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Exploiting cone approximations in an augmented Lagrangian method for conic optimization

Mituhiro Fukuda, Walter Gómez, Gabriel Haeser, Leonardo Makoto Mito

TL;DR

This work tackles nonlinear conic programming where projecting onto the full cone $\mathcal{K}$ is costly or intractable, by developing an augmented Lagrangian method that updates cheap polyhedral outer approximations $\{\mathcal{K}^k\}$ of $\mathcal{K}$ during iterations. It extends sequential optimality concepts to nonlinear conic settings via a relaxed gradient projection (R-AGP) framework and proves that feasible limit points satisfy R-AGP, with KKT guaranteed under Robinson's constraint qualification. The authors instantiate the approach on nonlinear copositive programming using Yıldırım's polyhedral outer approximations, showing that gradually refining the cone approximation outperforms standard fixed-approximation schemes in numerical experiments over a suite of test problems. The method enables practical solving of hard conic problems when the full cone is difficult to project onto, and the framework is adaptable to other convex cones beyond COP. Overall, the paper contributes a convergent, cone-approximation-based strategy that blends theoretical guarantees with empirical efficiency for nonlinear conic optimization.

Abstract

We propose an algorithm for general nonlinear conic programming which does not require the knowledge of the full cone, but rather a simpler, more tractable, approximation of it. We prove that the algorithm satisfies a strong global convergence property in the sense that it generates a strong sequential optimality condition. In particular, a KKT point is necessarily found when a limit point satisfies Robinson's condition. We conduct numerical experiments minimizing nonlinear functions subject to a copositive cone constraint. In order to do this, we consider a well known polyhedral approximation of this cone by means of refining the polyhedral constraints after each augmented Lagrangian iteration. We show that our strategy outperforms the standard approach of considering a close polyhedral approximation of the full copositive cone in every iteration.

Exploiting cone approximations in an augmented Lagrangian method for conic optimization

TL;DR

This work tackles nonlinear conic programming where projecting onto the full cone is costly or intractable, by developing an augmented Lagrangian method that updates cheap polyhedral outer approximations of during iterations. It extends sequential optimality concepts to nonlinear conic settings via a relaxed gradient projection (R-AGP) framework and proves that feasible limit points satisfy R-AGP, with KKT guaranteed under Robinson's constraint qualification. The authors instantiate the approach on nonlinear copositive programming using Yıldırım's polyhedral outer approximations, showing that gradually refining the cone approximation outperforms standard fixed-approximation schemes in numerical experiments over a suite of test problems. The method enables practical solving of hard conic problems when the full cone is difficult to project onto, and the framework is adaptable to other convex cones beyond COP. Overall, the paper contributes a convergent, cone-approximation-based strategy that blends theoretical guarantees with empirical efficiency for nonlinear conic optimization.

Abstract

We propose an algorithm for general nonlinear conic programming which does not require the knowledge of the full cone, but rather a simpler, more tractable, approximation of it. We prove that the algorithm satisfies a strong global convergence property in the sense that it generates a strong sequential optimality condition. In particular, a KKT point is necessarily found when a limit point satisfies Robinson's condition. We conduct numerical experiments minimizing nonlinear functions subject to a copositive cone constraint. In order to do this, we consider a well known polyhedral approximation of this cone by means of refining the polyhedral constraints after each augmented Lagrangian iteration. We show that our strategy outperforms the standard approach of considering a close polyhedral approximation of the full copositive cone in every iteration.
Paper Structure (10 sections, 3 theorems, 16 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 10 sections, 3 theorems, 16 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations
  3. General framework
  4. An augmented Lagrangian variant with projections onto $\mathcal{K}^k$
  5. Continuous (polyhedral) approximation of the cone of co-po-si-ti-ve matrices
  6. Numerical Experiments
  7. Test problems
  8. Details of the implementation
  9. Numerical results
  10. Concluding Remarks

Key Result

Lemma 2.2

Let $\mathcal{K}$ and $\{\mathcal{K}^k\}_{k\in \mathbb{N}}$ be nonempty closed convex cones in $\mathbb{E}$. Then:

Figures (3)

  • Figure 1: Performance profile for $m=3$ when solving 14 problems by the "proposed" method for values of $\zeta=15, 25, 35, 45, 55, 65,$ and $75$. It shows that $\zeta=45$ is the best choice.
  • Figure 2: Performance profile for $m=5$ when solving 14 problems by the "proposed" method for values of $\zeta=30, 50, 70, 90,$ and $110$. It shows that $\zeta=70$ is preferred.
  • Figure 3: Performance profile for $m=3$ (left) and $m=5$ (right) when solving 14 problems by the "proposed" method and the "standard" augmented Lagrangian method.

Theorems & Definitions (5)

  • Definition 2.1: Continuous approximation of $\mathcal{K}$
  • Lemma 2.2
  • Definition 2.3: R-AGP
  • Theorem 2.4
  • Theorem 2.5