论文信息 - A scaling algorithm for optimizing arbitrary functions over vertices of polytopes

A scaling algorithm for optimizing arbitrary functions over vertices of polytopes

In this paper we present a scaling algorithm for minimizing arbitrary functions over vertices of polytopes in an oracle model of computation which includes an augmentation oracle. For the binary case, when the vertices are 0–1 vectors, we show that the oracle time is polynomial. Also, this algorithm allows us to generalize some concepts of combinatorial optimization concerning performance bounds of greedy algorithms and leads to new bounds for the complexity of the simplex method.

Sergei Chubanov

[1] Friedrich Eisenbrand,et al. On Sub-determinants and the Diameter of Polyhedra , 2014, Discret. Comput. Geom..

[2] Shinji Mizuno,et al. A bound for the number of different basic solutions generated by the simplex method , 2011, Mathematical Programming.

[3] András Frank,et al. An application of simultaneous diophantine approximation in combinatorial optimization , 1987, Comb..

[4] Jesús A. De Loera,et al. On Augmentation Algorithms for Linear and Integer-Linear Programming: From Edmonds-Karp to Bland and Beyond , 2014, SIAM J. Optim..

[5] Andreas S. Schulz,et al. 0/1-Integer Programming: Optimization and Augmentation are Equivalent , 1995, ESA.

[6] Abraham P. Punnen,et al. Approximate local search in combinatorial optimization , 2004, SODA '04.

[7] Andreas S. Schulz. On the Relative Complexity of 15 Problems Related to 0/1-Integer Programming , 2008, Bonn Workshop of Combinatorial Optimization.

[8] Martin E. Dyer,et al. Random walks, totally unimodular matrices, and a randomised dual simplex algorithm , 1994, IPCO.

[9] Denis Naddef,et al. The hirsch conjecture is true for (0, 1)-polytopes , 1989, Math. Program..

[10] Peter Kleinschmidt,et al. On the diameter of convex polytopes , 1992, Discret. Math..