Solving Large-scale Quadratic Programs with Tensor-train-based Algorithms with Application to Trajectory Optimization

Abstract

In this paper, we develop an efficient interior point method (IPM) for convex quadratic programming (QP). The proposed algorithm keeps all relevant variables in a compressed format enabled by the tensor-train (TT) decomposition. The algorithm, called TT-IPM, requires much less memory, even when compared to IPMs that utilize sparse arrays, and is able to solve large-scale QPs. Under certain assumptions, we prove that the TT-IPM inherits the superlinear convergence property of traditional IPMs. Furthermore, the TT-IPM uses storage that scales polylogarithmically with the number of variables and the TT-ranks. Finally, we illustrate the computation time and storage savings of TT-IPM in a trajectory optimization problem. We show that even in fairly sparse optimization problems with percentage of nonzero elements in the problem data close to 0.0002 percent, the tensor-train representation allows TT-IPM to outperform state-of-the-art IPM that use sparse arrays.