Solving SDP Faster: A Robust IPM Framework and Efficient Implementation

This paper introduces a new robust interior point method analysis for semidefinite programming (SDP). This new robust analysis can be combined with either logarithmic barrier or hybrid barrier. Under this new framework, we can improve the running time of semidefinite programming (SDP) with variable size $n\times n$ and m constraints up to $\epsilon$ accuracy. We show that for the case $m=\Omega(n^{2})$, we can solve SDPs in $m^{\omega}$ time. This suggests solving SDP is nearly as fast as solving the linear system with equal number of variables and constraints. This is the first result that tall dense SDP can be solved in the nearly-optimal running time, and it also improves the stateof-the-art SDP solver [Jiang, Kathuria, Lee, Padmanabhan and Song, FOCS 2020]. In addition to our new IPM analysis, we also propose a number of techniques that might be of further interest, such as, maintaining the inverse of a Kronecker product using lazy updates, a general amortization scheme for positive semi-definite matrices.

Authors

• Baihe Huang
• Shunhua Jiang
• Zhao Song 0002
• Runzhou Tao 0001
• Ruizhe Zhang 0001

Keywords

• Linear systems
• Computer science
• Convex functions
• Interior Point Method
• Semidefinite Programming
• Running Time
• Interior Point
• Kronecker Product
• Positive Semidefinite Matrix
• Positive Semidefinite
• Barrier Function
• Symmetric Matrix
• Generative Adversarial Networks
• Matrix Multiplication
• Iteration Step
• Feasible Set
• Matrix Estimation
• Hessian Matrix
• Dual Space
• Straightforward Implementation
• Spectral Norm
• Robust Learning
• Cutting-plane
• Central Path
• Theoretical Computer Science
• Symmetric Positive Semidefinite Matrix
• Semi definite programming