Fitness threshold accepting over extremal optimization ranks.

We treat the problem of selecting the next degree of freedom for update in an extremal optimization algorithm designed to find the ground state of a system with a complex energy landscape. We show that there exists a best distribution for selecting the next degree of freedom in order to optimize any linear function of the state probabilities, e.g., the expected number of visits to the ground state. We dub the class of algorithms using this best distribution in conjunction with extremal optimization fitness threshold accepting. In addition, we construct an extended random walk and use it to show that fitness threshold accepting is optimal also for several other measures of algorithm performance, such as maximizing the expected probability of seeing the ground state and minimizing the expected value of the lowest energy seen.