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

Mehdi Karimi, Levent Tuncel

TL;DR

DDS 3.0, a new release of the Domain-Driven Solver, is presented, which provides an efficient interior-point implementation tailored for hyperbolic programming, and a new straight-line program (SLP) representation is introduced that enables compact representation and efficient computation of hyperbolic polynomials, their gradients, and Hessians.

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.

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

TL;DR

DDS 3.0, a new release of the Domain-Driven Solver, is presented, which provides an efficient interior-point implementation tailored for hyperbolic programming, and a new straight-line program (SLP) representation is introduced that enables compact representation and efficient computation of hyperbolic polynomials, their gradients, and Hessians.

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.
Paper Structure (19 sections, 5 theorems, 53 equations, 2 figures, 7 tables, 2 algorithms)

This paper contains 19 sections, 5 theorems, 53 equations, 2 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Different formats for Hyperbolic polynomials
  4. Derivative Evaluations via Straight-Line Programs (SLP)
  5. Hyperbolic Polynomials as Straight Line Programs
  6. Recursive Formula for Elementary Symmetric Polynomials
  7. Hyperbolic polynomials from matriods
  8. Derivatives of hyperbolic polynomials
  9. Product of linear forms
  10. Composition of hyperbolic polynomials
  11. Domain-Driven solver and a new corrector step
  12. Numerical results for Hyperbolic Polynomials
  13. Minimizing entropy combined with HP programming
  14. Projection onto the hyperbolicity cone
  15. DDS versus SDP relaxations and homotopy method
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $p(\pmb{x})$ be a homogeneous polynomial of degree $d$, which is hyperbolic in direction $\pmb{e}$. Then, the function $-\textup{ln}(p(\pmb{x}))$ is a $d$-logarithmicaly homogenous s.c. barrier for $\Lambda_+(p,\pmb{e})$.

Figures (2)

  • Figure 1: The SLP graph structure for the function $p(\pmb{x})=x_1^2 - x_2^2 - x_3^2$
  • Figure 2: Comparing DDS with the results presented in klingler2025homotopy-Figure 5 for the homotopy method and the SDP relaxation. As can be seen, DDS clearly outperforms both methods.

Theorems & Definitions (8)

  • Theorem 1.1: Güler guler1997hyperbolic
  • Theorem 3.1
  • proof
  • Definition 4.1
  • Lemma 4.1
  • Theorem 4.1
  • Theorem 4.2: bauschke2001hyperbolic-Theorem 3.1
  • Example 6.1