Informatica

We describe an adaptive algorithm for approximating the global minimum of a continuous univariate function. The convergence rate of the error is studied for the case of a random objective function distributed according to the Wiener measure.

In this paper, we consider the so-called structured low rank approximation (SLRA) problem as a problem of optimization on the set of either matrices or vectors. Briefly, SLRA is defined as follows. Given an initial matrix with a certain structure (for example, Hankel), the aim is to find a matrix of specified lower rank that approximates this initial matrix, whilst maintaining the initial structure. We demonstrate that the optimization problem arising is typically very difficult; in particular, the objective function is multiextremal even in simple cases. We also look at different methods of solving the SLRA problem. We show that some traditional methods do not even converge to a locally optimal matrix.

The most classical visualization methods, including multidimensional scaling and its particular case – Sammon's mapping, encounter difficulties when analyzing large data sets. One of possible ways to solve the problem is the application of artificial neural networks. This paper presents the visualization of large data sets using the feed-forward neural network – SAMANN. This back propagation-like learning rule has been developed to allow a feed-forward artificial neural network to learn Sammon's mapping in an unsupervised way. In its initial form, SAMANN training is computation expensive. In this paper, we discover conditions optimizing the computational expenditure in visualization even of large data sets. It is shown possibility to reduce the original dimensionality of data to a lower one using small number of iterations. The visualization results of real-world data sets are presented.

A well-known example of global optimization that provides solutions within fixed error limits is optimization of functions with a known Lipschitz constant. In many real-life problems this constant is unknown.

The FDA's Quality by Design initiative and associated design space construct (ICH, 2009), have stimulated the use of quantitative methods, mathematical and statistical models, and designed experiments in the process of drug development and manufacture. For a given drug product, the design space may be interpreted as the constrained region of the manufacturing operating variable space within which assurance can be provided that drug product quality specifications will be met. It is now understood, at least conceptually, that this assurance is not deterministic, rather it must be stated in probabilistic terms. In this paper, we report on the use of Bayesian methods to develop a suitable risk metric based on both mathematical and statistical models of the manufacturing processes and product properties. The Bayesian estimation is carried out to determine the joint posterior distribution of the probability of the product meeting quality specifications. The computations are executed using a novel Variational Bayes approximation. In this paper the direct computational approach using this approximation is compared to the widely used but computationally very intensive Markov Chain Monte Carlo method. The approach is illustrated using experimental data and models drawn from a recent QbD study on the drug gabapentin in which the authors were participants.

Many biological processes and objects can be described by fractals. The paper uses a new type of objects – blinking fractals – that are not covered by traditional theories considering dynamics of self-similarity processes. It is shown that both traditional and blinking fractals can be successfully studied by a recent approach allowing one to work numerically with infinite and infinitesimal numbers. It is shown that blinking fractals can be applied for modeling complex processes of growth of biological systems including their season changes. The new approach allows one to give various quantitative characteristics of the obtained blinking fractals models of biological systems.

In this paper, a novel approach involving the concepts from mathematical programming and number theory is proposed to find the D-optimal designs. In specific, we will propose a mathematical formulation for the D-optimal design. In addition to that, we will present the use of cyclotomic cosets in the mathematical formulation, in order to reduce the total number of binary variables. We will illustrate the validity of our proposed method by solving a difficult known instance (N=126) of the D-optimal design.

The concentration of a substrate in a solution can be measured using amperometric signals of biosensors: in fact the maximum (steady state) current is measured which is calibrated in the units of concentration. Such a simple method is not applicable in the case of several substrates. In the present paper, the problem of evaluation of concentrations of several substrates is tackled by minimizing the discrepancy between the observed and modeled transition processes of the amperometric signal.

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.