This paper introduces the design and implementation of two parallel dual simplex solvers for general large scale sparse linear programming problems. One approach, called PAMI, extends a relatively unknown pivoting strategy called suboptimization and exploits parallelism across multiple iterations. The other, called SIP, exploits purely single iteration parallelism by overlapping computational components when possible. Computational … Read more
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
We consider the application of mixed-integer linear programming (MILP) solvers to the minimization of submodular functions. We evaluate common large-scale linear-programming (LP) techniques (e.g., column generation, row generation, dual stabilization) for solving a LP reformulation of the submodular minimization (SM) problem. We present heuristics based on the LP framework and a MILP solver. We evaluated … Read more
Until now it has been an open question whether the Linear Programming (LP) problem can be solved in strong polynomial time. The simplex algorithm with its combinatorial nature does not even offer a polynomial bound, whereas the complexity of the polynomial algorithms by Khachiyan and Karmarkar is based on the number of variables n, and … Read more
This paper proposes a polynomial algorithm for linear programming which is strongly polynomial for linear optimization problems $\min\{c^Tx : Ax = b, x\ge {\bf 0}\}$ having optimal solutions where each non-zero component $x_j$ belongs to an interval of the form $[\alpha_j, \alpha_j\cdot 2^{p(n)}],$ where $\alpha_j$ is some positive value and $p(n)$ is a polynomial of … Read more
As a natural extension of the dual simplex algorithm, the dual face algorithm performed remarkably in computational experiments with a set of Netlib standard problems. In this paper, we generalize it to bounded-variable LP problems via local duality. CitationDepartment of Mathematics, Southeast University, Nanjing, 210096, China, 12/2014ArticleDownload View PDF
We present a clustering-based preconditioning strategy for KKT systems arising in stochastic programming within an interior-point framework. The key idea is to perform adaptive clustering of scenarios (inside-the-solver) based on their influence on the problem as opposed to cluster scenarios based on problem data alone, as is done in existing (outside-thesolver) approaches. We derive spectral … Read more
We propose a family of search directions based on primal-dual entropy in the context of interior-point methods for linear optimization. We show that by using entropy based search directions in the predictor step of a predictor-corrector algorithm together with a homogeneous self-dual embedding, we can achieve the current best iteration complexity bound for linear optimization. … Read more
We consider discounted Markov Decision Processes (MDPs) with countably-infinite state spaces, finite action spaces, and unbounded rewards. Typical examples of such MDPs are inventory management and queueing control problems in which there is no specific limit on the size of inventory or queue. Existing solution methods obtain a sequence of policies that converges to optimality … Read more
Recently, Todd got a new bound on the diameter of a polyhedron using an analysis due to Kalai and Kleitman in 1992. In this short note, we prove that the bound by Todd can further be improved. Although our bound is not valid when the dimension is 1 or 2, it is tight when the … Read more