Efficient LP warmstarting for linear modifications of the constraint matrix
Guillaume Derval, Bardhyl Miftari, Damien Ernst, Quentin Louveaux
TL;DR
This paper addresses linear programs whose constraint matrix changes linearly with a parameter $\lambda$ by leveraging a precomputed optimal basis to warmstart multiple evaluations. It introduces three reformulations based on the decomposition of $E_B = A_B^{-1} D_B$: an eigenvalue-based approach, a Schur-based approach, and a randomized tweaked eigen-decomposition to ensure diagonalizability, all providing exact $x_B(\lambda)$ and $o^*(\lambda)$ for a set of $\lambda$ with total complexity $O(n^\omega + p m^2)$. The authors establish existence, validity, and optimality conditions for the basis across $\lambda$ and derive a local bound on the objective to enable a piecewise-linear surrogate over a continuous range of $\lambda$. Collectively, the methods offer efficient, provable sensitivity analysis and fast evaluation for parametric LPs with linearly varying constraint matrices, with practical implications for real-time or iterative decision-making under uncertainty.
Abstract
We consider the problem of computing the optimal solution and objective of a linear program under linearly changing linear constraints. More specifically, we want to compute the optimal solution of a linear optimization where the constraint matrix linearly depends on a paramater that can take p different values. 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(m^3 + pm^2) where m is the number of constraints of the original problem and p the number of values of the parameter that we want to evaluate. We also provide theorems related to the optimality conditions to verify when a basis is still optimal and a local bound on the objective.