Vector Space Decomposition for Linear Programs

This paper describes a vector space decomposition algorithmic framework for linear programming guided by dual feasibility considerations. The resolution process moves from one basic solution to the next according to an exchange mechanism which is defined by a direction and a post-evaluated step size. The core component of this direction is obtained via the smallest … Read more

Decomposition theorems for linear programs

It is well known that any feasible arc-flow solution to a network problem defined on a graph $G = (N, A)$, where $N$ is the set of nodes whereas $A$ is the set of arcs, can be expressed using at most $|A| + |N|$ paths and cycles having nonzero flow, out of these, at most … Read more

Tools for primal degenerate linear programs: IPS, DCA, and PE

This paper describes three recent tools for dealing with primal degeneracy in linear programming. The first one is the Improved Primal Simplex (IPS) algorithm which turns degeneracy into a possible advantage. The constraints of the original problem are dynamically partitioned based on the numerical values of the current basic variables. The idea is to work … Read more

Improved Column Generation for Highly Degenerate Master Problems

Column generation for solving linear programs with a huge number of variables alternates between solving a master problem and a pricing subproblem to add variables to the master problem as needed. The method is known to suffer from degeneracy of the master problem, exposing what is called the tailing-off effect. Inspired by recent advances in … Read more