Projections onto polyhedral sets: An improved finite step method and new distributed projection methods

Abstract

A polyhedral set is the intersection of a finite number of closed half-spaces. It is very difficult to obtain the projection of any point onto a general polyhedral set, especially when the polyhedral set is formed by a large number of closed half-spaces. In this work, we focus on the theoretical aspects of the projection problem itself and of related methods for solving it. The first part of this work is to systematically study various optimality conditions on the projection problem by using the projection theorem. The second part of this work is to design a safe and verifiable screening rule to improve the computational efficiency of Rutkowski's finite step method. In the third part of this work, we introduce a graph-based parameterized operator and prove its conical averagedness. We then propose the convergent scheme of the Krasnosel'skiĭ–Mann fixed point iteration of this operator to find the projection. We also point out that, if we take incidence matrices of graphs as decomposition matrices in the graph-based scheme, the scheme has satisfactory distributability. Several special connected graph networks are provided and under their guidance, new explicit distributed projection methods are shown. These graph-based distributed schemes and methods are also extended to solve the problem of projecting onto finitely generated cones.