Logarithmic barriers for sparse matrix cones

TL;DR

This paper introduces efficient algorithms for computing gradients and Hessians of logarithmic barrier functions tailored for sparse semidefinite matrix cones, enhancing interior-point methods for large-scale sparse semidefinite programming.

Contribution

It develops novel large-scale algorithms based on multifrontal sparse Cholesky factorization for barrier functions in sparse semidefinite cones and their duals.

Findings

Algorithms enable efficient barrier evaluations for large sparse matrices.

Improved interior-point methods for sparse semidefinite programming.

Enhanced computational performance in nonsymmetric conic optimization.

Abstract

Algorithms are presented for evaluating gradients and Hessians of logarithmic barrier functions for two types of convex cones: the cone of positive semidefinite matrices with a given sparsity pattern, and its dual cone, the cone of sparse matrices with the same pattern that have a positive semidefinite completion. Efficient large-scale algorithms for evaluating these barriers and their derivatives are important in interior-point methods for nonsymmetric conic formulations of sparse semidefinite programs. The algorithms are based on the multifrontal method for sparse Cholesky factorization.