# Non-Stationary Stochastic Global Optimization Algorithms
## 1 Introduction  
In global optimization studies, a general question is which type of algorithms are more fit for which type of optimization problems, so, we can decide when to use an evolutionary algorithm to solve a class of optimization problems by understanding its theoretical properties and characterizing its observable behavior [1].  
According to [PERSON] and [PERSON] in [2], stochastic global optimization algorithms (SGoals in short) are inseparable from their presentation and analysis. Several researchers have done active development of the field from long time ago. Such as: [PERSON] and [PERSON] in [3], [PERSON] and [PERSON] in [4], or [PERSON] and [PERSON] in [5]. However, it remains an active field of research including mathematical analysis of problems.  
SGoalshave been also studied from a Markovian perspective: [PERSON] and [PERSON] in Sections 3.3 and 3.4 in [6] and [PERSON] in [7] studied the convergence rate of some homogeneous Markov monotone random search optimization algorithms. Also, [PERSON] et al. in [8] studied some random non-adaptive algorithms for finding the maximum of a continuous function on the unit interval. An analysis of selection algorithms was done by [PERSON] et al. in [9] and an analysis for evolutionary strategies for global minimization was described by [PERSON] in [10].  
As can be noticed, these studies do not use measure theory to formalize probabilistic concepts or are developed around a specific optimization problem. [PERSON] in [11], describes a formal and systematic approach for characterizing stochastic global optimization algorithms. There, the required theory of probability to characterize SGoalssuch as measure theory, Markov kernel, operations between kernels, products and conditions to studyconvergence is presented. In addition, it is proved that some algorithmic functions like projection and sort can be represented by kernels.  
Moreover, the notion of join-kernel has been introduced in that paper as a way to characterize the combination of stochastic methods. Also, it is defined a formal structure of an optimization space for studying SGoals. Finally, [PERSON] formalizes algorithms that have a next-population stochastic method, which does not change transition probabilities over time. Such algorithms can be viewed from the perspective of stationary Markov processes. This viewpoint applies among others for, standard versions of hill-climbing, parallel hill-climbing, steady-state genetic, generational genetic, and differential evolution algorithms.  
This work continues such a systematic formal approach. First, we review the theory done by [PERSON] in [11]. Next, we generalize the sufficient conditions convergence Lemma 71 in [11] from stationary to non-stationary Markov processes. Third, we develop arithmetic kernels to characterize arithmetic operations between measurable functions. We develop Markov kernels for some selection and recombination schemes. Finally, in order to show some applications of the concepts developed, we formalize both simulated-annealing and evolutionary-strategies using the systematic formal approach, which are classical algorithms and can be found several studies in the literature such as [PERSON] et al. in [12] or [PERSON] in [13].