Laboratory of Operations Research and Decision Systems

Operations Research comprises optimization theory, specifically in global-, linear and quadratic, smooth optimization; analysis of stochastic, furthermore, deterministic systems, while Decision Systems relate to multi-attribute group decision making, applications of operations research and mathematical physics. The activity in these two areas resulted in almost 50, real-life applications during the past 10 years.

Description

Research Projects

I. Operations research

(T. Rapcsák, K. Balla, Cs. Mészáros, Gy. Michaletzky, S. Z. Németh, Zs. Vágó)

Equilibrium systems and smooth optimization

The global Lagrange multiplier rule based on Riemannian geometry and introduced in smooth nonconvex optimization by Rapcsák is summarized in.

The convexification of continuous and smooth functions is related to the choice of Riemannian metrics originated from regular nonlinear coordinate transformations [26].

The minimization of a smooth function on a Stiefel manifold is a new and interesting global optimization problem with important theoretical and practical applications. By using the nice geometric structure, the optimality conditions are obtained by the global Lagrange multiplier rule and global optimality conditions based on local information.

In [31], the aim is to show that interesting and important statistical problems containing the principal component analysis, statistical visualization and singular value decomposition, furthermore, the matrix spectral theorem, the charaterization of the structural stability of dynamical systems and many others lead to the minimization of a smooth function on a Stiefel manifold where the question is to find optimal orthogonal matrices.

Based on the recently developed techniques in smooth optimization, a simple geometric proof of the van der Waerden theorem is given.

Minty's classical homeomorphism theorem for monotone operators was extended to Riemannian manifolds.

In [28], it is shown that the principle of virtual work considered an axiom of mechanics by Lagrange (1788) and Farkas (1906) can be embedded in a general equilibrium system, the quasi-variational inequalities introduced by Bensoussan and Lions in 1973, assuming force fields and holonomic-scleronomic constraints.

In [27], the path of life and the scientific activity of Jeno Egerváry is summarized.

Large-Scale Optimization

A new research direction was started on the methodology and the practical issues of the dual-simplex method. In the first step, some efficient techniques of linear algebra kernel were evolved. Research will be continued on the algorithmic issues of the algorithm.

For interior point methods, a new Cholesky-factorization scheme was developed. The method exploits better the commonly used hardware architectures by using multilevel blocking for data processing. A blocking scheme was also developed for sparse computations, which gives significant speed-up on large optimization problems.

Discrete optimization with randomization

The possibility of applying some suitable modifications of the SPSA method in discrete optimization problems is investigated. A stochastic search algorithm on Z^p where Z^p is the grid of vectors with integer-valued coordinates is developed. A fixed gain SPSA method is used where both the size of perturbation and the stepsize of the parameter update is fixed, followed by truncation. Extensive simulation results were also presented.

Differential algebraic equations

DAEs with minimal smoothness and a leading term composed of a pair of well matched matrices are treated in. Boundary value problems (BVP) for linear DAEs yielding a selfadjoint BVP for the essentially underlying ordinary differential equation and depending on the spectral parameter in a monotone way are dealt with in. A method for computing the number of eigenvalues on an interval of the parameter is constructed.

Analysis of stochastic and deterministic systems

One of the main practical problems in statistical analysis is distinguishing the observations from the "trash". Using an a priori bound on the possible percentage of "fake" data values, the problem leads to a divergence optimization problem analysed and solved in.

The 2nd, revised edition of risk processes contains a detailed analysis of ruin probabilities for classical risk processes, including the case losses with heavy tails belonging to the class of sub-exponential distributions.

Fuhrmann–Gombani proved that in the case of linear, stationary stochastic systems if the number of system zeros is less than the number of poles, the missing zeros appear in the dynamic structure of the external part of the state-process points out that this is essentially an operator-theoretic result.

Statistical analysis of financial time series

Statistical analysis of ARCH and GARCH models and HMM is an important subject of mathematical finance. Change point detection is studied in a HMM framework. The approach links the statistical theory of HMMs and linear stochastic systems.

II. Decision systems

(T. Rapcsák, K. Balla, S. Bozóki, J. Fülöp, G. Kéri, S. Márton, S. Z. Németh)

Multi-attribute decision making

In the evaluation of the efficiency of the Tender for Developing the Economy in Hungary regarding the years of 1996, 1997 and 1998 is presented. The task was completed as a multi-attribute group decision problem and solved by our software system WINDGSS 4.1.

Decision laws of multi-attribute decision problems were aggregated by the generalization of Gauss' arithmetic-geometric mean.

In [10] it is shown that the singular value decomposition is a good tool to provide a theoretically well formulated solution both for the scaling and consistency problems in a major multi-attribute decision model, the Analytic Hierarchy Process.

Environmental modelling

Due to any possible damage of water proofing, the penetration of waste-material into the soil may occur. Based on known models, we propounded ideas useful in simulations for two planned regional waste-material depositories, a planned sewage sludge composting plant and an active aluminium dross depository of a foundry.

In the river Tápió is characterized by mathematical statistics from hydrological point of view.

Based on the Hungarian National Standards, we have been studying the transmission of pollutants emitted by point sources (stacks), linear and surface sources in the field by the help of program-system ATP.

Data Envelopment Analysis

Data Envelopment Analysis (DEA) is a linear programming based methodology for the efficiency analysis of decision units producing outputs from inputs. In [*5] we give a brief survey on the basic models of DEA as well as their computational issues. We also report on an application of the DEA methodology for an analysis of the Hungarian industrial parks. Tools of statistics are also applied in the application.

Applications

Together with the members of the Department of Operations Research and Decision Systems, the members of the Laboratory took part in the application of mathematical and software engineering results related to decision systems.

An environmental impact assessment was elaborated by using multi-attribute group decision making techniques for the revitalization of Lake Sasfészek at the village of Páty. (G. Kéri, S. Márton and T. Rapcsák)

In order to rank developments at a leading bank in Budapest, a multi-attribute decision making method was elaborated and introduced. (S. Bozóki, S. Márton and T. Rapcsák)

The main goal of SIADCERO project (Strategic Integrated Assessment of Dynamic Carbon Emission Reduction Policies) of European Union is to analyze the strategic interests and policy options of the EU in international negotiations on climate change. Among other models, dynamic-sequential ones of game theory have been established. We developed an Excel add-in for the computation with the appropriate game theoretic models. The add-in allows several versions of the reaction function approach, models of correlated and tree-correlated equilibrium as well as the Nash bargaining and the Kalai-Smorodinsky models. (J. Fülöp, M. Prill)