A Newton Augmented Lagrangian Method for Symmetric Cone Programming with Complexity Analysis
Rui-Jin Zhang, Ruoyu Diao, Xin-Wei Liu, Yu-Hong Dai
TL;DR
A novel augmented Lagrangian function is constructed and a Newton augmented Lagrangian (NAL) method is developed, shown to achieve an $\mathcal{O}(1/{\epsilon})$ complexity bound.
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 $\mathcal{O}(1/ε)$ 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 $\mathcal{O}(1/μ)$, which is better than the $\mathcal{O}(1/{μ^2})$ 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.