Zorana Lužanin

Newton-like method with modification of the right-hand-side vector

This paper proposes a new Newton-like method which defines new iterates using a linear system with the same coefficient matrix in each iterate. while the correction is performed on the right-hand-side vector of the Newton system. In this way a method is obtained which is less costly than the Newton method and faster than the fixed Newton method. Local convergence is proved for nonsingular systems. The influence of the relaxation parameter is analyzed and explicit formulae for the selection of an optimal parameter are presented. Relevant numerical examples are used to demonstrate the advantages of the proposed method.

Practical Quasi-Newton algorithms for singular nonlinear systems

Quasi-Newton methods for solving singular systems of nonlinear equations are considered in this paper. Singular roots cause a number of problems in implementation of iterative methods and in general deteriorate the rate of convergence. We propose two modifications of QN methods based on Newton’s and Shamanski’s method for singular problems. The proposed algorithms belong to the class of two-step iterations. Influence of iterative rule for matrix updates and the choice of parameters that keep iterative sequence within convergence region are empirically analyzed and some conclusions are obtained.

Epidemiology of Cancers in Serbia and Possible Connection with Cyanobacterial Blooms

Cyanobacteria produce toxic metabolites known as cyanotoxins. These bioactive compounds can cause acute poisoning, and some of them may promote cancer through chronic exposure. Direct ingestion of and contact with contaminated water is one of the many exposure routes to cyanotoxins. The aim of this article was to review the incidence of 13 cancers during a 10-year period in Serbia and to assess whether there is a correlation between the cancer incidences and cyanobacterial bloom occurrence in reservoirs for drinking water supply. The types of cancers were chosen and subjected to epidemiological analyses utilizing previously published data. Based on the epidemiological and statistical analysis, the group of districts in which the incidences of cancers are significant, and may be considered as critical, include Nišavski, Toplički, and Šumadijski district. A significantly higher incidence of ten cancers was observed in the three critical districts as compared to the remaining 14 districts in Central Serbia. These elevated incidences of cancer include: brain cancer, heart, mediastinum and pleura cancer, ovary cancer, testicular cancer, gastric cancer, colorectal cancer, retroperitoneum and peritoneum cancer, leukemia, malignant melanoma of skin, and primary liver cancer. In addition, the mean incidence of five chosen cancers was the highest in the three critical regions, then in the rest of Central Serbia, while the lowest values were recorded in Vojvodina. Persistent and recurrent cyanobacterial blooms occur during summer months in reservoirs supplying water to waterworks in the three critical districts. People in Central Serbia mainly use surface water as water supply (but not all the water bodies are blooming) while in Vojvodina region (control region in this study) only groundwater is used. Among the 14 “noncritical” districts, reservoirs used for drinking water supply have been affected by recurrent cyanobacterial blooms in two districts (Rasinski and Zaječarski), but the waterworks in these districts have been performing ozonation for more than 30 years. We propose that the established statistical differences of cancer incidences in Serbia could be related to drinking water quality, which is affected by cyanobacterial blooms in drinking water reservoirs in certain districts. However, more detailed research is needed regarding cyanobacterial secondary metabolites as risk factors in tumor promotion and cancerogenesis in general.

Descent direction method with line search for unconstrained optimization in noisy environment

A two-phase descent direction method for unconstrained stochastic optimization problem is proposed. A line-search method with an arbitrary descent direction is used to determine the step sizes during the initial phase, and the second phase performs the stochastic approximation (SA) step sizes. The almost sure convergence of the proposed method is established, under standard assumption for descent direction and SA methods. The algorithm used for practical implementation combines a line-search quasi-Newton (QN) method, in particular the Broyden–Fletcher–Goldfarb–Shanno (BFGS) and Symmetric Rank 1 (SR1) methods, with the SA iterations. Numerical results show good performance of the proposed method for different noise levels.

A nonmonotone line search method for noisy minimization

A nonmonotone line search method for optimization in noisy environment is proposed. The method is defined for arbitrary search directions and uses only the noisy function values. Convergence of the proposed method is established under a set of standard assumptions. The computational issues are considered and the presented numerical results affirm that nonmonotone strategies are worth considering. Four different line search rules with three different directions are compared numerically. The influence of nonmonotonicity is discussed.

Adaptive stochastic approximation algorithm

In this paper, stochastic approximation (SA) algorithm with a new adaptive step size scheme is proposed. New adaptive step size scheme uses a fixed number of previous noisy function values to adjust steps at every iteration. The algorithm is formulated for a general descent direction and almost sure convergence is established. The case when negative gradient is chosen as a search direction is also considered. The algorithm is tested on a set of standard test problems. Numerical results show good performance and verify efficiency of the algorithm compared to some of existing algorithms with adaptive step sizes.

Stochastic Newton-like methods for computing equilibria in general equilibrium models

Calculating an equilibrium point in general equilibrium models in many cases reduces to solving a nonlinear system of equations. Taking model parameter values as random variables with a known distribution increases the level of information provided by the model but makes computation of equilibrium points even more challenging. We propose a computationally efficient procedure based on application of the fixed Newton method for a sequence of equilibrium problems generated by simulation of parameters values. The convergence conditions of the method are derived. The numerical results presented are obtained using the neoclassic exchange model and the spatial price equilibrium model. The results show a clear difference in the quality of information obtained by solving a sequence of problems if compared with the single equilibrium problem. At the same time the proposed numerical procedure is affordable.

Newton-like method for nonlinear banded block diagonal system

We present an efficient Newton-like method for solving systems of nonlinear equations with banded block diagonal structure. Such systems arise in economic models. The main idea of this method is to exploit the special structure of the Jacobian in a Newton-like method. The proposed algorithm uses an iterative method for calculation of the approximation of diagonal blocks in the Jacobian. Depending on dimensions of blocks significant reduction in computational cost is obtained and the algorithm can be easily parallelized. Local convergence of the proposed method is proved and some numerical results are presented.

Iterative method with modification of the right-hand side vector for nonlinear complementarity problems

Nonlinear complementarity problems (NCPs) are often solved by iterative methods based on a generalization of the classical Newton method and its modifications for smooth equations. We consider a method with modification of the right-hand side vector for a reformulation of the semi-smooth equation arising from NCPs and prove local convergence of the proposed method. Some numerical results are presented.