Probabilistic analysis of a differential equation for linear programming

TL;DR

This paper analyzes the probabilistic convergence behavior of differential equations used in linear programming, deriving asymptotic distributions and bounds for convergence rates and computation times under Gaussian input assumptions.

Contribution

It introduces a novel probabilistic framework using Random Matrix Theory to characterize the convergence properties of differential equations in linear programming.

Findings

Distribution of convergence rates is a scaling function of problem parameters.

Distribution of computation times also follows a scaling law.

High probability bounds on convergence and computation times are derived.

Abstract

In this paper we address the complexity of solving linear programming problems with a set of differential equations that converge to a fixed point that represents the optimal solution. Assuming a probabilistic model, where the inputs are i.i.d. Gaussian variables, we compute the distribution of the convergence rate to the attracting fixed point. Using the framework of Random Matrix Theory, we derive a simple expression for this distribution in the asymptotic limit of large problem size. In this limit, we find that the distribution of the convergence rate is a scaling function, namely it is a function of one variable that is a combination of three parameters: the number of variables, the number of constraints and the convergence rate, rather than a function of these parameters separately. We also estimate numerically the distribution of computation times, namely the time required to reach a vicinity of the attracting fixed point, and find that it is also a scaling function. Using the problem size dependence of the distribution functions, we derive high probability bounds on the convergence rates and on the computation times.