Worst-case analysis of restarted primal-dual hybrid gradient on totally unimodular linear programs

TL;DR

This paper provides a worst-case complexity analysis of restarted primal-dual hybrid gradient (PDHG) algorithms applied to totally unimodular linear programs, establishing bounds on the number of matrix-vector multiplications needed for ε-optimal solutions.

Contribution

The paper introduces a novel worst-case analysis of restarted PDHG for totally unimodular LPs, deriving explicit bounds based on problem parameters.

Findings

Restarted PDHG finds ε-optimal solutions in O( H m_1^{2.5} √nnz(A) log(H m_2 / ε) ) matrix-vector multiplications.

The analysis links algorithm complexity to problem-specific parameters such as matrix sparsity and coefficient bounds.

Provides theoretical guarantees for the efficiency of PDHG on a broad class of linear programs.

Abstract

We analyze restarted PDHG on totally unimodular linear programs. In particular, we show that restarted PDHG finds an $\epsilon$-optimal solution in $O( H m_1^{2.5} \sqrt{\textbf{nnz}(A)} \log(H m_2 /\epsilon) )$ matrix-vector multiplies where $m_1$ is the number of constraints, $m_2$ the number of variables, $\textbf{nnz}(A)$ is the number of nonzeros in the constraint matrix, $H$ is the largest absolute coefficient in the right hand side or objective vector, and $\epsilon$ is the distance to optimality of the outputted solution.