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.
Highlights
Optimization methods based on Bregman distances offer the possibility of matching the Bregman distance to the structure in the problem, with the goal of reducing the complexity per iteration
The paper is motivated by the difficulty of exploiting sparsity in large-scale semidefinite programming in general and, for proximal methods, Research supported in part by NSF Grant ECCS 1509789
Research on proximal methods for semidefinite programming has been largely based on the standard Euclidean proximal operators and the distance defined by the matrix entropy [6]
Summary
Optimization methods based on Bregman distances offer the possibility of matching the Bregman distance to the structure in the problem, with the goal of reducing the complexity per iteration. We show that if the Bregman divergence generated by the barrier function for the cone K is used, the generalized projections can be computed very efficiently, with a complexity dominated by the cost of a sparse Cholesky factorization with sparsity pattern E This is much cheaper than the eigenvalue decomposition needed to compute a Euclidean projection on the positive semidefinite cone. These properties are leveraged in the dual interior-point methods described in [9,10,11,12,13] In another line of research, techniques based on properties and algorithms for chordal sparsity patterns have been applied to semidefinite programming since the late 1990s [3, 13, 18, 29, 30, 34, 35, 42, 46, 50, 51, 58]; see [54, 60] for recent surveys.
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