# Computing the Best Approximation Over the Intersection of a Polyhedral Set and the Doubly Nonnegative Cone

## TL;DR

This paper presents an efficient inexact accelerated block coordinate descent algorithm for computing the best approximation of matrices over the intersection of linear constraints and the doubly nonnegative cone, with proven convergence and strong numerical results.

## Contribution

It introduces a novel inexact accelerated two-block coordinate descent method utilizing semismooth Newton for nonsmooth dual problems involving the doubly nonnegative cone.

## Key findings

- The algorithm achieves an $O(1/k^2)$ convergence rate.
- Numerical experiments show superior performance on large-scale semidefinite programming problems.
- The method effectively handles the intersection of linear constraints and the doubly nonnegative cone.

## Abstract

This paper introduces an efficient algorithm for computing the best approximation of a given matrix onto the intersection of linear equalities, inequalities and the doubly nonnegative cone (the cone of all positive semidefinite matrices whose elements are nonnegative). In contrast to directly applying the block coordinate descent type methods, we propose an inexact accelerated (two-)block coordinate descent algorithm to tackle the four-block unconstrained nonsmooth dual program. The proposed algorithm hinges on the efficient semismooth Newton method to solve the subproblems, which have no closed form solutions since the original four blocks are merged into two larger blocks. The $O(1/k^2)$ iteration complexity of the proposed algorithm is established. Extensive numerical results over various large scale semidefinite programming instances from relaxations of combinatorial problems demonstrate the effectiveness of the proposed algorithm.