Faster and Simpler Width-Independent Parallel Algorithms for Positive Semidefinite Programming

TL;DR

This paper introduces a faster, simpler parallel algorithm for positive semidefinite programming that is width-independent and nearly linear in the size of the input, improving efficiency for large-scale problems.

Contribution

It presents a new ext{NC} parallel algorithm for positive semidefinite programs with fewer iterations and simpler matrix operations, enhancing scalability and simplicity.

Findings

Requires $O(rac{1}{^3} \, \log^3 n)$ iterations

Each iteration involves simple matrix exponential and trace computations

Total work is nearly-linear in the size of the factorization

Abstract

This paper studies the problem of finding an $(1+\epsilon)$-approximate solution to positive semidefinite programs. These are semidefinite programs in which all matrices in the constraints and objective are positive semidefinite and all scalars are non-negative. We present a simpler \NC parallel algorithm that on input with $n$ constraint matrices, requires $O(\frac{1}{\epsilon^3} log^3 n)$ iterations, each of which involves only simple matrix operations and computing the trace of the product of a matrix exponential and a positive semidefinite matrix. Further, given a positive SDP in a factorized form, the total work of our algorithm is nearly-linear in the number of non-zero entries in the factorization.