Antanas Žilinskas

A review of statistical models for global optimization

A review of statistical models for global optimization is presented. Rationality of the search for a global minimum is formulated axiomatically and the features of the corresponding algorithm are derived from the axioms. Furthermore the results of some applications of the proposed algorithm are presented and the perspectives of the approach are discussed.

On global optimization in two-dimensional scaling

We consider multidimensional scaling for embedding dimension two, which allows the detection of structures in dissimilarity data by simply drawing two-dimensional figures. The corresponding objective function, called STRESS, is generally nondifferentiable and has many local minima. In this paper we investigate several features of this function, and discuss the application of different global optimization procedures. A method based on combining a local search algorithm with an evolutionary strategy of generating new initial points is proposed, and its efficiency is investigated by numerical examples.

On strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions

Abstract The implementation of global optimization algorithms, using the arithmetic of infinity, is considered. A relatively simple version of implementation is proposed for the algorithms that possess the introduced property of strong homogeneity. It is shown that the P-algorithm and the one-step Bayesian algorithm are strongly homogeneous.

Branch and bound algorithm for multidimensional scaling with city-block metric

A two level global optimization algorithm for multidimensional scaling (MDS) with city-block metric is proposed. The piecewise quadratic structure of the objective function is employed. At the upper level a combinatorial global optimization problem is solved by means of branch and bound method, where an objective function is defined as the minimum of a quadratic programming problem. The later is solved at the lower level by a standard quadratic programming algorithm. The proposed algorithm has been applied for auxiliary and practical problems whose global optimization counterpart was of dimensionality up to 24.

Global optimization based on a statistical model and simplicial partitioning

Abstract A statistical model for global optimization is constructed generalizing some properties of the Wiener process to the multidimensional case. An approach to the construction of global optimization algorithms is developed using the proposed statistical model. The convergence of an algorithm based on the constructed statistical model and simplicial partitioning is proved. Several versions of the algorithm are implemented and investigated.

On similarities between two models of global optimization: statistical models and radial basis functions

Construction of global optimization algorithms using statistical models and radial basis function models is discussed. A new method of data smoothing using radial basis function and least squares approach is presented. It is shown that the P-algorithm for global optimization in the presence of noise based on a statistical model coincides with the corresponding radial basis algorithm.

Two level minimization in multidimensional scaling

Multidimensional scaling with city block norm in embedding space is considered. Construction of the corresponding algorithm is reduced to minimization of a piecewise quadratic function. The two level algorithm is developed combining combinatorial minimization at upper level with local minimization at lower level. Results of experimental investigation of the efficiency of the proposed algorithm are presented as well as examples of its application to visualization of multidimensional data.

A statistical model-based algorithm for ‘black-box’ multi-objective optimisation

The problem of multi-objective optimisation with ‘expensive’ ‘black-box’ objective functions is considered. An algorithm is proposed that generalises the single objective P-algorithm constructed using the statistical model of multimodal functions and concepts of the theory of rational decisions under uncertainty. Computational examples are included demonstrating that the algorithm proposed possess several expected properties.

One-Dimensional P-Algorithm with Convergence Rate O(n−3+δ) for Smooth Functions

Algorithms based on statistical models compete favorably with other global optimization algorithms as shown by extensive testing results. A theoretical inadequacy of previously used statistical models for smooth objective functions was eliminated by the authors who, in a recent paper, have constructed a P-algorithm for a statistical model for smooth functions. In the present paper, a modification of that P-algorithm with an improved convergence rate is described.

On the worst-case optimal multi-objective global optimization

Multi-objective optimization problem with Lipshitz objective functions is considered. It is shown that the worst-case optimal passive algorithm can be reduced to the computation of centers of balls producing the optimal cover of a feasible region, where the balls are of equal minimum radius. It is also shown, that in the worst-case, adaptivity does not improve the guaranteed accuracy achievable by the passive algorithm.