Tropicalizing the simplex algorithm

We develop a tropical analog of the simplex algorithm for linear programming. In particular, we obtain a combinatorial algorithm to perform one tropical pivoting step, including the computation of reduced costs, in O(n(m+n)) time, where m is the number of constraints and n is the dimension.Comment: v1: 35 pages, 7 figures, 4 algorithms; v2: improved presentation, 39 pages, 9 figures, 4 algorithm

Computing projections for the Karmarkar algorithm

AbstractAlternatives are considered for computing the projection of a vector onto the nullspace of a matrix, as is required to compute the step direction for the Karmarkar projective algorithm. Among the possibilities considered are forming and factoring the normal matrix, and working with a larger but sparser extended matrix. The extended matrices are symmetric but indefinite. A modification to the Harwell MA27 set of subroutines to make them more efficient for indefinite matrices is presented. Computational results are given to support the conclusion that computing these projections using one of the extended matrices offers significant computational advantages over the other alternatives with which it is compared

175 Years of linear programming: 2. Pivots in column space

The simplex method has been the veritable workhorse of linear programming for five decades now. An elegant geometric interpretation of the simplex method can be visualised by viewing the animation of the algorithm in acolumn space representation. In fact, it is this interpretation that explains why it is called the simplex method. The extreme points of the feasible region (polyhedron) of the linear programme can be shown to correspond to an arrangement of simplices in this geometry and the pivoting operation to a physical pivot from one simplex to an adjacent one in the arrangement. This paper introduces this vivid description of the simplex method as a tutored dance of simplices performing 'pivots in column space'