Gaussian mixtures (i.e. linear combinations of multivariate Gaussian probability densities) appear in numerous applications due to their universal ability to approximate multimodal probability distributions. Finding the modes (maxima) of a Gaussian mixture is a fundamental problem arising in...

In practical applications related to, for instance, machine learning, data mining and pattern recognition, one is commonly dealing with noisy data lying near some low-dimensional manifold. A well-established tool for extracting the intrinsically low-dimensional structure from such data is...

Trust region algorithms are well known in the field of local continuous optimization. They proceed by maintaining a confidence region in which a simple, most often quadratic, model is substituted to the criterion to be minimized. The minimum of the model in the trust region becomes the next...

A new method of moving asymptotes for large scale minimization subject to linear inequality constraints is discussed in this paper. In each step of the iterative process, a descend direction is obtained by solving a convex separable subproblem with dual technique. The new rules for controlling...

In this paper we show how techniques from response surface methodology and mathematical programming can be combined into a new sequential derivative-free approach for solving unconstrained deterministic black-box optimization problems.In this sequential derivative-free optimization approach...

This paper presents a trust region subspace method for minimizing large-scale unconstrained problems. We choose a subspace that consists of some old directions which are invariable and some newest directions which are changed at each iteration. A restart technique is used when the old directions...

By means of a gradient strategy, the Moreau-Yosida regularization, limited memory BFGS update, and proximal method, we propose a trust-region method for nonsmooth convex minimization. The search direction is the combination of the gradient direction and the trust-region direction. The global...

The measure of the regional disparities in the European Union was widely approached on the literature. Recent years have seen a spate of studies taking into account the spatial aspect of data in the analysis of convergence from an empirical viewpoint, or integrating spatial interdependencies...

In the last decade, after accession to European Union, Poland has achieved a significant improvement of indicators relating to income on a national level. Polish GDP per capita (in PPP terms) increased from 49% of European Union average in 2004 to 67% in 2013 showing progress in each year...