Replacing projection on finitely generated convex cones with projection on bounded polytopes

TL;DR

This paper introduces an efficient method for projecting onto finitely generated convex cones by reformulating the problem as projection onto a bounded polytope, leveraging specialized algorithms for improved computational performance.

Contribution

It proposes a novel approach to cone projection using bounded polytopes and demonstrates its efficiency with a specialized algorithm outperforming traditional quadratic programming methods.

Findings

The new method reduces computational complexity for cone projections.

Specialized algorithms significantly outperform general quadratic programming.

Numerical experiments confirm the approach's efficiency and scalability.

Abstract

This paper is devoted to the general problem of projection onto a polyhedral convex cone generated by a finite set of generators.This problem is reformulated into projection onto the polytope obtained by simple truncation of the original cone. Then it can be solved with just two closely related projections onto the same bounded polytope. This approach's computational performance is conditioned by the crucial tool's efficiency for solving the fundamental problem of finding the least norm element in a convex hull of a given finite set of points. In our numerical experiments, we used for this purpose the specialized finite algorithm implemented in the open-source system for matrix-vector calculations octave. This algorithm is practically indifferent to the proportions between the number of generators and their dimensionality and significantly outperformed a general-purpose quadratic programming algorithm of the active-set variety built into octave when the number of points exceeds the dimensionality of the original problem.