M2ICAL: A technique for analyzing imperfect comparison algorithms using Markov chains

Abstract

Practical optimization problems are often not well-defined, in the sense that the quality of a solution cannot be easily calculated. Many algorithms that attempt to solve such problems are comparison-based, i.e., they have a comparison function that compares two (or more) solutions and returns the superior one. Due to the difficulty in calculating the quality of a solution, the comparison function employed is imperfect (i.e., it may make an error). This thesis describes a 4-step process to model imperfect comparison algorithms into Markov Chains, using Monte Carlo simulations to handle the effects of comparison errors. We call this process the Monte Carlo Markov Chain for Imperfect Comparison ALgorithms method, or the M2ICAL method for short. Information that can be extracted from the model include the estimated solution quality after t iterations; the standard deviation of the solutions' quality; and the time to convergence.