We apply a probabilistic approach to study the computational complexity of analog computers which solve linear programming problems. We numerically analyze various ensembles of linear programming problems and obtain, for each of these ensembles, the probability distribution functions of certain quantities which measure the computational complexity, known as the convergence rate, the barrier and the computation time. We find that in the limit of very large problems these probability distributions are universal scaling functions. In other words, the probability distribution function for each of these three quantities becomes, in the limit of large problem size, a function of a single scaling variable, which is a certain composition of the quantity in question and the size of the system. Moreover, various ensembles studied seem to lead essentially to the same scaling functions, which depend only on the variance of the ensemble. These results extend analytical and numerical results obtained recently for the Gaussian ensemble, and support the conjecture that these scaling functions are universal.
Bibliographical noteFunding Information:
It is a great pleasure to thank our colleagues Asa Ben-Hur and Hava Siegelmann for very useful advice and discussions. This research was supported in part by the Shlomo Kaplansky Academic Chair, by the Technion-Haifa University Collaboration Fund, by the U.S.-Israel Binational Science Foundation (BSF), by the Israeli Science Foundation (ISF), and by the Minerva Center of Nonlinear Physics of Complex Systems.
ASJC Scopus subject areas
- General Physics and Astronomy
- Applied Mathematics
- Statistical and Nonlinear Physics
- Mathematical Physics