Session: Recent advances in matrix optimization
Chair: Chao Ding
Cluster: Conic and Semidefinite Optimization
Talk 1: A quadratically convergent semismooth Newton method for nonlinear semidefinite programming without subdifferential regularity
Speaker: Fuxiaoyue Feng
Abstract: The non-singularity of generalized Jacobians of the Karush-Kuhn-Tucker (KKT) system is crucial for local convergence analysis of semismooth Newton methods. In this talk, we present a new approach that challenges this conventional requirement. Our discussion revolves around a methodology that leverages some newly developed variational properties, effectively bypassing the necessity for non-singularity of all elements in the generalized Jacobian. Quadratic convergence results of our Newton methods are established without relying on commonly assumed subdifferential regularity conditions. This discussion may offer fresh insights into semismooth Newton methods, potentially paving the way for designing robust and efficient second-order algorithms for general nonsmooth composite optimizations.
Talk 2: On efficient and scalable computation of the nonparametric maximum likelihood estimator in mixture models
Speaker: Yangjing Zhang
Abstract: The nonparametric maximum likelihood estimation (NPMLE) is a classic and important method to estimate the mixture models from finite observations. In this talk, we propose an efficient and scalable semismooth Newton based augmented Lagrangian method (ALM). By carefully exploring the structure of the ALM subproblem, we show that the computational cost of the generalized Hessian (second order information) is independent of the number of grid points. Extensive numerical experiments are conducted to show the effectiveness of our approach.
Talk 3: An Accelerated Proximal ADMM for ODC of Uncertain Systems
Speaker: Xinyuan Zhao
Abstract: To ensure the system stability of the H2-guaranteed cost optimal decentralized control (ODC) problem, we formulate an approximate semidefinite programming (SDP) problem that leverages the block diagonal structure of the decentralized controller's gain matrix. To minimize data storage requirements and enhance computational efficiency, we employ the Kronecker product to vectorize the SDP problem into a conic programming (CP) problem. We then propose a proximal alternating direction method of multipliers (PADMM) to solve the dual of the resulting CP problem. By using the equivalence between the semi-proximal ADMM and the (partial) proximal point algorithm, we identify the non-expansive operator of PADMM, enabling the use of Halpern fixed-point iteration to accelerate the algorithm. Finally, we demonstrate that the sequence generated by the proposed accelerated PADMM exhibits a fast convergence rate for the Karush-Kuhn-Tucker residual. Numerical experiments confirm that the accelerated algorithm outperforms the well-known COSMO, MOSEK, and SCS solvers in efficiently solving large-scale CP problems, particularly those arising from H2-guaranteed cost ODC problems.
