Une procédure de purification pour les problèmes de complémentarité linéaire, monotones
RAIRO - Operations Research - Recherche Opérationnelle, Volume 38 (2004) no. 1, pp. 63-83.

In this paper, we propose a new purification method for monotone linear complementarity problems. This method associates to each iterate of the sequence, generated by an interior point method, one basis which is not necessarily feasible. We show that, under the strict complementarity and non-degeneracy hypoteses, the sequence of bases converges on a finite number of iterations to an optimal basis which gives the exact solution of the problem. The adopted process also serves to preconditioning the conjugate gradient algorithm when computing the Newton direction.

Dans cet article, nous proposons une nouvelle méthode de purification pour les problèmes de complémentarité linéaire, monotones. Cette méthode associe à chaque itéré de la suite, générée par une méthode de points intérieurs, une base non nécessairement réalisable. Nous montrons que, sous les hypothèses de complémentarité stricte et de non dégénérescence, la suite des bases converge en un nombre fini d'itérations vers une base optimale qui donne une solution exacte du problème. Le procédé adopté sert également à préconditionner l'algorithme de gradient conjugué lors du calcul de la direction de Newton.

DOI: 10.1051/ro:2004012
Classification: 65K05, 90C33, 90C51
Keywords: problèmes de complémentarité linéaire, méthodes de points intérieurs, purification, solution exacte, convergence finie, gradient conjugué, préconditionnement, programmation quadratique convexe
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     title = {Une proc\'edure de purification pour les probl\`emes de compl\'ementarit\'e lin\'eaire, monotones},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {63--83},
     publisher = {EDP-Sciences},
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Kadiri, Abderrahim; Yassine, Adnan. Une procédure de purification pour les problèmes de complémentarité linéaire, monotones. RAIRO - Operations Research - Recherche Opérationnelle, Volume 38 (2004) no. 1, pp. 63-83. doi : 10.1051/ro:2004012. http://archive.numdam.org/articles/10.1051/ro:2004012/

[1] J.F. Bonnans, J.C. Gilbert, C. Lemarechal and C. Sagastizabal, Optimisation Numérique. Aspects théoriques et pratiques. Springer-Verlag (1997). | MR | Zbl

[2] J.F. Bonnans and C.C. Gonzaga, Convergence of interior point algorithms for the monotone linear complementarity problem. Math. Oper. Res. 21 (1996) 1-25. | MR | Zbl

[3] R.W. Cottle, J.S. Pang and V. Venkateswaran, Sufficient matrices and the linear complementarity problem. Linear Algebra Appl. 114/115 (1989) 231-249. | MR | Zbl

[4] F. Facchinei, A. Fischer and C. Kanzow, On the identification of zero variables in a interior-point framework. SIAM J. Optim. 10 (2000) 1058-1078. | MR | Zbl

[5] C.C. Gonzaga, Path-following methods for linear programming. SIAM Rev. 34 (1992) 167-224. | MR | Zbl

[6] T. Illes, J. Peng, C. Roos and T. Terlaky, A strongly polynomial rounding procedure yielding a maximally complementary solution for P * (κ) linear complementarity problems. SIAM J. Optim. 11 (2000) 320-340. | MR | Zbl

[7] J. Ji and A. Potra, Tapia indicators and finite termination of infeasible-interior-point methods for degenerate LCP, edited by J. Renegar, M. Shub and S. Smale. AMS, Providence, RI. Math. Numer. Anal., Lect. Appl. Math. 32 (1996) 443-454. | MR | Zbl

[8] J. Ji, A. Potra and S. Huang, Predictor-corrector method for linear complementarity problems with polynomial complexity and superlinear convergence. JOTA 85 (1995) 187-199. | MR | Zbl

[9] A. Kadiri, Analyse numérique des méthodes de points intérieurs pour les problèmes de complémentarité linéaire et la programmation quadratique convexe. Thèse de Doctorat, INSA de Rouen (2001).

[10] C.T. Kelley, Iterative methods for linear and nonlinear equations. Frontiers Appl. Math. 16 (1995). | MR | Zbl

[11] M. Kojima, S. Mizuno and A. Yoshise, A polynomial-time algorithm for a class of linear complementarity problems. Math. Program. 44 (1989) 1-26. | MR | Zbl

[12] M. Kojima, Y. Kurita and S. Mizuno, Large-step interior point algorithmsfor linear complementarity problems. SIAM J. Optim. 3 (1993) 398-412. | MR | Zbl

[13] K. Kortanek and J. Zhu, New purification algorithms for linear programming. Naval Res. Logist 35 (1988) 571-583. | MR | Zbl

[14] K. Mcshane, Superlineary convergent O(nL)-iteration interior-point algorithms for LP and the monotone LCP. SIAM J. Optim. 4 (1994) 247-261. | MR | Zbl

[15] R. Monteiro and I. Adler, Interior path-following primal-dual algorithms, part II: Convex quadratic programming. Math. Program. 44 (1989) 43-66. | MR | Zbl

[16] R. Monteiro and S. Wright, Local convergence of interior-point algorithms for degenerate monotone LCP. Comput. Optim. Appl. 3 (1994) 131-155. | MR | Zbl

[17] C.R. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall. Englewood Cliffs, New Jersey (1982). | MR | Zbl

[18] F.A. Potra and R. Sheng, A superlineary convergent infeasible-interior-point algorithm for degenerate LCP. J. Optim. Theory Appl. 97 (1998) 249-269. | MR | Zbl

[19] Y. Ye, On the finite convergence of interior point algorithms for linear programming. Math. Program. 57 (1992) 325-335. | MR | Zbl

[20] Y. Ye, Interior Point Algorithms: Theory and Analysis. John Wiley, New York (1997). | MR | Zbl

[21] Y. Ye and K.M. Anstreicher, On quadratic and O(nL) convergence of a predictor-corrector algorithm for LCP. Math. Program. 62 (1993) 537-551. | MR | Zbl

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