A Newton Augmented Lagrangian Method for Symmetric Cone Programming with Complexity Analysis

TL;DR

This paper introduces a Newton augmented Lagrangian method for symmetric cone programming that combines barrier functions and augmented Lagrangian techniques, achieving better complexity bounds and condition number properties, with demonstrated numerical improvements.

Contribution

The paper develops a novel Newton augmented Lagrangian method for symmetric cone programming, leveraging barrier functions for smoothing and achieving improved complexity and condition number bounds.

Findings

Achieves an $ ext{O}(1/\epsilon)$ complexity bound.

Condition numbers of Schur complement matrices are of order $ ext{O}(1/\mu)$, better than classical IPMs.

Numerical experiments show significant performance improvements over existing methods.

Abstract

Symmetric cone programming covers a broad class of convex optimization problems, including linear programming, second-order cone programming, and semidefinite programming. Although the augmented Lagrangian method (ALM) is well-suited for large-scale problems, its subproblems are often not twice continuously differentiable, preventing the direct use of classical Newton methods. To address this issue, we observe that barrier functions used in interior-point methods (IPMs) naturally serve as effective smoothing terms to alleviate such nonsmoothness. By combining the strengths of ALM and IPMs, we construct a novel augmented Lagrangian function and subsequently develop a Newton augmented Lagrangian (NAL) method. By leveraging the self-concordance property of the barrier function, the proposed method is shown to achieve an $\mathcal{O}(1/{\epsilon})$ complexity bound. In addition, a spectral analysis reveals that the condition numbers of the Schur complement matrices arising in the NAL method are of order $\mathcal{O}(1/{\mu})$, which is better than the $\mathcal{O}(1/{\mu^2})$ order of classical IPMs. This improvement is further illustrated by a heatmap of condition numbers. Numerical experiments conducted on standard benchmarks indicate that the NAL method exhibits significant performance improvements compared to several existing methods.