A Parallel Linear-Constraint Active Set Method
E. Dov Neimand, Serban Sabau
TL;DR
This work introduces two parallel optimization algorithms for convex objectives: one constrained by linear inequalities in a Hilbert space and another constrained by a non-convex polyhedron in $\mathbb{R}^n$. It centers on a closed-form recursive expression for minima on $P$-cones $P_A$ and a parallel algorithm that explores affine spaces in order of codimension, reusing previous minimizers to prune the search and avoid unnecessary calls to a black-box minimizer. The authors establish necessary and sufficient criteria to identify the active affine space, prove correctness, and derive non-asymptotic parallel complexity bounds for both convex and non-convex settings, including memory considerations for $m_B$ and efficient representation of $P_A$. Numerical experiments on pseudo-random polyhedra demonstrate substantial practical speedups, with the fraction of spaces requiring the black-box minimizer shrinking as problem size grows. Overall, the paper contributes a scalable, assumption-light framework for parallel linear-inequality constrained optimization and its non-convex extension, with potential for significant impact in large-scale optimization tasks.
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 $ν(\cdot)$ 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 $O(ν(\langle \cdot,\cdot \rangle) + ν(\min_A f)))$ 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.