Tomonari Kitahara

A bound for the number of different basic solutions generated by the simplex method

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems (LP) having optimal solutions. The bound is polynomial of the number of constraints, the number of variables, and the ratio between the minimum and the maximum values of all the positive elements of primal basic feasible solutions. When the problem is primal nondegenerate, it becomes a bound for the number of iterations. The result includes strong polynomiality for Markov Decision Problem by Ye (http://www.stanford.edu/~yyye/simplexmdp1.pdf, 2010) and utilize its analysis. We also apply our result to an LP whose constraint matrix is totally unimodular and a constant vector b of constraints is integral.

Klee-Minty's LP and upper bounds for Dantzig's simplex method

Kitahara and Mizuno (2010) [2] get two upper bounds for the number of different basic feasible solutions generated by Dantzigs simplex method. The size of the bounds highly depends on the ratio between the maximum and the minimum values of all the positive elements of basic feasible solutions. We show that the ratio for a simple variant of Klee-Mintys LP is equal to the number of iterations by Dantzigs simplex method for solving it.

An upper bound for the number of different solutions generated by the primal simplex method with any selection rule of entering variables

Recently, Kitahara, and Mizuno derived an upper bound for the number of different solutions generated by the primal simplex method with Dantzigs (the most negative) pivoting rule. In this paper, we obtain an upper bound with any pivoting rule which chooses an entering variable whose reduced cost is negative at each iteration. The upper bound is applied to a linear programming problem with a totally unimodular matrix. We also obtain a similar upper bound for the dual simplex method.

On the number of solutions generated by the dual simplex method

Abstract In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the dual simplex method with Dantzig’s rule for LP. The bound is comparable with the bound given by Kitahara and Mizuno (in press) [3] for the primal simplex method. We apply the result to the maximum flow problem and get a strong polynomial bound.

An extension of Chubanov's polynomial-time linear programming algorithm to second-order cone programming

In this paper, we extend Chubanovs new polynomial-time algorithm for linear programming to second-order cone programming based on the idea of cutting plane method. The algorithm finds an -dimensional vector x which satisfies , where and is a direct product of n second-order cones and half lines. Like Chubanovs algorithm, one iteration of the proposed algorithm consists of two phases: execution of a basic procedure and scaling. Within calls of the basic procedure, the algorithm either (i) finds an interior feasible solution, (ii) finds a non-zero dual feasible solution, or (iii) verifies that there is no interior feasible solution whose minimum eigenvalue is greater than or equal to ϵ. Each basic procedure requires arithmetic operations, where is the dimension of each second-order cone. If the problem is interior feasible, then the algorithm finds an interior feasible solution in calls of the basic procedure, where is a condition number associated with the system.

Proximity of Weighted and Layered Least Squares Solutions

In this paper, we analyze the limiting behavior of the weighted least squares (WLS) problem

An extension of Chubanov’s algorithm to symmetric cones

\min_{x\in\Re^n}\sum_{i=1}^p\|D_i(A_ix-b_i)\|^2

A refinement of Todd's bound for the diameter of a polyhedron

, where each

A Simple Variant of the Mizuno-Todd-Ye Predictor-Corrector Algorithm and its Objective-Function-Free Complexity

D_i

The LP-Newton method for standard form linear programming problems

is a positive definite diagonal matrix. We consider the situation where the magnitude of the weights differs drastically from one block to the next so that