Efficient LP warmstarting for linear modifications of the constraint matrix

TL;DR

This paper introduces three efficient warm-start algorithms for solving linear programs with linearly changing constraints, leveraging eigenvalue and Schur decompositions to improve computational efficiency.

Contribution

The paper presents novel warm-starting algorithms for LPs with linearly varying constraints, utilizing eigenvalue and Schur decompositions for faster solutions.

Findings

Algorithms achieve overall complexity of O(pm^2+pmn)

The methods efficiently evaluate solutions for multiple constraint variations

Theoretical conditions for basis optimality are established.

Abstract

We consider the problem of computing the optimal solution and objective of a linear program under linearly changing linear constraints. The problem studied is given by $\min c^t x \text{ s.t } Ax + \lambda Dx \leq b$ where $\lambda$ belongs to a set of predefined values $\Lambda$. Based on the information given by a precomputed basis, we present three efficient LP warm-starting algorithms. Each algorithm is either based on the eigenvalue decomposition, the Schur decomposition, or a tweaked eigenvalue decomposition to evaluate the optimal solution and optimal objective of these problems. The three algorithms have an overall complexity $O(pm^2+pmn)$ where $m$ (resp. $n$) is the number of constraints (resp. variables) of the original problem and $p$ the number of values in $\Lambda$ after an initial preprocessing step. We also provide theorems related to the optimality conditions to verify when a basis is still optimal and a local bound on the objective.