In this paper we establish necessary and sufficient conditions for linear convergence of operator splitting methods for a general class of convex optimization problems where the associated fixed-point operator is averaged. We also provide a tight bound on the achievable convergence rate. Most existing results establishing linear convergence in such methods require restrictive assumptions regarding … Read more
One of the most efficient interior-point methods for some classes of block-angular structured problems solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient for, respectively, the block and linking constraints. In this work we show that the choice of a good preconditioner depends on geometrical properties of the constraints structure. … Read more
In a recent work, Muter et al. (2013a) identified and characterized a general class of linear programming (LP) problems – known as problems with column-dependent-rows (CDR-problems). These LPs feature two sets of constraints with mutually exclusive groups of variables in addition to a set of structural linking constraints, in which variables from both groups appear … Read more
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