Finite Horizon Optimization: Framework and Applications

TL;DR

This paper introduces a finite horizon optimization framework for algorithm performance under strict iteration limits, applying it to linear programming and proposing a novel convex SDP-based stepsize rule that accelerates convergence in real-world instances.

Contribution

The work develops a finite horizon optimization framework, reveals hidden convexity in stepsize design for primal-dual methods, and proposes an efficient SDP reformulation for practical acceleration.

Findings

Achieves an average 3.9× speed-up over optimal constant stepsize on LP instances.

Provides theoretical acceleration guarantees at fixed iteration T.

Demonstrates significant real-world computational time savings.

Abstract

In modern engineering scenarios, there is often a strict upper bound on the number of algorithm iterations that can be performed within a given time limit. This raises the question of optimal algorithmic configuration for a fixed and finite iteration budget. In this work, we introduce the framework of finite horizon optimization, which focuses on optimizing the algorithm performance under a strict iteration budget $T$. We apply this framework to linear programming (LP) and propose Finite Horizon stepsize rule for the primal-dual method. The main challenge in the stepsize design is controlling the singular values of $T$ cumulative product of non-symmetric matrices, which appears to be a highly nonconvex problem, and there are very few helpful tools. Fortunately, in the special case of the primal-dual method, we find that the optimal stepsize design problem admits hidden convexity, and we propose a convex semidefinite programming (SDP) reformulation. This SDP only involves matrix constraints of size $4 \times 4$ and can be solved efficiently in negligible time. Theoretical acceleration guarantee is also provided at the pre-fixed $T$-th iteration, but with no asymptotic guarantee. On more than 90 real-world LP instances, Finite Horizon stepsize rule reaches an average 3.9$\times$ speed-up over the optimal constant stepsize, saving 75\% wall-clock time. Our numerical results reveal substantial room for improvement when we abandon asymptotic guarantees, and instead focus on the performance under finite horizon. We highlight that the benefits are not merely theoretical - they translate directly into computational speed-up on real-world problems.