Recently, Pena and Soheili presented a deterministic rescaling perceptron algorithm and proved that it solves a feasible perceptron problem in $O(m^2n^2\log(\rho^{-1}))$ perceptron update steps, where $\rho$ is the radius of the largest inscribed ball. The original stochastic rescaling perceptron algorithm of Dunagan and Vempala is based on systematic increase of $\rho$, while the proof of … Read more
The simplex method has its own problems related to degenerate basic feasible solutions. While such solutions are infrequent, from a theoretical standpoint a proof of the strong duality theorem that uses the simplex method is not complete until it has taken a few extra steps. Further, for economists the duality theorem is extremely important whereas … Read more
Interior-point methods are known to be sensitive to ill-conditioning and to scaling of the data. This paper presents new asymptotically sharp bounds on the condition numbers of the linear systems at each iteration of an interior-point method for solving linear or semidefinite programs and discusses a stopping test which leads to a problem-independent “a priori” … Read more
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
The Johnson-Lindenstrauss lemma allows dimension reduction on real vectors with low distortion on their pairwise Euclidean distances. This result is often used in algorithms such as $k$-means or $k$ nearest neighbours since they only use Euclidean distances, and has sometimes been used in optimization algorithms involving the minimization of Euclidean distances. In this paper we … Read more
We propose a cutting-plane approach (namely, Benders decomposition) for a class of capacitated multi-period facility location problems. The novelty of this approach lies on the use of a specialized interior-point method for solving the Benders subproblems. The primal block-angular structure of the resulting linear optimization problems is exploited by the interior-point method, allowing the (either … Read more
Recently the alternating direction method of multipliers (ADMM) has been widely used for various applications arising in scientific computing areas. Most of these application models are, or can be easily reformulated as, linearly constrained convex minimization models with separable nonlinear objective functions. In this note we show that ADMM can also be easily used for … Read more
We describe an iterative refinement procedure for computing extended precision or exact solutions to linear programming problems (LPs). Arbitrarily precise solutions can be computed by solving a sequence of closely related LPs with limited precision arithmetic. The LPs solved share the same constraint matrix as the original problem instance and are transformed only by modification … Read more
The dual face algorithm uses Cholesky factorization, as would be not very suitable for sparse computations. The purpose of this paper is to present a dual face algorithm using Gauss-Jordan elimination for solving bounded-variable LP problems. ArticleDownload View PDF
Zhang et al recently propose another approach to the face algorithm. This note gives a modification of the result. ArticleDownload View PDF