An Inexact Tensor-Train Primal-Dual Interior-Point Method for Semidefinite Programs

Abstract

In this work, we introduce an interior-point method that employs tensor decompositions to efficiently represent and manipulate the variables and constraints of semidefinite programs, targeting problems where the solutions may not be low-rank but admit low-tensor-train rank approximations. Our method leverages a primal-dual infeasible interior-point framework with global and local convergence estimates. In experiments on Maximum Cut, Maximum Stable Set, and Correlation Clustering, the tensor-train interior-point method handles problems up to size $2^{12}$ with certified primal-dual accuracy, reported gaps of approximate order $10^{-4}$, and remains the only primal-dual SDP solver reported at the largest dimensions across all three benchmarks. Moreover, numerical evidence indicates that tensor-train ranks of the iterates typically remain structured and moderate along the interior-point trajectory, with instance-dependent growth for harder high-rank cases, supporting the scalability of the approach. Tensor-train interior-point methods offer a promising avenue for problems that lack traditional sparsity or low-rank structure, exploiting tensor-train structures instead.