Rescaled Coordinate Descent Methods for Linear Programming

We propose two simple polynomial-time algorithms to find a positive solution to $$Ax=0$$. Both algorithms iterate between coordinate descent steps similar to von Neumann's algorithm, and rescaling steps. In both cases, either the updating step leads to a substantial decrease in the norm, or we can infer that the condition measure is small and rescale in order to improve the geometry. We also show how the algorithms can be extended to find a solution of maximum support for the system $$Ax=0$$, $$x\ge 0$$. This is an extended abstract. The missing proofs will be provided in the full version.

[1]  Albert B Novikoff,et al.  ON CONVERGENCE PROOFS FOR PERCEPTRONS , 1963 .

[2]  László A. Végh,et al.  A polynomial projection-type algorithm for linear programming , 2013, Oper. Res. Lett..

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

[4]  Ulrich Betke.  Relaxation, New Combinatorial and Polynomial Algorithms for the Linear Feasibility Problem , 2004, Discret. Comput. Geom..

[5]  I. J. Schoenberg,et al.  The Relaxation Method for Linear Inequalities , 1954, Canadian Journal of Mathematics.

[6]  Robert M. Freund,et al.  Condition number complexity of an elementary algorithm for computing a reliable solution of a conic linear system , 2000, Math. Program..

[7]  Javier Peña,et al.  A deterministic rescaled perceptron algorithm , 2016, Math. Program..

[8]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[9]  sergei. chubanov,et al.  A polynomial algorithm for linear optimization which is strongly polynomial under certain conditions on optimal solutions , 2015 .

[10]  Jean-Louis Goffin,et al.  The Relaxation Method for Solving Systems of Linear Inequalities , 1980, Math. Oper. Res..

[11]  Sergei Chubanov,et al.  A polynomial projection algorithm for linear feasibility problems , 2015, Math. Program..

[12]  Kees Roos,et al.  On Chubanov’s Method for Solving a Homogeneous Inequality System , 2015 .

[13]  Javier Peña,et al.  A Smooth Perceptron Algorithm , 2012, SIAM J. Optim..

[14]  Jaime G. Carbonell,et al.  Saddle Points and Accelerated Perceptron Algorithms , 2014, ICML.

[15]  Sergei Chubanov.  A strongly polynomial algorithm for linear systems having a binary solution , 2012, Math. Program..

[16]  J. Sherman,et al.  Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix , 1950 .

[17]  F ROSENBLATT,et al.  The perceptron: a probabilistic model for information storage and organization in the brain. , 1958, Psychological review.

[18]  Santosh S. Vempala,et al.  A simple polynomial-time rescaling algorithm for solving linear programs , 2004, STOC '04.

[19]  Philip Wolfe,et al.  Finding the nearest point in A polytope , 1976, Math. Program..

[20]  Sergei Chubanov.  A polynomial projection algorithm for linear programming , 2013 .

[21]  Arkadi Nemirovski,et al.  Prox-Method with Rate of Convergence O(1/t) for Variational Inequalities with Lipschitz Continuous Monotone Operators and Smooth Convex-Concave Saddle Point Problems , 2004, SIAM J. Optim..

[22]  Jesús A. De Loera,et al.  On Chubanov's Method for Linear Programming , 2012, INFORMS J. Comput..