Plenary Lecture

Finding All the Minimizers of Higly Multimodal Functions by using a Monte-Carlo Technique

Professor Andre A. Keller
Laboratoire d'Informatique Fondamentale de Lille/SMAC, UMR 8022/CNRS
Université de Lille Nord de France
France
E-mail: Andre.Keller@univ-lille1.fr

Abstract: Multimodal benchmark functions are essential for testing and comparing the effectiveness of global meta-heuristic optimization methods, such as with the genetic algorithms or the particle swarm optimization methods. In many test-functions, a large number of local minimizers coexist with a single (or few) global minimizer(s). Anyway, this situation increases the difficulty to find the global optimum points. Finding the local minimizers may be also profitable in the real-world problems, since some local minimizers may be a “best” choice for economic or computational cost reasons. The two-dimensional Shubert test functions with box constraints from B.O. Shubert (1972) belong to a cosine class with a high number of minimizers. The Shubert function I has 760 minimum points, 18 of which are global optimum points in 9 regular distributed clusters. However, the Shubert function III with additive square expressions has a single global minimizer. The n-dimensional Levy sine functions for n= 4, 5, 6, 7 from A.V. Levy et al. (1981) have respectively 7.1E4, 1E5, 1E6 and 1E8 local minimizers.
The purpose of this study is to estimate the distribution of multiple local minimizers. A simple Monte-Carlo method consists in choosing at random the starting points of the local minimization problems. The stochastic search iterative procedure is made of different steps for finding all the minimizers. At the first step, the boxed search space is regularly divided into small areas. At each iteration, a starting point is selected at random in each sub-box. Thereafter a local minimizer is searched in these small areas. In the following steps, all the minimizers are sorted according to their objective value, the doubles are eliminated automatically and a number of local minimizers is deduced. (minus the number of global minimizers). The whole process is being repeated a hundred of times or more. An empirical distribution of the minimizer numbers is obtained, for which we deduce the value statistical parameters. The computations are carried out by using the software Wolfram Mathematica ® 7, which allows interactive applications with controls, to vary the value of parameters.

Brief Biography of the Speaker: André A. Keller (Prof.) is at present an associate researcher from the “Multi-agent Systems and Behaviors” division of LIFL (Lille Fundamental Computer Science Laboratory), a research unit UMR8022 of the French Centre National de la Recherche Scientifique (CNRS) by the Université de Lille 1, Sciences et Technologies. He received a PhD in Economics (Operations Research) in 1977 from the Université de Paris Panthéon-Sorbonne. He is a WSEAS Member since 2010 and a Reviewer for the ELSEVIER journal Ecological Modelling, the Journal Mathematical Analysis and Applications (jmaa) and WSEAS Transactions on Information Science and Applications. He taught applied mathematics (optimization techniques) and econometric modeling, microeconomics, theory of games and dynamic macroeconomic analysis. His experience centers are on building and analyzing large scale macro-economic models, as well as simulating economic policies, and forecasting. His research interest has concentrated on: high frequency time-series modeling with application to the foreign exchange market, on discrete mathematics (graph theory), stochastic differential games and tournaments, circuit analysis, optimal control in a fuzzy context. His publications consist in writing articles, books and book chapters. The book chapters are e.g. on semi-reduced forms (Martinus Nijhoff, 1984), econometrics of technical change (Springer and IIASA, 1989), advanced time-series analysis (Woodhead Faulkner, 1989), circuits enumeration (Springer, 2008), stochastic differential games (Nova Science, 2009), optimal fuzzy control (InTech, 2009), fuzzy games (Nova Science, 2010). One book is on “Time-Delay Systems:with Applications to Economic Dynamics & Control” (LAP, 2010). One another book is on “Nonconvex Optimization in Practice: Theory, Algorithms and Applications” (WSEAS Press, under review).

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