Session: Moment-SOS Hierarchy: From Theory to Computation in the Real World (II)
Chair: Jie Wang
Cluster: Conic and Semidefinite Optimization
Talk 1: Bregman primal--dual first-order method and application to sparse semidefinite programming
Speaker: Xin Jiang
Abstract: We discuss the centering problem in large-scale semidefinite programming with sparse coefficient matrices. The logarithmic barrier function for the cone of positive semidefinite completable sparse matrices is used as the distance-generating kernel. For this distance, the complexity of evaluating the Bregman proximal operator is shown to be roughly proportional to the cost of a sparse Cholesky factorization. This is much cheaper than the standard proximal operator with Euclidean distances, which requires an eigenvalue decomposition. Then primal-dual proximal algorithm with Bregman distances are applied to solve large-scale sparse semidefinite programs efficiently.
Talk 2: Moment-SOS hierarchies for variational problems and PDE control
Speaker: Giovanni Fantuzzi
Abstract: Moment-SOS hierarchies are an established tool to compute converging sequences of lower bounds on the global minimum of finite-dimensional polynomial optimization problems. In this talk, I will show that they can be combined with finite-element discretizations to give a "discretize-then-relax" framework for solving two classes of infinite-dimensional problems: minimization problems for nonconvex integral functionals over functions from a Sobolev space, and some optimal control problems for nonlinear partial differential equations. For each class of problems, I will discuss conditions ensuring that this "discretize-then-relax" framework produces converging approximations (sometimes, bounds) to the global minimum and to the corresponding optimizer. Gaps between theory and practice will be illustrated by means of examples.
Talk 3: Inexact Augmented Lagrangian Methods for Semidefinite Optimization: Quadratic Growth and Linear Convergence
Speaker: Feng-Yi Liao
Abstract: Augmented Lagrangian Methods (ALMs) are widely employed in solving constrained optimizations, and some efficient solvers are developed based on this framework. Under the quadratic growth assumption, it is known that the dual iterates and the Karush–Kuhn–Tucker (KKT) residuals of ALMs applied to semidefinite programs (SDPs) converge linearly. In contrast, the convergence rate of the primal iterates has remained elusive. In this talk, we resolve this challenge by establishing new quadratic growth and error-bound properties for primal and dual SDPs under the strict complementarity condition. Our main results reveal that both primal and dual iterates of the ALMs converge linearly contingent solely upon the assumption of strict complementarity and a bounded solution set. This finding provides a positive answer to an open question regarding the asymptotically linear convergence of the primal iterates of ALMs applied to semidefinite optimization.
