Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices

SIAM Journal on Optimization

Volumn 7, Issue 1, 1997, Pages 86-125

  Kojima, Masakazu ^a   Shindoh, Susumu ^b   Hara, Shinji ^a  

^a TOKYO INSTITUTE OF TECHNOLOGY   (Japan)

^b NATIONAL DEFENSE ACADEMY   (Japan)

Author keywords

Interior point method; Linear complementarity problem; Linear matrix inequality; Linear program; Semidefinite program

Indexed keywords

EID: 0031542833     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623494269035     Document Type: Article

References (49)

1. F. ALIZADEH, Optimization over the positive-definite cone: Interior point methods and combinatorial applications, in Advances in Optimization and Parallel Computing, P. Pardalos, ed., North-Holland, Amsterdam, 1992, pp. 1-25.
2. F. ALIZADEH, Interior point, methods in semidefinite, programming with application to combinatorial optimization, SIAM J. Optim., 5 (1995), pp. 13-51.
3. F. ALIZADEH, J.-P. A. HAEBERLY, AND M. L. OVERTON, Primal-dual interior-point methods for semidefinite programming, Presented at the Mathematical Programming Symposium, Ann Arbor, MI, 1994, manuscript.
4. J. F. BONNANS AND C. C. GONZAGA, Convergence of interior point algorithms for the monotone linear complementarity problem, Technical Report 2074, INRIA, Roequencourt, France, 1993.
5. S. BOYD, L. E. GHAOUT, E. FERON, AND V. BALAKRISHINAN, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994.
6. R. W. COTTLE J. S PANG, AND R. E. STONE, The Linear Complementarity Problem, Academic Press, New York, 1992.
7. A. V. FIACCO AND G. P. McCORMIK, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley & Sons, New York, 1968.
8. G. H. GOLUB AND C. H. VAN LOAN, Matrix Computations, Johns Hopkins University Press, Baltimore, London, 1989.
9. C. C. GONZAGA AND M. J. TODD, An O (√nL)-iteration large-step primal-dual affine algorithm for linear programming problems, SIAM J. Optim., 2 (1992), pp. 349- 359.
10. M. S. GOWDA AND T. I. SEIDMAN, Generalized linear complementarity problems, Math. Programming, 46 (1990), pp. 329-340.
11. O. GÜLLER, Generalized linear complementarity problems and interior point algorithms for their solutions, Internal Report, Dept. of Industrial Engineering and Operations Research, University of California, Berkeley, CA, 1993.
12. C. HELMBERG, F. RENDL, R. J. VANDERBEI, AND H. WOLKOWICZ, An interior-point, method for semidefinite programming, Program in Statistics and Operations Research, Princeton University, Princeton, NJ, 1994.
13. R. D. HlLL AND S. R. WATERS, On the cone of positive semidefinite matrices, Linear Algebra Appl., 90 (1987), pp. 81-88.
14. M. W. HIRSCH AND S. SMALE, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, 1974.
15. F. JARRE, An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices, SIAM J. Control Optim., 31 (1993), pp. 1360-1377.
16. N. KARMARKAR, A New Polynomial-Time Algorithm for Linear Programming, Cotnbinatorica, 4 (1984), pp. 373-395.
17. M. KOJlMA, Basic lemmas in polynomial-time infeasible-interior-point methods for linear programs, Ann. Oper. Res., 62 (1996), pp. 1-28.
18. M. KOJIMA, N. MEGIDDO, AND S. MIZUNO, A primal-dual infeasible-interior-point algorithm for linear programming, Math. Programming, 61 (1993), pp. 263-280.
19. M. KOJIMA, N. MEGIDDO, T. NOMA, AND A. YOSHISE, A unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Lecture Notes in Computer Science 538, Springer-Verlag, New York, 1991.
20. M. KOJIMA. S. MIZUNO, AND A. YOSHISE, A primal-dual interior point algorithm for linear programming, in Progress in Mathematical Programming, Interior-Point and Related Methods, N. Megiddo, ed., Springer-Verlag, New York, 1989, pp. 29-47.
21. M. KOJIMA, S. MIZUNO, AND A. YOSHISE, A polynomial-time algorithm for a class of linear complementary problems, Math. Programming, 44 (1989), pp. 1-26.
22. M. KOJIMA. S. MIZUNO, AND A. YOSHISE, An O(√n L) iteration potential reduction algorithm for linear camplementarity problems, Math. Programming, 50 (1991), pp. 331-342.
23. M. KOJIMA, T. NOMA, AND A. YOSHISE, Global convergence in infeasible-interior-point algorithms, Math. Programming, 65 (1994), pp. 43-72.
24. M. KOJIMA, M. SHIDA, AND S. SHINDOH, Reduction of monotone linear complementarity problems over cones to linear programs over cones, Research Report #296, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro, Tokyo 152, Japan, 1995.
25. I. J. LUSTlG, Feasibility issues in a primal-dual interior-point method for linear programming, Math. Programming, 49 (1990/91), pp. 145-162.
26. I. J. LUSTIG, R. E. MARSTEN, AND D. F. SHANNO, Computational experience with a primaldual interior point method for linear programming, Linear Algebra Appl., 152 (1991), pp. 191-222.
27. R. MARSTEN, R. SUBRAMANIAN, M. SALTZMAN, I. J. LUSTIG, AND D. SHANNO, Interior point methods for linear programming: Just call Newton, Lagrange, and Fiacco and McCormick!, Interfaces, 20 (1990), pp. 105-116.
28. L. McLlNDEN, The complementarity problem for maximal monotone, multifunctions, in Variational Inequalities and Complementarity Problems, R. W. Cottle, F. Giannessi, and J.- L. Lions, eds., John Wiley & Sons, New York, 1980, pp. 251-270.
29. N. MEGIDDO, A monotone complementarity problem with feasible solutions but not complementarity solutions, Math. Programming, 12 (1977), pp. 131-132.
30. N. MEGIDDO, Pathways to the optimal, set in linear programming, in Progress in Mathematical Programming, Interior-Point and Related Methods, N. Megiddo, ed., Springer-Verlag, New York, 1989, pp. 131-158.
31. S. MIZUNO, F. JARRE, AND J. STOER, A unified approach to infeasible-interior-point algorithms via geometrical linear complementarity problems, Inst. Statist. Math., 1994.
32. S. MIZUNO, M. KOJlMA, AND M. J. TODD, Infeasible-interior-point primal-dual potential-reduction algorithms for linear programming, SIAM J. Optim., 5 (1995), pp. 52- 67.
33. S. MIZUNO, M. J. TODD, AND Y. YE, On adaptive-step primal-dual interior-point algorithms for linear programming, Math. Oper. Res., 18 (1993), pp. 964-981.
34. R. D. C. MONTEIRO AND I. ADLER, Interior path following primal-dual algorithms, Part I: Linear programming, Math. Programming, 44 (1989), pp. 27-41.
35. JU. E. NESTEROV AND A. S. NEMIROVSKII, Self-concordant functions and polynomial-time methods in convex programming, Report, Central Economical and Mathematical Institute, USSR Acad. Sci., Moscow, USSR, 1989.
36. JU. E. NESTEROV AND A. S. NEMIROVSKII, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM, Philadelphia, PA, 1994.
37. JU. E. NESTEROV AND M. J. TODD, Self-Scaled Cones and Interior-Point Methods in Nonlinear Programming, Working Paper, CORE, Catholic University of Louvain- Louvain-la-Neuve, Belgium, 1994.
38. A. M. OSTROWSKI AND O. TAUSSKY, On the variation of the determinant of a positive definite matrix, Nederl, Akad. Wetensch, Proc. Ser. A, 54 (1951), pp. 383-385.
39. 0- and P- properties; extended vertical and horizontal LCP's, Linear Algebra Appl., to appear.
40. K. TANABE, Centered Newton methods for mathematical in system Modeling and Optimisation, M. Iri and K. Yajima, eds., Springer-Verlag, New York, 1988, pp. 197-206.
41. M. J. TODD, Recent developments and new directions in linear programming, in Recent Developments and Applications, M. Iri and K. Tanabe, eds., Kluwer Academic Publishers, London, 1998, pp. 109- 157.
42. M. J. TODD AND Y. YE., A centered projective algorithm for linear programming, Math. Oper. Res., 15 (1990), pp. 508-529.
43. R. H. TÜTÜNCÜ AND M. J. TODD, Reducing horizontal linear complementarity problems, Technical Report, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, 1993.
44. L. VANDENBERGHE AND S. BOYD, A primal-dual potential reduction method for problems involving matrix inequalities, Math. Programming, Series B, 69 (1995), pp. 205-236.
45. L. VANDENBERGHE AND S. BOYD, Semidefinite programming, Working Paper, Information Systems Laboratory, Stanford University, Stanford, CA, 1995.
46. 3 L] potential reduction algorithm for linear programming, Math. Programming, 50 (1991), pp. 239-258.
47. Y. YE, A Fully polynomial-time approximation algorithm for computing a stationary point of the general linear complementarity problem, Math. Oper. Res., 18 (1993), pp. 334-345.
48. Y. ZHANG, On the convergence of a class of interior-point algorithms for the horizontal linear complementarity problem, SIAM J. Optim., 4 (1994), pp. 208-227.
49. Y. ZHANG, R. A. T'APIA, AND F. POTRA, On the superlinear convergence of interior point algorithms for a general class of problems, SIAM J. Optim., 3 (1993), pp. 413-422.