Informatica

Multidimensional scaling with city-block distances is considered in this paper. The technique requires optimization of an objective function which has many local minima and can be non-differentiable at minimum points. This study is aimed at developing a fast and effective global optimization algorithm spanning the whole search domain and providing good solutions. A multimodal evolutionary algorithm is used for global optimization to prevent stagnation at bad local optima. Piecewise quadratic structure of the least squares objective function with city-block distances has been exploited for local improvement. The proposed algorithm has been compared with other algorithms described in literature. Through a comprehensive computational study, it is shown that the proposed algorithm provides the best results. The algorithm with fine-tuned parameters finds the global minimum with a high probability.

Recent publications on multidimensional scaling express contradicting opinion on multimodality of STRESS criterion. An example has been published with rigorously provable multimodality of STRESS. We present an example of data and the rigorous proof of multimodality of SSTRESS for this data. Some comments are included on widely accepted opinion that minimization of SSTRESS is easier than minimization of STRESS.

In this paper we are concerned with global optimization, which can be defined as the problem of finding points on a bounded subset of R^m, in which some real-valued function f(x) assumes its optimal value. We consider here a global optimization algorithm. We present a stochastic approach, which is based on the simulated annealing algorithm. The optimization function f(x) here is discrete and with noise.

In well-known statistical models of global optimization only values of objective functions are taken into consideration. However, efficient algorithms of local optimization are also based on the use of gradients of objective functions. Thus, we are interested in a possibility of the use of gradients in statistical models of multimodal functions, aiming to create productive algorithms of global optimization.

In the paper the global optimization is described from the point of an interactive software design. The interactive software that implements numeric methods and other techniques to solve global optimization problems is presented. Some problems of such a software design are formulated and discussed.