5 research outputs found

    Relaxation, New Combinatorial and Polynomial Algorithms for the Linear Feasibility Problem

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    We consider the homogenized linear feasibility problem, to find an xx on the unit sphere, satisfying nn line ar inequalities aiTxβ‰₯0a_i^Tx\ge 0. To solve this problem we consider the centers of the insphere of spherical simpl ices, whose facets are determined by a subset of the constraints. As a result we find a new combinatorial algor ithm for the linear feasibility problem. If we allow rescaling this algorithm becomes polynomial. We point out that the algorithm solves as well the more general convex feasibility problem. Moreover numerical experiments s how that the algorithm could be of practical interest

    A Sampling Kaczmarz-Motzkin Algorithm for Linear Feasibility

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    We combine two iterative algorithms for solving large-scale systems of linear inequalities, the relaxation method of Agmon, Motzkin et al. and the randomized Kaczmarz method. In doing so, we obtain a family of algorithms that generalize and extend both techniques. We prove several convergence results, and our computational experiments show our algorithms often outperform the original methods

    A Simple Method for Convex Optimization in the Oracle Model

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    We give a simple and natural method for computing approximately optimal solutions for minimizing a convex function ff over a convex set KK given by a separation oracle. Our method utilizes the Frank--Wolfe algorithm over the cone of valid inequalities of KK and subgradients of ff. Under the assumption that ff is LL-Lipschitz and that KK contains a ball of radius rr and is contained inside the origin centered ball of radius RR, using O((RL)2Ξ΅2β‹…R2r2)O(\frac{(RL)^2}{\varepsilon^2} \cdot \frac{R^2}{r^2}) iterations and calls to the oracle, our main method outputs a point x∈Kx \in K satisfying f(x)≀Ρ+min⁑z∈Kf(z)f(x) \leq \varepsilon + \min_{z \in K} f(z). Our algorithm is easy to implement, and we believe it can serve as a useful alternative to existing cutting plane methods. As evidence towards this, we show that it compares favorably in terms of iteration counts to the standard LP based cutting plane method and the analytic center cutting plane method, on a testbed of combinatorial, semidefinite and machine learning instances.Comment: Major revisio

    Greed Works: An Improved Analysis of Sampling Kaczmarz-Motzkin

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    Stochastic iterative algorithms have gained recent interest in machine learning and signal processing for solving large-scale systems of equations, Ax=bAx=b. One such example is the Randomized Kaczmarz (RK) algorithm, which acts only on single rows of the matrix AA at a time. While RK randomly selects a row of AA to work with, Motzkin's Method (MM) employs a greedy row selection. Connections between the two algorithms resulted in the Sampling Kaczmarz-Motzkin (SKM) algorithm which samples a random subset of Ξ²\beta rows of AA and then greedily selects the best row of the subset. Despite their variable computational costs, all three algorithms have been proven to have the same theoretical upper bound on the convergence rate. In this work, an improved analysis of the range of random (RK) to greedy (MM) methods is presented. This analysis improves upon previous known convergence bounds for SKM, capturing the benefit of partially greedy selection schemes. This work also further generalizes previous known results, removing the theoretical assumptions that Ξ²\beta must be fixed at every iteration and that AA must have normalized rows

    Relaxation, New Combinatorial and Polynomial Algorithms for the Linear Feasibility Problem

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    We consider the homogenized linear feasibility problem, to find an x on the unit sphere, satisfying n linear inequalities aT i x β‰₯ 0. To solve this problem we consider the centers of the insphere of spherical simplices, whose facets are determined by a subset of the constraints. As a result we find a new combinatorial algorithm for the linear feasibility problem. If we allow rescaling this algorithm becomes polynomial. We point out that the algorithm solves as well the more general convex feasibility problem. Moreover numerical experiments show that the algorithm could be of practical interest
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