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Properties and construction of NCP functions

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Abstract

The nonlinear complementarity or NCP functions were introduced by Mangasarian and these functions are proved to be useful in constrained optimization and elsewhere. Interestingly enough there are only two general methods to derive such functions, while the known or used NCP functions are either individual constructions or modifications of the few individual NCP functions such as the Fischer-Burmeister function. In the paper we analyze the elementary properties of NCP functions and the various techniques used to obtain such functions from old ones. We also prove some new nonexistence results on the possible forms of NCP functions. Then we develop and analyze several new methods for the construction of nonlinear complementarity functions that are based on various geometric arguments or monotone transformations. The appendix of the paper contains the list and source of the known NCP functions.

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References

  1. Aczél, J.: Lectures on Functional Equations and Their Applications. Academic Press, New York (1966)

    MATH  Google Scholar 

  2. Alsina, C., Frank, M.J., Schweizer, B.: Associative Functions: Triangular Norms and Copulas. World Scientific, Singapore (2006)

    Book  MATH  Google Scholar 

  3. Arnold, V.I.: On the representability of functions of two variables in the form χ(ϕ(x)+ψ(y)). Usp. Mat. Nauk 12, 119–121 (1957). (Russian)

    Google Scholar 

  4. Castillo, E., Iglesias, A., Ruíz-Cobo, R.: Functional Equations in Applied Sciences. Elsevier, Amsterdam (2005)

    MATH  Google Scholar 

  5. Chen, J.-S.: On Some NCP-functions based on the generalized Fischer-Burmeister function. Asia-Pac. J. Oper. Res. 24, 401–420 (2007)

    Article  MathSciNet  Google Scholar 

  6. Chen, J.-S., Pan, S.: A family of NCP functions and a descent method for the nonlinear complementarity problem. Comput. Optim. Appl. 40, 389–404 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, B., Chen, X., Kanzow, C.: A penalized Fischer-Burmeister NCP-function: theoretical investigation and numerical results. Hamburger Beiträge zur Angewandten Math. Reihe A, prepr. 126 (1997)

  8. Chen, B., Chen, X., Kanzow, C.: A penalized Fischer-Burmeister NCP-function. Math. Program. 88, 211–216 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, X., Qi, L., Yang, Y.F.: Lagrangian globalization methods for nonlinear complementarity problems. J. Optim. Theory Appl. 112, 77–95 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Evtushenko, Yu.G., Purtov, V.A.: Sufficient conditions for a minimum for nonlinear programming problems. Sov. Math. Dokl. 30, 313–316 (1984)

    MATH  Google Scholar 

  11. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I–II. Springer, Berlin (2003)

    Google Scholar 

  12. Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fischer, A., Jiang, H.: Merit functions for complementarity and related problems: a survey. Comput. Optim. Appl. 17, 159–182 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Floudas, C.A., Pardalos, P.M. (eds.): Encyclopedia of Optimization. Springer, Berlin (2009)

    MATH  Google Scholar 

  16. Gibson, C.G.: Elementary Geometry of Algebraic Curves. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  17. Hu, S.-L., Huang, Z.-H., Chen, J.-S.: Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems. J. Comput. Appl. Math. 230, 69–82 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jiménez-Fernández, V.M., Agustín-Rodríguez, J., Marcelo-Julián, P., Agamennoni, O.: Evaluation algorithm for a decomposed simplicial piecewise linear formulation. J. Appl. Res. Technol. 6, 159–169 (2008)

    Google Scholar 

  19. Kanzow, C., Kleinmichel, H.: A class of Newton-type methods for equality and inequality constrained optimization. Optim. Methods Softw. 5, 173–198 (1995)

    Article  Google Scholar 

  20. Kanzow, C., Kleinmichel, H.: A new class of semismooth Newton-type methods for nonlinear complementarity problems. Comput. Optim. Appl. 11, 227–251 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kanzow, C., Yamashita, N., Fukushima, M.: New NCP-functions and their properties. J. Optim. Theory Appl. 94, 115–135 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Khavinson, S.Ya.: Best Approximation by Linear Superpositions (Approximate Nomography). AMS, Providence (1997)

    MATH  Google Scholar 

  23. Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization. Kluwer Academic, Dordrecht (2002)

    MATH  Google Scholar 

  24. Lorentz, G.G.: Approximation of Functions. AMS, Providence (1986)

    MATH  Google Scholar 

  25. Luo, Z.Q., Tseng, P.: A new class of merit functions for the nonlinear complementarity problem. In: Ferris, M.C., Pang, J.S. (eds.) Complementarity and Variational Problems: State of the Art, pp. 204–225. SIAM, Philadelphia (1997)

    Google Scholar 

  26. Mangasarian, O.L.: Equivalence of the complementarity problem to a system of nonlinear equations. SIAM J. Appl. Math. 31, 89–92 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  27. Magasarian, O.L., Solodov, M.V.: Nonlinear complementarity as unconstrained and constrained minimization. Math. Program. 62, 277–297 (1993)

    Article  Google Scholar 

  28. Mulholland, J., Monagan, J.: Algorithms for trigonometric polynomials. In: Kaltofen, E., Villard, G. (eds.) ISSAC’01: Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, pp. 245–252. ACM, New York (2001)

    Chapter  Google Scholar 

  29. Pang, J.-S.: A B-differentiable equation based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems. Math. Program. 51, 101–131 (1991)

    Article  MATH  Google Scholar 

  30. Pu, D., Zhou, Y.: Piecewise linear NCP function for QP free feasible methods. Appl. Math. J. Chin. Univ. 21, 289–301 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Qi, L.: Regular almost smooth NCP and BVIP functions and globally and quadratically convergent generalized Newton methods for complementarity and variational inequality problems. Technical Rep. AMR 97/14, University of New South Wales (1997)

  32. Qi, L., Yang, Y.-F.: NCP functions applied to Lagrangian globalization for the nonlinear complementarity problem. J. Glob. Optim. 24, 261–283 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sun, D.: A regularization Newton method for solving nonlinear complementarity problems. Appl. Math. Optim. 40, 315–339 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sun, D., Qi, L.-Q.: On NCP-functions. Comput. Optim. Appl. 13, 201–220 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sun, D., Womersley, R.S.: A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Gauss-Newton method. SIAM J. Optim. 9, 388–413 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ulbrich, M.: Semismooth Newton methods for operator equations in function spaces. Technical Rep. 00-11, Department of Computational and Applied Mathematics. Rice University (2000)

  37. Vitushkin, A.G., Khenkin, G.M.: Linear superpositions of functions. Russ. Math. Surv. 22, 77–125 (1967)

    Article  MATH  Google Scholar 

  38. Walker, R.J.: Algebraic Curves. Springer, Berlin (1978)

    Book  MATH  Google Scholar 

  39. Wierzbicki, A.P.: Note on the equivalence of Kuhn-Tucker complementarity conditions to an equation. J. Optim. Theory Appl. 37, 401–405 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yamada, K., Yamashita, N., Fukushima, M.: A new derivative-free descent method for the nonlinear complementarity problems. In: Pillo, G.D., Gianessi, F. (eds.) Nonlinear Optimization and Related Topics, pp. 463–489. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

  41. Yamashita, N.: Properties of restricted NCP functions for nonlinear complementarity problems. J. Optim. Theory Appl. 98, 701–717 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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  1. John von Neumann Faculty of Informatics, Óbuda University, Bécsi út 96/b., 1034, Budapest, Hungary

    Aurél Galántai

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  1. Aurél Galántai
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Galántai, A. Properties and construction of NCP functions. Comput Optim Appl 52, 805–824 (2012). https://doi.org/10.1007/s10589-011-9428-9

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