Computing Estimators of Dantzig Selector type via Column and Constraint Generation

We consider a class of linear-programming based estimators in reconstructing a sparse signal from linear measurements. Specific formulations of the reconstruction problem considered here include Dantzig selector, basis pursuit (for the case in which the measurements contain no errors), and the fused Dantzig selector (for the case in which the underlying signal is piecewise constant). … Read more

On the use of the simplex method for a type of allocation problems

In this study we discuss the use of the simplex method to solve allocation problems whose flow matrices are doubly stochastic. Although these problems can be solved via a 0-1 integer programming method, H.W. Kuhn [4] suggested the use of linear programming in addition to the Hungarian method. Specifically, we use the Birkhoff’s theorem to … Read more

Permutations in the factorization of simplex bases

The basis matrices corresponding to consecutive iterations of the simplex method only differ in a single column. This fact is commonly exploited in current LP solvers to avoid having to compute a new factorization of the basis at every iteration. Instead, a previous factorization is updated to reflect the modified column. Several methods are known … Read more

Closing the gap in pivot methods for linear programming

We propose pivot methods that solve linear programs by trying to close the duality gap from both ends. The first method maintains a set $\B$ of at most three bases, each of a different type, in each iteration: a primal feasible basis $B^p$, a dual feasible basis $B^d$ and a primal-and-dual infeasible basis $B^i$. From … Read more

A primal-simplex based Tardos’ algorithm

In the mid-eighties Tardos proposed a strongly polynomial algorithm for solving linear programming problems for which the size of the coefficient matrix is polynomially bounded by the dimension. Combining Orlin’s primal-based modification and Mizuno’s use of the simplex method, we introduce a modification of Tardos’ algorithm considering only the primal problem and using simplex method … Read more

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 … Read more

Steepest Edge as Applied to the Standard Simplex Method

In this paper we discuss results and advantages of using steepest edge column choice rules and their derivatives. We show empirically, when we utilize the steepest edge column choice rule for the tableau method, that the density crossover point at which the tableau method is more efficient than the revised method drops to 5%. This … Read more

Computational aspects of risk-averse optimisation in two-stage stochastic models

In this paper we argue for aggregated models in decomposition schemes for two-stage stochastic programming problems. We observe that analogous schemes proved effective for single-stage risk-averse problems, and for general linear programming problems. A major drawback of the aggregated approach for two-stage problems is that an aggregated master problem can not contain all the information … Read more

Parallel distributed-memory simplex for large-scale stochastic LP problems

We present a parallelization of the revised simplex method for large extensive forms of two-stage stochastic linear programming (LP) problems. These problems have been considered too large to solve with the simplex method; instead, decomposition approaches based on Benders decomposition or, more recently, interior-point methods are generally used. However, these approaches do not provide optimal … Read more

A Proof by the Simplex Method for the Diameter of a (0,1)-Polytope

Naddef shows that the Hirsch conjecture is true for (0,1)-polytopes by proving that the diameter of any $(0,1)$-polytope in $d$-dimensional Euclidean space is at most $d$. In this short paper, we give a simple proof for the diameter. The proof is based on the number of solutions generated by the simplex method for a linear … Read more