Solving LP Problems via Weighted Centers

The feasibility problem for a system of linear inequalities can be converted into an unconstrained optimization problem by using ideas from the ellipsoid method, which can be viewed as a very simple minimization technique for the resulting nonlinear function. This function is related to the volume of an ellipsoid containing all feasible solutions, which is parametrized by certain weights which we choose to minimize the function. The center of the resulting ellipsoid turns out to be a feasible solution to the inequalities. Using more sophisticated nonlinear minimization algorithms, we develop and investigate more efficient methods, which lead to two kinds of weighted centers for the feasible set. Using these centers, we develop new algorithms for solving linear programming problems.

[1]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[2]  Pravin M. Vaidya,et al.  A new algorithm for minimizing convex functions over convex sets , 1996, Math. Program..

[3]  Michael J. Todd,et al.  Improved Bounds and Containing Ellipsoids in Karmarkar's Linear Programming Algorithm , 1988, Math. Oper. Res..

[4]  Michael J. Todd,et al.  Modifications and implementation of the ellipsoid algorithm for linear programming , 1982, Math. Program..

[5]  Y. Ye.  Karmarkar's algorithm and the ellipsoid method , 1987 .

[6]  James Renegar,et al.  A polynomial-time algorithm, based on Newton's method, for linear programming , 1988, Math. Program..

[7]  V. Chvátal,et al.  Notes on Bland’s pivoting rule , 1978 .

[8]  Leonid Khachiyan,et al.  On the complexity of approximating the maximal inscribed ellipsoid for a polytope , 1993, Math. Program..

[9]  L. G. H. Cijan.  A polynomial algorithm in linear programming , 1979 .

[10]  D. J. Newman,et al.  Location of the Maximum on Unimodal Surfaces , 1965, JACM.

[11]  Michael J. Todd,et al.  The Ellipsoid Method Generates Dual Variables , 1985, Math. Oper. Res..

[12]  S. Agmon.  The Relaxation Method for Linear Inequalities , 1954, Canadian Journal of Mathematics.

[13]  Thomas F. Coleman,et al.  Handbook for matrix computations , 1988 .

[14]  N. Z. Shor.  Cut-off method with space extension in convex programming problems , 1977, Cybernetics.

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

[16]  Michael J. Todd,et al.  The Ellipsoid Method: A Survey , 1980 .

[17]  N. Z. Shor.  Utilization of the operation of space dilatation in the minimization of convex functions , 1972 .

[18]  Ai-Ping Liao.  Algorithms for linear programming via weighted centers , 1992 .

[19]  Michael J. Todd.  On Minimum Volume Ellipsoids Containing Part of a Given Ellipsoid , 1982, Math. Oper. Res..

[20]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[21]  Michael J. Todd,et al.  The ellipsoid algorithm using parallel cuts , 1993, Comput. Optim. Appl..

[22]  M. Akgül,et al.  Topics in relaxation and ellipsoidal methods , 1984 .

[23]  Walter L. Smith.  Probability and Statistics , 1959, Nature.

[24]  J. L. Nazareth.  Pricing criteria in linear programming , 1988 .

[25]  G. Sonnevend.  An "analytical centre" for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming , 1986 .

[26]  Jean-Louis Goffin,et al.  On the non-polynomiality of the relaxation method for systems of linear inequalities , 1982, Math. Program..