Stochastic optimisation

A general statement of the problem of global optimisation is as follows. Let f be a function given on a 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.

In stochastic optimisation algorithms, the points where f is evaluated are selected with the help of probabilistic models. 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 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.

Related software has been developed. A number of applied problems of engineering and design have been solved using the stochastic global optimisation algorithms.

Cardiff Investigators

• Prof. Anatoly Zhigljavsky
• Prof. Duc Pham
• Dr. Jonathan Gillard
• Dr. Michael Packianather

Collaborators

Prof. Panos Pardalos, Prof. Antanas Zilinskas, Prof. Yaroslav Sergeyev, Prof. James Calvin, Dr. Julius Zilinskas, Dr. Dmitrii Kvasov, Dr. Ivan Markovsky, Dr. Vladimir Nekrutkin.

Selected publications

• Zhigljavsky A. (2011) Stochastic Global Optimization. In: International Encyclopedia of Statistical Science, Springer, 1521-1524.
• Gillard J., Zhigljavsky A. (2011) Analysis of structured low rank approximation as an optimisation problem, Informatica (to appear)
• Zhigljavsky A., Hamilton E. (2010) Stopping rules in k-adaptive global random search algorithms. Journal of Global Optimization, v. 48, No. 1, 87-97.
• Hamilton E., Savani V., Zhigljavsky A. (2007) Estimating the minimal value of a function in global random search: comparison of estimation procedures. In: Models and algorithms for global optimization, Springer, p. 193-214.
  
• Zhigljavsky A.A., Zilinskas A.G. (2008) Stochastic Global Optimization. Springer-Verlag, Berlin
• Zhigljavsky A.A., Zilinskas A.G. (1991) Search for Global Extremum. Nauka, Moscow. (in Russian).
• Zhigljavsky A.A. (1991) Theory of Global Random Search. Kluwer Academic Press, Dordrecht e.a. xviii+342 pp.
• Zhigljavsky A.A. (1985) Mathematical Theory of the Global Random Search. St.Petersburg University Press, 296 pp. (in Russian)