Informatica

In this paper public available C and C++ packages for interval arithmetic are investigated and experimentally compared. The results of comparison give suggestions which packages and when are preferable.

We consider a geographical region with spatially separated customers, whose demand is currently served by some pre-existing facilities owned by different firms. An entering firm wants to compete for this market locating some new facilities. Trying to guarantee a future satisfactory captured demand for each new facility, the firm imposes a constraint over its possible locations (a finite set of candidates): a new facility will be opened only if a minimal market share is captured in the short-term. To check that, it is necessary to know the exact captured demand by each new facility. It is supposed that customers follow the partially binary choice rule to satisfy its demand. If there are several new facilities with maximal attraction for a customer, we consider that the proportion of demand captured by the entering firm will be equally distributed among such facilities (equity-based rule). This ties breaking rule involves that we will deal with a nonlinear constrained discrete competitive facility location problem. Moreover, minimal attraction conditions for customers and distances approximated by intervals have been incorporated to deal with a more realistic model. To solve this nonlinear model, we first linearize the model, which allows to solve small size problems because of its complexity, and then, for bigger size problems, a heuristic algorithm is proposed, which could also be used to solve other constrained problems.

A new heuristic algorithm for solution of bi-objective discrete competitive facility location problems is developed and experimentally investigated by solving different instances of a facility location problem for firm expansion. The proposed algorithm is based on ranking of candidate locations for the new facilities, where rank values are dynamically adjusted with respect to behaviour of the algorithm. Results of the experimental investigation show that the proposed algorithm is suitable for the latter facility location problems and provides good results in sense of accuracy of the approximation of the true Pareto front.

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.

Copositivity plays an important role in combinatorial and quadratic optimization since setting up a linear optimization problem over the copositive cone leads to exact reformulations of combinatorial and quadratic programming problems. A copositive programming problem may be approached checking copositivity of several matrices built with different values of the variable and the solution is the extreme value for which the matrix is copositive. However, this approach has some shortcomings. In this paper, we develop a simplicial partition algorithm for copositive programming to overcome the shortcomings. The algorithm has been investigated experimentally on a number of problems.

The optimization problems occurring in nonlinear regression normally cannot be proven unimodal. In the present paper applicability of global optimization algorithms to this problem is investigated with the focus on interval arithmetic based algorithms.

Multidimensional scaling is a technique for exploratory analysis of multidimensional data widely usable in different applications. By means of this technique the image points in a low-dimensional embedding space can be found whose inter-point distances fit the given dissimilarities between the considered objects. In this paper dependence of relative visualization error on the dimensionality of embedding space is investigated. Both artificial and practical data sets have been used. The images in three-dimensional embedding space normally show the structural properties of sets of considered objects with acceptable accuracy, and widening of applications of stereo screens makes three-dimensional visualization very attractive.

New ways to estimate ranges of values of functions from standard and inner interval arithmetic have been proposed. Using the proposed ways ranges of values of mathematical test functions for global optimization and of objective functions for practical global optimization problems have been estimated and compared. Results of the experiments show that it is promising to use proposed balanced interval arithmetic in interval global optimization.

The results of experimental testing of balanced random interval arithmetic with typical mathematical test functions and practical problem are presented and discussed. The possibility of evaluation ranges of functions using balanced random interval arithmetic is investigated. The influence of the predefined probabilities of standard and inner interval operations to the ranges of functions is experimentally investigated.