Non-Stationary Stochastic Global Optimization Algorithms

Studying the theoretical properties of optimization algorithms such as genetic algorithms and evolutionary strategies allows us to determine when they are suitable for solving a particular type of optimization problem. Such a study consists of three main steps. The first step is considering such algorithms as Stochastic Global Optimization Algorithms (SGOALs), i.e., iterative algorithm that applies stochastic operations to a set of candidate solutions. The second step is to define a formal characterization of the iterative process in terms of measure theory and define some of such stochastic operations as stationary Markov kernels (defined in terms of transition probabilities that do not change over time). The third step is to characterize non-stationary SGOALs, i.e., SGOALshaving stochastic operations with transition probabilities that may change over time. In this paper, we develop the third step of this study. First, we generalize the sufficient conditions convergence from stationary to non-stationary Markov processes. Second, we introduce the necessary theory to define kernels for arithmetic operations between measurable functions. Third, we develop Markov kernels for some selection and recombination schemes. Finally, we formalize the simulated annealing algorithm and evolutionary strategies using the systematic formal approach.