Efficient LP warmstarting for linear modifications of the constraint matrix

2025-01-07

AI Summary

This paper addresses parametric multiple-instance linear programming (LP), where the constraint matrix varies linearly across $p$ parameter values and the problem has $m$ constraints. We propose three matrix-decomposition-based warm-start algorithms—novel applications of eigenvalue decomposition, Schur decomposition, and an enhanced eigen-decomposition for basis reuse—with time complexity $O(m^3 + p m^2)$. Theoretical contributions include a basis optimality verification theorem, local bounds on the objective function, and a sufficient condition guaranteeing successful basis reuse. Experiments demonstrate near cubic-to-quadratic speedups over cold-start reoptimization across instances, substantially reducing computational overhead. The approach is particularly effective in optimization settings requiring repeated solution of similar LP instances.

Technology Category

• Machine Learning: Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Linear Programming
• Parameterized Constraints
• Optimization

Innovation

• Linear Programming
• Parameterized Algorithms
• Eigenvalue Decomposition