We present a simple, scaleable, distributed simplex implementation for large linear programs. It is designed for coarse grained computation, particularly, readily available networks of workstations. Scalability is achieved by using the standard form of the simplex rather than the revised method. Virtually all serious implementations are based on the revised method because it is much … Read more
The purpose of this paper is to derive computational checks for the simplex method of Linear Programming which can be applied at any iteration. The paper begins with a quick review of the simplex algorithm. It then goes through a theoretical development of the simplex method and its dual all the time focusing on the … Read more
Support set invariancy sensitivity analysis deals with finding the range of the parameter variation where there are optimal solutions with the same positive variables for all parameter values throughout this range. This approach to sensitivity analysis has been studied for Linear Optimization (LO) and Convex Quadratic Optimization (CQO) problems, when they are in standard form. … Read more
It is well known that a wide-neighborhood interior-point algorithm for linear programming performs much better in implementation than those small-neighborhood counterparts. In this paper, we provide a unified way to enlarge the neighborhoods of predictor-corrector interior-point algorithms for linear programming. We prove that our methods not only enlarge the neighborhoods but also retain the so-far … Read more
We present a full-Newton step infeasible interior-point algorithm. It is shown that at most $O(n)$ (inner) iterations suffice to reduce the duality gap and the residuals by the factor $\frac1{e}$. The bound coincides with the best known bound for infeasible interior-point algorithms. It is conjectured that further investigation will improve the above bound to $O(\sqrt{n})$. … Read more
Consider solving a linear program in standard form, where the constraint matrix $A$ is $m \times n$, with $n \gg m \gg 1$. Such problems arise, for example, as the result of finely discretizing a semi-infinite program. The cost per iteration of typical primal-dual interior-point methods on such problems is $O(m^2n)$. We propose to reduce … Read more
We obtain a class of primal ane scaling algorithms which generalize some known algorithms. This class, depending on a r-parameter, is constructed through a family of metrics generated by ��r power, r 1, of the diagonal iterate vector matrix. We prove the so-called weak convergence of the primal class for nondegenerate linearly constrained convex … Read more
The Primal-Dual Second Order Corrector (PDSOC) algorithm that we investigate computes on each iteration a corrector direction in addition to the direction of the standard primal-dual path-following interior point method (Kojima et al., 1989) for Linear Programming (LP), in an attempt to improve performance. The corrector is multiplied by the square of the stepsize in … Read more
The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NP-hard, and contains the maximum stable set problem in graphs as a special case. In this note we show that the SQO problem may be reformulated as an (exponentially sized) linear program. CitationCentER … Read more
In this paper we study the behavior of Convex Quadratic Optimization problems when variation occurs simultaneously in the right-hand side vector of the constraints and in the coefficient vector of the linear term in the objective function. It is proven that the optimal value function is piecewise-quadratic. The concepts of transition point and invariancy interval … Read more