Bregman primal–dual first-order method and application to sparse semidefinite programming

Abstract

We present a new variant of the Chambolle–Pock primal–dual algorithm with Bregman distances, analyze its convergence, and apply it to the centering problem in sparse semidefinite programming. The novelty in the method is a line search procedure for selecting suitable step sizes. The line search obviates the need for estimating the norm of the constraint matrix and the strong convexity constant of the Bregman kernel. As an application, 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.

Bregman three-operator splitting methods

“…The Bregman Condat-Vũ algorithms ( 21) and ( 22) can be viewed as applications of the Bregman proximal point algorithm to the optimality conditions (3). This interpretation extends the derivation of the Bregman PDHG algorithm from the Bregman proximal point algorithm given in [29]. The idea originates with He and Yuan's interpretation of PDHG as a "preconditioned" proximal point algorithm [27].…”

“…(see [7, Bregman prox-grad To show convergence of the entire sequence (x (k) , z (k) ), we substitute (x, ẑ) in (29):…”

“…The proposed backtracking procedure is similar to the technique in [35] for the special setting of PDHG with Euclidean proximal operators, but has some important differences even in this special case. We give a detailed analysis of the algorithm with line search and recover the O(1/k) ergodic rate of convergence for related algorithms in [35,29].…”

“…As a second benefit, a Bregman proximal operator of a function may be easier to compute than the standard Euclidean proximal operator, and therefore reduce the complexity per iteration of an optimization algorithm. Recent applications of this kind include optimal transport problems [16], optimization over nonnegative trigonometric polynomials [13], and sparse semidefinite programming [29].…”

Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization

“…Solutions of the regularized problems for fixed finite µ are usually found using Newton's method, leading to so-called primal scaling and dual scaling interiorpoint methods. Other methods can also be used; for instance, Jiang & Vandenberghe (2021) recently suggested solving (3.21) with a Bregman first-order method, where the complexity of evaluating the Bregman proximal operator can be reduced using a sparse Cholesky factorization. When Newton's method is applied to (3.21), the KKT optimality conditions are…”