Sparse Convex Optimization Toolkit: A Mixed-Integer Framework

TL;DR

This paper introduces SCOT, an open-source distributed solver for Sparse Convex Optimization problems, utilizing a mixed-integer approach and a novel algorithm to efficiently find exact solutions on distributed networks.

Contribution

The paper presents a new mixed-integer framework and the DiHOA algorithm for solving SCO problems efficiently in distributed computing environments.

Findings

SCOT outperforms existing MINLP solvers in benchmarks.

The DiHOA algorithm effectively solves large-scale SCO problems.

Distributed datasets benefit from the parallel capabilities of SCOT.

Abstract

This paper proposes an open-source distributed solver for solving Sparse Convex Optimization (SCO) problems over computational networks. Motivated by past algorithmic advances in mixed-integer optimization, the Sparse Convex Optimization Toolkit (SCOT) adopts a mixed-integer approach to find exact solutions to SCO problems. In particular, SCOT brings together various techniques to transform the original SCO problem into an equivalent convex Mixed-Integer Nonlinear Programming (MINLP) problem that can benefit from high-performance and parallel computing platforms. To solve the equivalent mixed-integer problem, we present the Distributed Hybrid Outer Approximation (DiHOA) algorithm that builds upon the LP/NLP based branch-and-bound and is tailored for this specific problem structure. The DiHOA algorithm combines the so-called single- and multi-tree outer approximation, naturally integrates a decentralized algorithm for distributed convex nonlinear subproblems, and utilizes enhancement techniques such as quadratic cuts. Finally, we present detailed computational experiments that show the benefit of our solver through numerical benchmarks on 140 SCO problems with distributed datasets. To show the overall efficiency of SCOT we also provide performance profiles comparing SCOT to other state-of-the-art MINLP solvers.