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A conditional gradient homotopy method with applications to Semidefinite Programming

Pavel Dvurechensky, Gabriele Iommazzo, Shimrit Shtern, Mathias Staudigl

TL;DR

The paper addresses solving large-scale conic-constrained convex programs, notably SDP relaxations, by introducing a projection-free, double-loop homotopy method. It couples a barrier-based inner loop, solved with a conditional gradient (CG) step to approximate the analytic central path, with an outer loop that updates the homotopy parameter $t$ and the accuracy to trace the central path toward the optimum. The authors establish iteration-complexity guarantees that are competitive with state-of-the-art SDP solvers, and extend the framework with line-search and inexact LMO variants, achieving robust performance under ill-conditioning and scaling. Numerical experiments on SDP relaxations and mixing-time estimation demonstrate both theoretical soundness and practical robustness to constraint scaling, with the line-search and inexact variants offering tangible gains.

Abstract

We propose a new homotopy-based conditional gradient method for solving convex optimization problems with a large number of simple conic constraints. Instances of this template naturally appear in semidefinite programming problems arising as convex relaxations of combinatorial optimization problems. Our method is a double-loop algorithm in which the conic constraint is treated via a self-concordant barrier, and the inner loop employs a conditional gradient algorithm to approximate the analytic central path, while the outer loop updates the accuracy imposed on the temporal solution and the homotopy parameter. Our theoretical iteration complexity is competitive when confronted to state-of-the-art SDP solvers, with the decisive advantage of cheap projection-free subroutines. Preliminary numerical experiments are provided for illustrating the practical performance of the method.

A conditional gradient homotopy method with applications to Semidefinite Programming

TL;DR

The paper addresses solving large-scale conic-constrained convex programs, notably SDP relaxations, by introducing a projection-free, double-loop homotopy method. It couples a barrier-based inner loop, solved with a conditional gradient (CG) step to approximate the analytic central path, with an outer loop that updates the homotopy parameter and the accuracy to trace the central path toward the optimum. The authors establish iteration-complexity guarantees that are competitive with state-of-the-art SDP solvers, and extend the framework with line-search and inexact LMO variants, achieving robust performance under ill-conditioning and scaling. Numerical experiments on SDP relaxations and mixing-time estimation demonstrate both theoretical soundness and practical robustness to constraint scaling, with the line-search and inexact variants offering tangible gains.

Abstract

We propose a new homotopy-based conditional gradient method for solving convex optimization problems with a large number of simple conic constraints. Instances of this template naturally appear in semidefinite programming problems arising as convex relaxations of combinatorial optimization problems. Our method is a double-loop algorithm in which the conic constraint is treated via a self-concordant barrier, and the inner loop employs a conditional gradient algorithm to approximate the analytic central path, while the outer loop updates the accuracy imposed on the temporal solution and the homotopy parameter. Our theoretical iteration complexity is competitive when confronted to state-of-the-art SDP solvers, with the decisive advantage of cheap projection-free subroutines. Preliminary numerical experiments are provided for illustrating the practical performance of the method.
Paper Structure (31 sections, 9 theorems, 112 equations, 3 figures, 5 algorithms)

This paper contains 31 sections, 9 theorems, 112 equations, 3 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Self-Concordant Functions
  4. The Optimization Problem
  5. Algorithm
  6. The Proposed CndG Solver
  7. Updating the Homotopy Parameter
  8. Modifications and Extensions
  9. Line Search
  10. Algorithm with inexact LMO
  11. Updating the Homotopy Parameter in the Inexact Case
  12. Complexity Analysis
  13. Analysis of procedure $\mathop{\mathrm{\texttt{CG}}}\nolimits(x^{0},\varepsilon,t)$
  14. Proof of Proposition \ref{['prop:Delta']}
  15. Proof of Proposition \ref{['prop:gap']}
  16. ...and 16 more sections

Key Result

Lemma 2.2

For all $t>0$, it holds that $z^{\ast}(t)\in\mathsf{C}$. In particular,

Figures (3)

  • Figure 1: Mixing time problem: relative gap from SDPT3 solution vs. iteration.
  • Figure 2: Randomly scaled SDP problem: relative optimality gap vs. iterations for several parameter choices (averaged over 30 instances for each $p$)
  • Figure 3: Randomly scaled SDP problem: relative optimality gap vs. time for several parameter choices (averaged over 30 instances for each $p$)

Theorems & Definitions (16)

  • Example 2.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Remark 3.1
  • Definition 4.1
  • Proposition 4.2
  • ...and 6 more