Sami Viitanen

Stochastic Global Optimization: Problem Classes and Solution Techniques

There is a lack of a representative set of test problems for comparing global optimization methods. To remedy this a classification of essentially unconstrained global optimization problems into unimodal, easy, moderately difficult, and difficult problems is proposed. The problem features giving this classification are the chance to miss the region of attraction of the global minimum, embeddedness of the global minimum, and the number of minimizers. The classification of some often used test problems are given and it is recognized that most of them are easy and some even unimodal. Global optimization solution techniques treated are global, local, and adaptive search and their use for tackling different classes of problems is discussed. The problem of fair comparison of methods is then adressed. Further possible components of a general global optimization tool based on the problem classes and solution techniques is presented.

A Numerical Comparison of Some Modified Controlled Random Search Algorithms

In this paper we propose a new version of the Controlled Random Search(CRS) algorithm of Price. The new algorithmhas been tested on thirteen global optimization test problems. Numericalexperiments indicate that the resulting algorithm performs considerablybetter than the earlier versions of the CRS algorithms. The algorithm,therefore, could offer a reasonable alternative to many currently availablestochastic algorithms, especially for problems requiring ’direct search‘type methods. Also a classification of the CRS algorithms is made based on’global technique‘ – ’local technique‘ and the relative performance ofclasses is numerically explored.

Topographical global optimization using pre-sampled points

A method for global minimization of a functionf(x), x εA ⊂Rn by using presampled global points inA is presented. The global points are obtained by uniform sampling, discarding points too near an already accepted point to obtain a very uniform covering. The accepted points and their nearest-neighbours matrix are stored on a file. When optimzing a given function these pre-sampled points and the matrix are read from file. Then the function value of each point is computed and itsk nearest neighbours that have larger function values are marked. The points for which all its neighbours are marked are extracted as promising starting points for local minimizations. Results from a parallel implementation are presented. The working of a sequential version in Fortran is illustrated.

A direct search variant of the simulated annealing algorithm for optimization involving continuous variables

Abstract A memory-based simulated annealing algorithm is proposed which fundamentally differs from the previously developed simulated annealing algorithms for continuous variables by the fact that a set of points rather than a single working point is used. The implementation of the new method does not need differentiability properties of the function being optimized. The method is well tested on a range of problems classified as easy, moderately difficult and difficult. The new algorithm is compared with other simulated annealing methods on both test problems and practical problems. Results showing an improved performance in finding the global minimum are given. Scope and purpose The inherent difficulty of global optimization problems lies in finding the very best optimum (maximum or minimum) from a multitude of local optima. Many practical global optimization problems of continuous variables are non-differentiable and noisy and even the function evaluation may involve simulation of some process. For such optimization problems direct search approaches are the methods of choice. Simulated annealing is a stochastic global optimization algorithm, initially designed for combinatorial (discrete) optimization problems. The algorithm that we propose here is a simulated annealing algorithm for optimization problems involving continuous variables. It is a direct search method. The strengths of the new algorithm are: it does not require differentiability or any other properties of the function being optimized and it is memory-based. Therefore, the algorithm can be applied to noisy and/or not exactly known functions. Although the algorithm is stochastic in nature, it can memorise the best solution. The new simulated annealing algorithm has been shown to be reliable, fast, general purpose and efficient for solving some difficult global optimization problems.

Iterative Topographical Global Optimization

In topographical global optimization a sample of points that super-uniformly cover the region of interest, A, is used in combination with the function evaluations f(x) in these points to obtain a topographical graph of/on A from which candidate points are easily extracted for local minimizations. This paper discusses some of the problems in obtaining such a cover and presents some solutions. These solutions are based on an iterative use of the topographical method. Several iterations of the topographical algorithm are run and the information gathered is collected into a single graph. Using multiple iterations speeds up the sampling process and also allows using the topographical method for constrained problems.

Parallel continuous simulated annealing for global optimization simulated annealing

In this paper a parallel algorithm for simulated annealing (SA) in the continuous case, the Multiple Trials and Adaptive Supplementary Search, MTASS algorithm, is presented. It is based on a combination of multiple trials, local improved searches and an adaptive cooling schedule. The results in optimizing some standard test problems are compared with some SA sequential algorithms and another parallel probabilistic algorithm