Probabilistic analysis of a differential equation for linear programming

  • Authors:

  • Asa Ben-Hur;Joshua Feinberg;Shmuel Fishman;Hava T. Siegelmann

  • Affiliations:

  • Biochemistry Department, Stanford University, Stanford, CA and Faculty of Industrial Engineering and Management, Technion, Haifa 32000, Israel;Physics Department, University of Haifa at Oranim, Tivon 36006, Israel and Physics Department, Technion, Israel Institute of Technology, Haifa 32000, Israel;Physics Department, Technion, Israel Institute of Technology, Haifa 32000, Israel and Institute for Theoretical Physics, University of California, Santa Barbara, CA;Laboratory of Bio-computation, Department of Computer Science, University of Massachussets at Amherst, Amherst, MA

  • Venue:
  • Journal of Complexity
  • Year:
  • 2003

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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.