A Parallel Linear-Constraint Active Set Method

TL;DR

This paper introduces two parallel algorithms for convex optimization over linear and polyhedral constraints, leveraging constraint reduction and parallel processing to efficiently find optima without requiring initial feasible points.

Contribution

The paper presents novel parallel algorithms that optimize convex functions over linear and polyhedral constraints, capable of handling empty feasible spaces and improving computational efficiency.

Findings

Achieves optimal solutions in O(ν(⟨·,·⟩)) + ν(min_A f) time with sufficient threads.

Handles constrained spaces with empty interiors and no initial feasible point.

Recognizes infeasibility of the feasible region during the process.

Abstract

We present two parallel optimization algorithms for a convex function $f$. The first algorithm optimizes over linear inequality constraints in a Hilbert space, $\mathbb H$, and the second over a non convex polyhedron in $\mathbb R^n$. The algorithms reduce the inequality constraints to equality constraints, and garner information from subsets of constraints to speed up the process. Let $r \in \mathbb N$ be the number of constraints and $\nu (\cdot)$ be the time complexity of some process, then given enough threads, and information gathered earlier from subsets of the given constraints, we compute an optimal point of a polyhedral cone in $O(\nu (\langle \cdot,\cdot \rangle) + \nu (\min_A f)))$ for affine space $A$, the intersection of the faces of the cone. We then apply the method to all the faces of the polyhedron to find the linear inequality constrained optimum. The methods works on constrained spaces with empty interiors, furthermore no feasible point is required, and the algorithms recognize when the feasible space is empty. The methods iterate over surfaces of the polyhedron and the corresponding affine hulls using information garnered from previous iterations of superspaces to speed up the process.