## 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 the surprising result that the distribution of the convergence rate is a scaling function of a single variable. This scaling variable combines the convergence rate with the problem size (i.e., the number of variables and the number of constraints). We also estimate numerically the distribution of the computation time to an approximate solution, which is the time required to reach a vicinity of the attracting fixed point. We 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 to the approximate solution.

Original language | English |
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Pages (from-to) | 474-510 |

Number of pages | 37 |

Journal | Journal of Complexity |

Volume | 19 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2003 |

### Bibliographical note

Funding Information:It is our great pleasure to thank Arkadi Nemirovski, Eduardo Sontag and Ofer Zeitouni for stimulating and informative discussions. This research was supported in part by the US–Israel Binational Science Foundation (BSF), by the Israeli Science Foundation Grant Number 307/98 (090-903), by the US National Science Foundation under Grant No. PHY99-07949, by the Minerva Center of Nonlinear Physics of Complex Systems and by the fund for Promotion of Research at the Technion.

## Keywords

- Dynamical systems
- Linear programming
- Random Matrix Theory
- Scaling
- Theory of analog computation

## ASJC Scopus subject areas

- Algebra and Number Theory
- Statistics and Probability
- Numerical Analysis
- General Mathematics
- Control and Optimization
- Applied Mathematics