Stochastic methods in global optimization
Let f be a
function given on an d-dimensional compact set X and belonging to a
suitable functional class F of multiextremal continuous functions. We
consider the problem of its minimization, that is approximation of a point x'
such that f(x')=min f(x), using evaluations of f at specially selected points.
Global random search algorithms are the optimization methods
where the procedure of selection of the these points includes random decisions.
We develop and study various algorithms such that on every iteration
statistical procedures are used concerning the objective function and its
maximum and an iteration consists in a transition from a bunch of points to
another bunch so that there is enough information in the bunches to make a
proper adaptation.
We study optimality properties of suitable statistical
procedures based on the the asymptotic theory of extreme order statistics. We
demonstrate, for example, that semi-random points, such as obtained by the
stratified sampling, provide more precise statistical inference concerning the
maximum of f than the points from an independent sample.
Another activity in this field is the study of convergence
of the sequence of probability distributions {P(n)} which generate points
of iteration n. There is a strong link between the corresponding limit theorems
and the theory of Markov Chain Monte-Carlo methods widely used in Bayesian
statistics to sample from aposteriori distributions.
Selected Publications
- Zhigljavsky A.A. (1991)
Theory of Global Random Search. Kluwer Academic Press,
- Zhigljavsky A.A. (1985)
Mathematical Theory of the Global Random Search.
- Zhigljavsky A.A., Zilinskas
A.G. (1991) Search for Global Extremum. Nauka,
- Zhigljavsky A.A. (1993) Semiparametric statistical inference in global random search. Acta Applicandae Mathematicae, 33, 69-88.
- Zhigljavsky A.A., Chekmasov M.V. (1996) Comparison of independent, stratified and random covering sample schemes in optimization problems. Mathematical and Computer Modelling, 23, 97-110.
- Kondratovich M., Zhigljavsky A.A. (1998) Comparison of independent and stratified sampling schemes in problems of global optimization, Monte Carlo and Quasi-Monte-Carlo Methods, (eds H. Niederrreiter, P. Hellekalek, G. Larcher, P. Zinterhof), Lecture notes in statistics, v. 127, pp. 292-299.
- Zhigljavsky A.A., Zilinskas A.G. (2007) Stochastic Global Optimization. Springer-Verlag, Berlin e.a.