doi.org/10.1023/A:1026598214921

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1026598214921

Search

Navigation