 # Ivan Vazler PhD
Department of Mathematics
Josip Juraj Strossmayer University of Osijek
Trg Ljudevita Gaja 6
Osijek, HR-31000, Croatia¸
 phone: +385-31-224-805 fax: +385-31-224-801 email: ivazler @ mathos.hr office: 7 (ground floor)

## Research Interests

Numerical mathematics
Functional programming

## Degrees

PhD in Electrical Engineering, Faculty of Electrical Engineering, University of Osijek, Croatia, 2012.
BSc in Mathematics and Computer Science, Department of Mathematics, University of Osijek, Croatia, 2006.

## Publications

Journal Publications

1. M. Essert, I. Kuzmanović, I. Vazler, T. Žilić, Theory of M-system, Logic Journal of the IGPL 25/5 (2017), 836-858
This paper introduces a new formal theory, the theory of M-system, which represents a many-valued logic system. Basic terms and concepts are defined which are the foundation for their further application. Though the theory comes from the field of electric circuitry, an attempt will be made to extend it to applications in linguistics. To check the examples used in this paper, a Haskell program was made, which can be found at https://www.schoolofhaskell.com/user/ivazler/linguistic.
2. K. Sabo, R. Scitovski, I. Vazler, One-dimensional center-based \$l_1\$-clustering method, Optimization Letters 7/1 (2013), 5-22
Motivated by the method for solving center-based Least Squares - clustering problem (Kogan(2007), Teboulle(2007)), we construct a very efficient iterative process for solving a one-dimensional center-based \$l_1\$ -clustering problem, on the basis of which it is possible to determine the optimal partition. We analyze the basic properties and convergence of our iterative process, which converges to a stationary point of the corresponding objective function for each choice of the initial approximation. Given is also a corresponding algorithm, which in only few steps gives a stationary point and the corresponding partition. The method is illustrated and visualized on the example of looking for an optimal partition with two clusters, where we check all stationary points of the corresponding minimizing functional. Also, the method is tested on the basis of large numbers of data points and clusters and compared with the method for solving the center-based Least Squares - clustering problem described in Kogan(2007) and Teboulle (2007).
3. I. Vazler, K. Sabo, R. Scitovski, Weighted median of the data in solving least absolute deviations problems, Communications is Statistics - Theory and Methods 41/8 (2012), 1455-1465
We consider the weighted median problem for a given set of data and analyze its main properties. As an illustration, an efficient method for searching for a weighted Least Absolute Deviations (LAD)-line is given, which is used as the basis for solving various linear and nonlinear LAD-problems occurring in applications. Our method is illustrated by an example of hourly natural gas consumption forecast.
4. K. Sabo, R. Scitovski, I. Vazler, M. Zekić-Sušac, Mathematical models of natural gas consumption, Energy conversion and management 52/3 (2011), 1721-1727
In this paper we consider the problem of hourly forecast of natural gas consumption on the basis of hourly movement of temperature and natural gas consumption in the preceding period. There are various methods and approaches for solving this problem in the literature. Some mathematical models with linear and nonlinear model functions relating forecast of natural gas consumption with the past natural gas consumption data, temperature data and temperature forecast data, are mentioned. The methods are tested on concrete examples referring to temperature and natural gas consumption for the area of the city of Osijek (Croatia) from the beginning of the year 2008.
5. K. Sabo, I. Vazler, R. Scitovski, Searching for a best LAD-solution of an overdetermined system of linear equations motivated by searching for a best LAD-hyperplane on the basis of given data, Journal of optimization theory and applications 149 (2011), 293-314
We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations \$mathbf{Xa}=mathbf{z}\$, \$mathbf{X}inmathbb{R}^{mtimes n}\$, \$mgeq n\$, \$mathbf{a}inmathbb{R}^n, mathbf{z}inmathbb{R}^m\$. This problem is equivalent to the problem of determining a best LAD-hyperplane \$mathbf{x}mapsto mathbf{a}^Tmathbf{x}\$, \$mathbf{x}inmathbb{R}^n\$ on the basis of given data \$(mathbf{x}_i,z_i),,mathbf{x}_i=(x_1^{(i)},ldots,x_n^{(i)})^Tinmathbb{R}^n,,z_iinmathbb{R},,i=1,ldots,m\$, whereby the minimizing functional is of the form [ F(mathbf{a})=|mathbf{z}-mathbf{Xa}|_1=sum_{i=1}^m|z_i-mathbf{a}^Tmathbf{x}_i|. ] An iterative procedure is constructed as a sequence of weighted median problems, which gives the solution in finitely many steps. A criterion of optimality follows from the fact that the minimizing functional \$F\$ is convex, and therefore the point \$mathbf{a}^*inmathbb{R}^n\$ is the point of a global minimum of the functional \$F\$ if and only if \$mathbf{0}inpartial F(mathbf{a}^*)\$. Motivation for the construction of the algorithm was found in a geometrically visible algorithm for determining a best LAD-plane \$(x,y)mapsto alpha x+beta y\$, passing through the origin of the coordinate system, on the basis of the data \$(x_i,y_i,z_i),,i=1,ldots,m\$.
6. Z. Krpić, G. Martinović, I. Vazler, Data Clustering: Applications in Engineering., Croatian Operational Research Review 1/1 (2010), 180-189
Dividing a set \$S={; ; x_i=(x_1^{; ; (i)}; ; , x_2^{; ; (i)}; ; , ldots, x_n^{; ; (i)}; ; )^Tin mathbb{; ; R}; ; ^n}; ; \$ (a set of vectors from a vector space \$mathbb{; ; R}; ; ^n\$) into disjunct subsets \$pi_1, ldots, pi_k\$ \$1leq kleq m\$, such that [ bigcuplimits_{; ; i=1}; ; ^k pi_i=S, qquad pi_icap pi_j=emptyset, quad ineq j, qquad |pi_j|geq 1, quad j=1, dots, k, ] determines a partition of the set \$S\$ . The elements of such partition \$pi_1, ldots, pi_k\$ are called clusters. For practical clustering applications the number of all clusters is too big and the problem of determining the optimal partition in the least-squares sense is an NP-hard problem. In this paper we will consider some well-known algorithms for searching for an optimal LS-partition, list some of the numerous applications of cluster analysis in engineering and give some practical applications.

Refereed Proceedings

1. G. Kralik, K. Sabo, R. Scitovski, I. Vazler, Solving parameter identification problem by the moving least absolute deviations method, 12th International Conference on Operational Research, Pula, Croatia, 2010, 297-307
On the basis of measured data, among which a significant number of outliers might appear, we introduce one method for parameter identification in a mathematical model given by the ordinary differential equation of the first order. The method consists of two steps. In the first step, we construct a smooth function by applying the moving least absolute deviations method. In the second step, by applying the least absolute deviations method we estimate unknown parameters of mathematical models. The method is applied to and tested on the problem of estimating saturation level and asymmetry coefficient in the mathematical model with saturation. The mathematical model described by a generalized Verhulst differential equation [frac{;dy(t)};{;dt};= c, y(t)left(1-left(frac{;y(t)};{;A};right)^gammaright), quad c, , gamma, , A > 0, ] is considered especially. In this case the parameter estimation problem is reduced to the nonlinear least absolute deviations problem for a 3-parametric exponential regression model. For solving this problem an efficient method is developed. The method is tested on real measurement data of weights of 60 pigs in the period of 26 weeks.
2. I. Kuzmanović, R. Scitovski, K. Sabo, I. Vazler, The least absolute deviation linear regression: properties and two efficient methods, Aplimat 2008, Bratislava, 2008, 227-240

Others

1. K. Sabo, R. Scitovski, I. Vazler, Grupiranje podataka: klasteri, Osječki matematički list 10 (2010), 149-176
U ovom radu razmatramo problem grupiranja elemenata skupa A u disjunktne neprazne podskupove - klastere, pri cemu pretpostavljamo da su elementi skupa A odreeni s jednim ili dva obilježja. Za rješavanje problema koristi se kriterij najmanjih kvadrata te kriterij najmanjih apsolutnih udaljenosti. Naveden je niz primjera koji ilustriraju razlike meu tim kriterijima. Izraena je odgovarajuca programska podrška s ciljem da zainteresirani strucnjaci u svom znanstvenom ili strucnom radu mogu olakšano koristiti ovu metodologiju i pristup.

## Professional Activities

Editorial Boards

Committee Memberships

Refereeing/Reviewing

Service Activities

## Teaching (in Croatian)

Konzultacije (Office Hours): Ponedjeljak (Mon) 14:00am-15:00pm, Srijeda (Wed) 10:00am-11:00pm. Konzultacije su moguće i po dogovoru.

## Personal

Ivan Vazler was born in Požega in 1982. Today he is married and has a beautiful baby girl.