A Fully Polynomial-Time Approximation Algorithm for Computing a Stationary Point of the General Linear Complementarity Problem

We apply a potential reduction algorithm to solve the general linear complementarity problem GLCP minimize xTy subject to Ax + By + Cz = q and x, y, z ≥ 0. We show that the algorithm is a fully polynomial-time approximation scheme FPTAS for computing an e-approximate stationary point of the GLCP. Note that there are some GLCPs in which every stationary point is a solution xTy = 0. These include the LCPs with row sufficient matrices. We also show that the algorithm is a polynomial-time algorithm for a special class of GLCPs.

[1]  C. E. Lemke,et al.  Bimatrix Equilibrium Points and Mathematical Programming , 1965 .

[2]  M. Todd.  Improving the convergence of fixed-point algorithms , 1978 .

[3]  B. Eaves.  Computing stationary points , 1978 .

[4]  Masakazu Kojima,et al.  Variable dimension algorithms: Basic theory, interpretations and extensions of some existing methods , 1982, Math. Program..

[5]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[6]  Panos M. Pardalos,et al.  Constrained Global Optimization: Algorithms and Applications , 1987, Lecture Notes in Computer Science.

[7]  Katta G. Murty,et al.  Linear complementarity, linear and nonlinear programming , 1988 .

[8]  R. Cottle,et al.  Sufficient matrices and the linear complementarity problem , 1989 .

[9]  Dolf Talman,et al.  A Simplicial Algorithm for Stationary Point Problems on Polytopes , 1989, Math. Oper. Res..

[10]  Michael J. Todd,et al.  A Centered Projective Algorithm for Linear Programming , 1990, Math. Oper. Res..

[11]  Y. Ye,et al.  A Class of Linear Complementarity Problems Solvable in Polynomial Time , 1991 .

[12]  Yinyu Ye,et al.  An O(n3L) potential reduction algorithm for linear programming , 1991, Math. Program..

[13]  Shinji Mizuno,et al.  An $$O(\sqrt n L)$$ iteration potential reduction algorithm for linear complementarity problems , 1991, Math. Program..

[14]  Nimrod Megiddo,et al.  A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems , 1991, Lecture Notes in Computer Science.

[15]  J. Pang.  Iterative descent algorithms for a row sufficient linear complementary problem , 1991 .

[16]  Clóvis C. Gonzaga,et al.  Polynomial affine algorithms for linear programming , 1990, Math. Program..

[17]  Yinyu Ye,et al.  On the finite convergence of interior-point algorithms for linear programming , 1992, Math. Program..

[18]  Nimrod Megiddo,et al.  An interior point potential reduction algorithm for the linear complementarity problem , 1992, Math. Program..

[19]  Y. Ye,et al.  Solution of $P_0 $-Matrix Linear Complementarity Problems Using a potential Reduction Algorithm , 1993, SIAM J. Matrix Anal. Appl..