## Stochastic Global 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
- Mrs Emily Hamilton

#### Collaborators

- Prof Antanas Zilinskas
- Dr Vladimir Nekrutkin
- Dr Alexey Tikhomirov

#### Selected publications

- 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)