Efficient Interior-Point Methods for Hyperbolic Programming via Straight-Line Programs

TL;DR

This paper introduces DDS 3.0, an efficient interior-point solver for hyperbolic programming that leverages a novel straight-line program representation to reduce computational costs and improve scalability.

Contribution

The paper presents a new straight-line program approach and an improved corrector step, enabling faster and more stable interior-point methods for large-scale hyperbolic programs.

Findings

Significant performance improvements over existing methods.

Efficient computation of hyperbolic polynomials, gradients, and Hessians.

Enhanced stability and convergence in numerical experiments.

Abstract

Hyperbolic (HB) programming generalizes many popular convex optimization problems, including semidefinite and second-order cone programming. Despite substantial theoretical progress on HB programming, efficient computational tools for solving large-scale hyperbolic programs remain limited. This paper presents DDS 3.0, a new release of the Domain-Driven Solver, which provides an efficient interior-point implementation tailored for hyperbolic programming. A key innovation lies in a new straight-line program (SLP) representation that enables compact representation and efficient computation of hyperbolic polynomials, their gradients, and Hessians. The SLP structure significantly reduces computational cost, allowing the Hessian to be computed in the same asymptotic complexity as the gradient through a batched reverse-over-forward differentiation scheme. We further introduce an improved corrector step for the primal-dual interior-point method, enhancing stability and convergence on convex sets where only the primal self-concordant barrier is efficiently computable. We create a comprehensive benchmark library beyond the elementary symmetric polynomials, using several different techniques. Numerical experiments demonstrate substantial performance gains of DDS 3.0 compared to first-order Frank-Wolfe algorithm, homotopy method, and SDP software utilizing SDP relaxations.