Klee–Minty’s LP and upper bounds for Dantzig’s simplex method

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

Kitahara and Mizuno (2010) get two upper bounds for the number of different basic feasible solutions generated by Dantzig’s simplex method. The size of the bounds highly depends on the ratio between the maximum and minimum values of all the positive elements of basic feasible solutions. In this paper, we show some relations between the ratio and the number of iterations by using an example of L...

Upper and lower bounds for numerical radii of block shifts

For an n-by-n complex matrix A in a block form with the (possibly) nonzero blocks only on the diagonal above the main one, we consider two other matrices whose nonzero entries are along the diagonal above the main one and consist of the norms or minimum moduli of the diagonal blocks of A. In this paper, we obtain two inequalities relating the numeical radii of these matrices and also determine ...

Lecture 2 : LP Duality , Simplex Method Complexity , Ellipsoid Method

An interesting concept associated with linear programs is that of duality. For every LP problem there is a second LP problem with which it is intrinsically paired. This second LP problem can bound the feasible values of the first problem and even help find the optimal value to that problem. The initial LP problem is called the primal LP while the second is referred to as the dual LP. Consider t...

A Strongly Polynomial Simplex Method for Totally Unimodular LP

Kitahara and Mizuno get new bounds for the number of distinct solutions generated by the simplex method for linear programming (LP). In this paper, we combine results of Kitahara and Mizuno and Tardos’s strongly polynomial algorithm, and propose an algorithm for solving a standard form LP problem. The algorithm solves polynomial number of artificial LP problems by the simplex method with Dantzi...

Technical Note - Generalized Upper Bounds and Triangular Decomposition in the Simplex Method