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A projected–gradient interior–point algorithm for complementarity problems

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Abstract

Interior–point algorithms are among the most efficient techniques for solving complementarity problems. In this paper, a procedure for globalizing interior–point algorithms by using the maximum stepsize is introduced. The algorithm combines exact or inexact interior–point and projected–gradient search techniques and employs a line–search procedure for the natural merit function associated with the complementarity problem. For linear problems, the maximum stepsize is shown to be acceptable if the Newton interior–point search direction is employed. Complementarity and optimization problems are discussed, for which the algorithm is able to process by either finding a solution or showing that no solution exists. A modification of the algorithm for dealing with infeasible linear complementarity problems is introduced which, in practice, employs only interior–point search directions. Computational experiments on the solution of complementarity problems and convex programming problems by the new algorithm are included.

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References

  1. Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: Augmented lagrangian methods under the constant positive linear dependence constraint qualification. Math. Program. 111, 5–32 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andreani, R., Friedlander, A., Santos, S.A.: On the resolution of the generalized nonlinear complementarity problem. SIAM J. Optim. 12, 303–321 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Andreani, R., Júdice, J., Martinez, J.M., Patrício, J.: On the natural merit function for solving complementarity problems. Math. Program. (to appear)

  4. Birgin, E.G., Martinez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196–1211 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Birgin, E.G., Martinez, J.M., Raydan, M.: Algorithm 813 SPG: software for convex–constrained optimization. ACM Trans. Math. Softw. 27, 340–349 (2001)

    Article  MATH  Google Scholar 

  6. Birgin, E.G., Martinez, J.M., Raydan, M.: Inexact spectral projected gradient methods on convex sets. IMA J. Numer. Anal. 23, 539–559 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brooke, A., Kendrick, D., Meeraus, A., Raman, R.: GAMS—A User’s Guide. GAMS Development Corporation (1998)

  8. Cottle, R., Pang, J., Stone, R.: The Linear Complementarity Problem. Academic, New York (1992)

    MATH  Google Scholar 

  9. Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice Hall Series in Computational Mathematics, Englewood Cliffs, New Jersey (1983)

  10. Facchinei, F., Pang, J.S.: Finite-Dimensional Inequalities and Complementarity Problems. Springer, New York (2003)

    Google Scholar 

  11. Fernandes, L., Friedlander, A., Guedes, M.C., Júdice, J.: Solution of a general linear complementarity problem using smooth optimization and its applications to bilinear programming and LCP. Appl. Math. Optim. 43, 1–19 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fernandes, L., Júdice, J., Figueiredo, I.: On the solution of a finite element approximation of a linear obstacle plate problem. Int. J. Appl. Math. Comput. Sci. 12, 27–40 (2002)

    MATH  MathSciNet  Google Scholar 

  13. Ferris, M.C., Kanzow, C., Munson, T.S.: Feasible descent algorithms for mixed complementarity problems. Math. Program. 86, 475–497 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ferris, M.C., Munson, T.S.: Interfaces to PATH 3.0: design, implementation and usage. Comput. Optim. Appl. 12, 207–227 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fischer, A.: New constrained optimization reformulation of complementarity problems. J. Optim. Theory Appl. 12, 303–321 (2001)

    Google Scholar 

  16. Floudas, C.A., Pardalos, P.M.: A Collection of Test Problems for Constrained Global Optimization Algorithms, vol. 455. Springer, New York (1990)

    Google Scholar 

  17. Gay, D.M.: Electronic Mail Distribution of Linear Programming Test Problems. Technical report, Mathematical Programming Society COAL Newsletter (1985)

  18. Golub, G., Van Loan, C.: Matrix Computations. Johns Hopkins University Press, Baltimore (1983)

    MATH  Google Scholar 

  19. Gould, N.I., Orban, D., Toint, P.: General CUTEr Documentation. Technical Report TR/PA/02/13, CERFACS (2003)

  20. Haslinger, J., Hlavacek, I., Necas, J.: Numerical methods for unilateral problems in solid mechanics. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. IV. North–Holland, Amsterdam (1996)

    Google Scholar 

  21. Intel Corporation: Intel Fortran Compiler User’s Guide (2002)

  22. Júdice, J., Raydan, M., Rosa, S., Santos, S.: On the solution of the symmetric eigenvalue complementarity problem by the spectral projected gradient algorithm. Numer. Algorithms 45, 391–407 (2008)

    Article  Google Scholar 

  23. Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Elements. SIAM, Philadelphia (1988)

    MATH  Google Scholar 

  24. Kojima, M., Megiddo, N., Noma, T., Yoshise, A.: A unified approach to interior-point algorithms for linear complementarity problems. IN: Lecture Notes in Computer Science, vol. 538. Springer, Berlin (1991)

    Google Scholar 

  25. Li, D.-H., Fukushima, M.: Derivative–free line search and global convergence of Broyden–like method for nonlinear equations. Optim. Methods Softw. 13, 181–201 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Murtagh, B.A., Saunders, M.A.: MINOS 5.1 User Guide. Technical Report, Department of Operations Research, Stanford University (1987)

  27. Murty, K.: Linear Complementarity, Linear and Nonlinear Programming. Heldermann, Berlin (1988)

    MATH  Google Scholar 

  28. Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (2000)

    Google Scholar 

  29. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  30. Ostrowski, A.M.: Solution of Equations and Systems of Equations. Academic, New York (1967)

    Google Scholar 

  31. Patrício, J.: Algoritmos de Pontos Interiores para Problemas Complementares Monótonos e Suas Aplicações. PhD thesis, Universidade de Coimbra, Portugal (2007). In Portuguese

  32. Portugal, L.F., Resende, M.G.C., Veiga, G., Júdice, J.J.: A truncated primal-infeasible dual-feasible network interior point method. Networks 35, 91–108 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  33. Schwetlick, H.: Numerische Lösung Nichtlinearer Gleichungen. Deutscher Verlag der Wissenschaften, Berlin (1979)

    Google Scholar 

  34. Solodov, M.V.: Stationary points of bound constrained minimization reformulations. J. Optim. Theory Appl. 94, 449–467 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  35. Vanderbei, R.J.: LOQO: an interior point code for quadratic programming. Optim. Methods Softw. 12, 451–484 (1999)

    Article  MathSciNet  Google Scholar 

  36. Vanderbei, R.J.: LOQO’s User Manual, Version 4.05. School of Engineering and Applied Science, Princeton University (2000)

  37. Wright, S.: Primal–Dual Interior–Point Methods. SIAM, Philadelphia (1997)

    MATH  Google Scholar 

  38. Wright, S.J., Ralph, D.: A superlinear infeasible interior-point algorithm for monotone complementarity problems. Math. Oper. Res. 21, 815–838 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Campinas, Brazil

    Roberto Andreani & José Mario Martínez

  2. Departamento de Matemática, Universidade de Coimbra, Coimbra, Portugal

    Joaquim J. Júdice

  3. Instituto de Telecomunicações, Polo de Coimbra, Coimbra, Portugal

    Joaquim J. Júdice & Joao Patrício

  4. Instituto Politécnico de Tomar, Tomar, Portugal

    Joao Patrício

Authors
  1. Roberto Andreani
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  2. Joaquim J. Júdice
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  3. José Mario Martínez
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  4. Joao Patrício
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Correspondence to Joao Patrício.

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Andreani, R., Júdice, J.J., Martínez, J.M. et al. A projected–gradient interior–point algorithm for complementarity problems. Numer Algor 57, 457–485 (2011). https://doi.org/10.1007/s11075-010-9439-0

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  • DOI: https://doi.org/10.1007/s11075-010-9439-0

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