{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:28:07Z","timestamp":1760243287550,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T00:00:00Z","timestamp":1664409600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Universidad Nacional de Colombia"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>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., SGoals having 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.<\/jats:p>","DOI":"10.3390\/a15100362","type":"journal-article","created":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T21:03:10Z","timestamp":1664485390000},"page":"362","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Non-Stationary Stochastic Global Optimization Algorithms"],"prefix":"10.3390","volume":"15","author":[{"given":"Jonatan","family":"Gomez","sequence":"first","affiliation":[{"name":"Departamento de Ingenier\u00eda de Sistemas e Industrial, Facultad de Ingenier\u00eda, Universidad Nacional de Colombia, Bogot\u00e1 11001, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5715-4420","authenticated-orcid":false,"given":"Andres","family":"Rivera","sequence":"additional","affiliation":[{"name":"Departamento de Ingenier\u00eda de Sistemas e Industrial, Facultad de Ingenier\u00eda, Universidad Nacional de Colombia, Bogot\u00e1 11001, Colombia"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1007\/s11047-021-09843-5","article-title":"On the class of hybrid adaptive evolutionary algorithms (chavela)","volume":"20","year":"2021","journal-title":"Nat. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"229","DOI":"10.15388\/Informatica.2016.83","article-title":"Stochastic global optimization: A review on the occasion of 25 years of Informatica","volume":"27","author":"Zhigljavsky","year":"2016","journal-title":"Informatica"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"T\u00f6rn, A., and \u017dilinskas, A. (1989). Global Optimization, Springer.","DOI":"10.1007\/3-540-50871-6"},{"key":"ref_4","first-page":"2","article-title":"The application of Bayesian methods for seeking the extremum","volume":"2","author":"Mockus","year":"1978","journal-title":"Towards Glob. Optim."},{"key":"ref_5","first-page":"101","article-title":"Function extremum search with the use of information maximum principle","volume":"27","author":"Neimark","year":"1966","journal-title":"Autom. Remote Control"},{"key":"ref_6","unstructured":"Zhigljavsky, A., and Zilinskas, A. (2007). Stochastic Global Optimization, Springer Science & Business Media."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"780","DOI":"10.1134\/S0965542507050053","article-title":"On the convergence rate of the Markov homogeneous monotone optimization method","volume":"47","author":"Tikhomirov","year":"2007","journal-title":"Comput. Math. Math. Phys."},{"key":"ref_8","first-page":"81","article-title":"Optimal random non-adaptive algorithm for global optimization of Brownian motion","volume":"8","author":"Calvin","year":"1996","journal-title":"J. Glob. Optim."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1162\/evco.1996.4.2.133","article-title":"Analysis of selection algorithms: A Markov chain approach","volume":"4","author":"Chakraborty","year":"1996","journal-title":"Evol. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1109\/4235.735430","article-title":"An evolutionary strategy for global minimization and its Markov chain analysis","volume":"2","year":"1998","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.ins.2018.09.021","article-title":"Stochastic global optimization algorithms: A systematic formal approach","volume":"472","author":"Gomez","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1017\/S0269964800003624","article-title":"Simulated annealing and adaptive search in global optimization","volume":"8","author":"Romeijn","year":"1994","journal-title":"Probab. Eng. Inform. Sci."},{"key":"ref_13","unstructured":"Weise, T. (2009). Global optimization algorithms-theory and application. Self-Publ. Thomas Weise, 361."},{"key":"ref_14","unstructured":"Russell, S., and Norvig, P. (2009). Artificial Intelligence: A Modern Approach, Prentice Hall Press. [3rd ed.]."},{"key":"ref_15","unstructured":"De Jong, K. (1975). An Analysis of the Behavior of a Class of Genetic Adaptive Systems. [Ph.D. Thesis, University of Michigan]."},{"key":"ref_16","unstructured":"Holland, J.H. (1975). Adaptation in Natural and Artificial Systems, The University of Michigan Press."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Mitchell, M. (1996). An Introduction to Genetic Algorithms, MIT Press.","DOI":"10.7551\/mitpress\/3927.001.0001"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Goldberg, D.E., and Deb, K. (1991). A comparative analysis of selection schemes used in genetic algorithms. Foundations of Genetic Algorithms, Morgan Kaufmann.","DOI":"10.1016\/B978-0-08-050684-5.50008-2"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1109\/TEVC.2010.2059031","article-title":"Differential Evolution: A Survey of the State-of-the-Art","volume":"15","author":"Das","year":"2011","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1023\/A:1008202821328","article-title":"Differential Evolution\u2013A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces","volume":"11","author":"Storn","year":"1997","journal-title":"J. Glob. Optim."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1126\/science.220.4598.671","article-title":"Optimization by Simulated Annealing","volume":"220","author":"Kirkpatrick","year":"1983","journal-title":"Science"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1023\/A:1015059928466","article-title":"Evolution strategies\u2013A comprehensive introduction","volume":"1","author":"Beyer","year":"2002","journal-title":"Nat. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1109\/4235.771166","article-title":"Parameter Control in Evolutionary Algorithms","volume":"3","author":"Eiben","year":"1999","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Rudolph, G. (1996). Convergence of Evolutionary Algorithms in General Search Spaces. Third IEEE Conference on Evolutionary Computation, IEEE Press.","DOI":"10.1109\/ICEC.1996.542332"},{"key":"ref_25","unstructured":"Simon, D. (2013). Evolutionary Optimization algorithms, John Wiley & Sons."},{"key":"ref_26","unstructured":"Bowerman, B.L. (1974). Nonstationary Markov Decision Processes and Related Topics in Nonstationary Markov Chains. [Ph.D. Thesis, University of Iowa]."},{"key":"ref_27","unstructured":"Royden, H.L., and Fitzpatrick, P. (2010). Real Analysis, Pearson Education Press. [4th ed.]."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1162\/evco.1996.4.4.361","article-title":"A Comparison of Selection Schemes Used in Evolutionary Algorithms","volume":"4","author":"Blickle","year":"1996","journal-title":"Evol. Comput."},{"key":"ref_29","first-page":"14","article-title":"Reducing bias and inefficiency in the selection algorithm","volume":"Volume 206","author":"Baker","year":"1987","journal-title":"Proceedings of the Second International Conference on Genetic Algorithms"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Angeline, P.J. (1994). Genetic Programming: On the Programming of Computers by Means of Natural Selection: John R. Koza, a Bradford Book, Elsevier.","DOI":"10.1016\/0303-2647(94)90062-0"},{"key":"ref_31","first-page":"116","article-title":"The GENITOR algorithm and selection pressure: Why rank-based allocation of reproductive trials is best","volume":"89","author":"Whitley","year":"1989","journal-title":"Icga"},{"key":"ref_32","first-page":"193","article-title":"Genetic Algorithms, Tournament Selection, and the Effects of Noise","volume":"9","author":"Miller","year":"1995","journal-title":"Complex Syst."},{"key":"ref_33","unstructured":"Koza, J. (1992). Genetic Programming: On the Programming of Computers by Means of Natural Selection, The MIT Press."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1162\/evco.1997.5.3.347","article-title":"Empirical investigation of multiparent recombination operators in evolution strategies","volume":"5","author":"Eiben","year":"1997","journal-title":"Evol. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1023\/A:1006504901164","article-title":"Tackling real-coded genetic algorithms: Operators and tools for behavioural analysis","volume":"12","author":"Herrera","year":"1998","journal-title":"Artif. Intell. Rev."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1016\/0360-8352(96)00037-X","article-title":"Evolutionary algorithms for constrained engineering problems","volume":"30","author":"Michalewicz","year":"1996","journal-title":"Comput. Ind. Eng."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/10\/362\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:41:54Z","timestamp":1760143314000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/10\/362"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,29]]},"references-count":36,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["a15100362"],"URL":"https:\/\/doi.org\/10.3390\/a15100362","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2022,9,29]]}}}