5 research outputs found
Relaxation, New Combinatorial and Polynomial Algorithms for the Linear Feasibility Problem
We consider the homogenized linear feasibility problem, to find an on the
unit sphere, satisfying line ar inequalities . 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
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
We give a simple and natural method for computing approximately optimal
solutions for minimizing a convex function over a convex set given by a
separation oracle. Our method utilizes the Frank--Wolfe algorithm over the cone
of valid inequalities of and subgradients of . Under the assumption that
is -Lipschitz and that contains a ball of radius and is
contained inside the origin centered ball of radius , using
iterations and calls to
the oracle, our main method outputs a point satisfying .
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
Stochastic iterative algorithms have gained recent interest in machine
learning and signal processing for solving large-scale systems of equations,
. One such example is the Randomized Kaczmarz (RK) algorithm, which acts
only on single rows of the matrix at a time. While RK randomly selects a
row of 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 rows
of 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
must be fixed at every iteration and that must have normalized
rows
Relaxation, New Combinatorial and Polynomial Algorithms for the Linear Feasibility Problem
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