2025/06/05 by Ruijin Zhang, Ruoyu Diao, Zhang, Rui-Jin +5
Mathematics · Computer Science · #Advanced Optimization Algorithms Research #Stochastic Gradient Optimization Techniques #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2506.04802
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 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 O(1/μ), which is better than the O(1/μ2) 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.