Abstract
This paper deals with the LCP (linear complementarity problem) with a positive semi-definite matrix. Assuming that a strictly positive feasible solution of the LCP is available, we propose ellipsoids each of which contains all the solutions of the LCP. We use such an ellipsoid for computing a lower bound and an upper bound for each coordinate of the solutions of the LCP. We can apply the lower bound to test whether a given variable is positive over the solution set of the LCP. That is, if the lower bound is positive, we know that the variable is positive over the solution set of the LCP; hence, by the complementarity condition, its complement is zero. In this case we can eliminate the variable and its complement from the LCP. We also show how we efficiently combine the ellipsoid method for computing bounds for the solution set with the path-following algorithm proposed by the authors for the LCP. If the LCP has a unique non-degenerate solution, the lower bound and the upper bound for the solution, computed at each iteration of the path-following algorithm, both converge to the solution of the LCP.
Similar content being viewed by others
References
I. Adler and D. Gale, “On the solutions of the positive semi-definite complementarity problem,” Technical Report ORC 75-12, Operations Research Center, University of California (Berkeley, CA, 1975).
R.W. Cottle, “Solution rays for a class of complementarity problems,”Mathematical Programming Study 1 (1974) 59–70.
G.B. Dantzig and R.W. Cottle, “Positive (semi-) definite programming,” in: J. Abadie, ed.,Nonlinear Programming (North-Holland, Amsterdam, 1967).
R.M. Freund, “Projective transformations for interior point methods, part I: Basic theory and linear programming,” OR 180-08, Sloan School of Management, Massachusetts Institute of Technology (Cambridge, MA, 1988).
R.M. Freund, “Projective transformations for interior point methods, part II: Analysis of an algorithm for finding the weighted center of a polyhedral system,” OR 180-88, Sloan School of Management, Massachusetts Institute of Technology (Cambridge, MA, 1988).
C.C. Gonzaga, “An algorithm for solving linear programming programs in O(n 3 L) operations,” in: N. Megiddo, ed.,Progress in Mathematical Programming, Interior Point and Related Methods (Springer, New York, 1989) pp. 1–28.
M. Grötschel, L. Lovász and A. Schrijver,Geometric Algorithms and Combinatorial Optimization (Springer, Berlin, Heidelberg, 1988).
N. Karmarkar, “A new polynomial-time algorithm for linear programming,”Combinatorica 4 (1984) 373–395.
M. Kojima, “Determining basic variables of optimal solutions in Karmarkar's new LP algorithm,”Algorithmica 1(1986) 499–515.
M. Kojima, S. Mizuno and A. Yoshise, “A polynomial-time algorithm for a class of linear complementary problems,”Mathematical Programming 44 (1989) 1–26.
M. Kojima, S. Mizuno and A. Yoshise, “A primal-dual interior point algorithm for linear programming,” in: N. Megiddo, ed.,Progress in Mathematical Programming, Interior-Point and Related Methods (Springer, New York, 1989), pp. 29–47.
M. Kojima and K. Tone, “An efficient implementation of Karmarkar's new LP algorithm,” Research Reports on Information Sciences B-180, Department of Information Sciences, Tokyo Institute of Technology (Tokyo, Japan, 1986).
C.E. Lemke, “Bimatrix equilibrium points and mathematical programming,”Management Science 11 (1965) 681–689.
C.E. Lemke and J.T. Howson Jr., “Equilibrium points of bimatrix games,”SIAM Journal on Applied Mathematics 12 (1964) 413–423.
O.L. Mangasarian, “Characterization of bounded solution sets of linear complementarity problems,”Mathematical Programming Study 19 (1982) 153–166.
O.L. Mangasarian, “Simple computable bounds for solutions of linear complementarity problems and linear programs,”Mathematical Programming Study 25 (1985) 1–12.
N. Megiddo, “Pathways to the optimal set in linear programming,” in: N. Megiddo, ed.,Progress in Mathematical Programming, Interior-Point and Related Methods (Springer, New York, 1989) pp. 131–158.
R.D.C. Monteiro and I. Adler, “Interior path following primal-dual algorithms. Part I: Linear programming,”Mathematical Programming 44 (1989) 27–41.
R.D.C. Monteiro and I. Adler, “Interior path following primal-dual algorithms. Part II: Convex quadratic programming,”Mathematical Programming 44 (1989) 43–66.
J.M. Ortega and W.G. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, Orlando, FL, 1970).
J.S. Pang, I. Kaneko and W.P. Hallman, “On the solution of some (parametric) linear complementarity problems with application to portfolio selection, structural engineering and actuarial graduation,”Mathematical Programming 16 (1978) 325–347.
J. Renegar, “A polynomial-time algorithm based on Newton's method for linear programming,”Mathematical Programming 40 (1988) 59–94.
G. Sonnevend, “An “analytical centre” for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming,” in:Lecture Notes in Control and Information Sciences, Vol. 84 (Springer, New York, 1985) pp. 866–876.
G. Sonnevend, “New algorithms in convex programming based on a notion of “centre” (for systems of analytic inequalities) and on rational extrapolation,” in: K.H. Hoffmann and J. Zowe, eds.,Trends in Mathematical Optimization, ISNM Vol. 84 (Birkhäuser, Basel, 1988) pp. 311–327.
M.J. Todd, “Improved bounds and containing ellipsoids in Karmarkar's linear programming algorithm,”Mathematical of Operations Research 13 (1988) 650–659.
P.M. Vaidya, “An algorithm for linear programming which requires O(((m + n)n 2 + (m + n)1.5 n)L) arithmetic operations,” Technical Report, AT&T Bell Laboratories (Murray Hill, NJ, 1987).
Y. Ye, “Further development on the interior algorithm for convex quadratic programming,” Technical Report, Department of Management Sciences, The University of Iowa (Iowa City, IA, 1987).
Y. Ye, “Karmarkar's algorithm and the ellipsoid method,”Operations Research Letters 6 (1987) 177–182.
Y. Ye and M.J. Todd, “Containing and shrinking ellipsoids in the path-following algorithm,” Technical Report, Department of Management Sciences, The University of Iowa (Iowa City, IA, 1987).
Author information
Authors and Affiliations
Additional information
Supported by Grant-in-Aids for General Scientific Research (63490010) of The Ministry of Education, Science and Culture.
Supported by Grant-in-Aids for Young Scientists (63730014) and for General Scientific Research (63490010) of The Ministry of Education, Science and Culture.
Rights and permissions
About this article
Cite this article
Kojima, M., Mizuno, S. & Yoshise, A. Ellipsoids that contain all the solutions of a positive semi-definite linear complementarity problem. Mathematical Programming 48, 415–435 (1990). https://doi.org/10.1007/BF01582266
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01582266