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On the von Neumann and Frank-Wolfe Algorithms with Away Steps

Javier Pena (jfp***at***andrew.cmu.edu)
Daniel Rodriguez (drod***at***cmu.edu)
Negar Soheili (nazad***at***uic.edu)

Abstract: The von Neumann algorithm is a simple coordinate-descent algorithm to determine whether the origin belongs to a polytope generated by a finite set of points. When the origin is in the interior of the polytope, the algorithm generates a sequence of points in the polytope that converges linearly to zero. The algorithm's rate of convergence depends on the radius of the largest ball around the origin contained in the polytope. We show that under the weaker condition that the origin is in the polytope, possibly on its boundary, a variant of the von Neumann algorithm that includes generates a sequence of points in the polytope that converges linearly to zero. The new algorithm's rate of convergence depends on a certain geometric parameter of the polytope that extends the above radius but is always positive. Our linear convergence result and geometric insights also extend to a variant of the Frank-Wolfe algorithm with away steps for minimizing a strongly convex function over a polytope.

Keywords: Von Neumann, Frank-Wolfe, Away Steps, Linear Convergence, Coordinate Descent

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation: Technical Report, Tepper School, Carnegie Mellon University

Download: [PDF]

Entry Submitted: 07/14/2015
Entry Accepted: 07/14/2015
Entry Last Modified: 07/16/2015

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