2015/07/15 by Javier Peña, Javier Pena, Daniel Rodríguez +5 · 1 citation
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Complexity and Algorithms in Graphs #FOS: Mathematics #Optimization and Control (math.OC) #Stochastic Gradient Optimization Techniques #math.OC
paper · pdf · doi:10.48550/arxiv.1507.04073
openalex publication_date 2015/07/15 · arxiv created 2015/11/25 · arxiv updated 2015/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.