On a parallelised diffusion induced stochastic algorithm with pure random search steps for global optimisation

Funding Information: Funding: For the second author, this work was undertaken with partial financial support of RFBR (Grant n. 19-01-00451). For the first and third author, this work was partially supported through the project of the Centro de Matemática e Aplicações, UID/MAT/00297/2020, financed by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology). The APC was by supported the New University of Lisbon through the PhD program in Statistics and Risk Management of the FCT Nova Faculty.We propose a stochastic algorithm for global optimisation of a regular function, possibly unbounded, defined on a bounded set with regular boundary; a function that attains its extremum in the boundary of its domain of definition. The algorithm is determined by a diffusion process that is associated with the function by means of a strictly elliptic operator that ensures an adequate maximum principle. In order to preclude the algorithm to be trapped in a local extremum, we add a pure random search step to the algorithm. We show that an adequate procedure of parallelisation of the algorithm can increase the rate of convergence, thus superseding the main drawback of the addition of the pure random search step.publishersversionpublishe

New bounds for ternary covering arrays using a parallel simulated annealing

A covering array (CA) is a combinatorial structure specified as a matrix of N rows and k columns over an alphabet on v symbols such that for each set of t columns every t-tuple of symbols is covered at least once. Given the values of t, k, and v, the optimal covering array construction problem (CAC) consists in constructing a CA (N; t, k, v) with the minimum possible value of N. There are several reported methods to attend the CAC problem, among them are direct methods, recursive methods, greedy methods, and metaheuristics methods. In this paper, There are three parallel approaches for simulated annealing: the independent, semi-independent, and cooperative searches are applied to the CAC problem. The empirical evidence supported by statistical analysis indicates that cooperative approach offers the best execution times and the same bounds as the independent and semi-independent approaches. Extensive experimentation was carried out, using 182 well-known benchmark instances of ternary covering arrays, for assessing its performance with respect to the best-known bounds reported previously. The results show that cooperative approach attains 134 new bounds and equals the solutions for other 29 instances. © 2012 Himer Avila-George et al.The authors thankfully acknowledge the computer resources and assistance provided by Spanish Supercomputing Network (TIRANT-UV). This research work was partially funded by the following projects: CONACyT 58554; Calculo de Covering Arrays; 51623-Fondo Mixto CONACyT; Gobierno del Estado de Tamaulipas.Avila-George, H.; Torres-Jimenez, J.; Hernández García, V. (2012). New bounds for ternary covering arrays using a parallel simulated annealing. Mathematical Problems in Engineering. 2012:1-19. doi:10.1155/2012/897027S119201

STUDIES ON STOCHASTIC ALGORITHMS INFORMATION CONTENT, PARALLELISATION AND DIFFUSION INDUCED STOCHASTIC ALGORITHMS FOR GLOBAL OPTIMISATION

This thesis presents the main results of two articles published by the authors in the field of stochastic optimization. We dedicated the chapter 1 to the article introduction to the information content of some stochastic algorithms written by Esquível, Machado, Krasii, and Mota, 2021. In this chapter, we formulate an optimization stochastic algorithm convergence theorem, of Solis and Wets type, and we show several instances of its application to concrete algorithms. In this convergence theorem the algorithm is a sequence of random variables and, in order to describe the increasing flow of information associated to this sequence we define a filtration – or flow of σ-algebras – on the probability space, depending on the sequence of random variables and on the function being optimized. We compare the flow of information of two convergent algorithms by comparing the associated filtrations by means of the Cotter distance of σ-algebras. The main result is that two convergent optimization algorithms have the same information content if both their limit minimization functions generate the full σ-algebra of the probability space. The article On a Parallelised Diffusion Induced Stochastic Algorithm with Pure Random Search Steps for Global Optimisation written by Esquível, Krasii, Mota, and Machado, 2021 was broken down into 2 chapters: the chapter 2 is related to parallelisation and the chapter 3 is related to Diffusion Induced Stochastic Algorithms. In the chapter 2 we show that an adequate procedure of parallelisation of the algorithm can increase the rate of convergence, thus superseding the main drawback of the addition of the pure random search step. Finally, in the chapter 3 we propose a stochastic algorithm for global optimisation of a regular function, possibly unbounded, defined on a bounded set with regular boundary; a function that attains its extremum in the boundary of its domain of definition. The algorithm is determined by a diffusion process that is associated with the function by means of a strictly elliptic operator that ensures an adequate maximum principle. In order to preclude the algorithm to be trapped in a local extremum, we add a pure random search step to the algorithm. As the two articles have their own introductions, we decided to create a glossary that together with the annexes and appendices, include the concepts, definitions and theorems that are relevant to the understanding of the thesis