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1
Content available Interval Methods for Computing Strong Nash Equilibria of Continuous Games
Kubica B. J., Woźniak A.
Decision Making in Manufacturing and Services
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2015
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Vol. 9, no. 1
63--78
EN
The problem of seeking strong Nash equilibria of a continuous game is considered. For some games, these points cannot be found analytically, only numerically. Interval methods provide us with an approach to rigorously verify the existence of equilibria in certain points. A proper algorithm is presented. We formulate and prove propositions, that give us features which have to be used by the algorithm (to the best knowledge of the authors, these propositions and properties are original). Parallelization of the algorithm is also considered, and numerical results are presented. As a particular example, we consider the game of “misanthropic individuals”, a game, invented by the first author, that may have several strong Nash equilibria depending on the number of players. Our algorithm is able to localize and verify these equilibria.
2
Content available The "Second Derivative" of a Non-Differentiable Function and its Use in Interval Optimization Methods
Kubica B. J.
Journal of Telecommunications and Information Technology
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2011
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nr 4
81-85
EN
The paper presents an idea to use weak derivatives in interval global optimization. It allows using the Newton operator to narrow domains of non-differentiable functions. Preliminary computational experiments are also presented.
3
Content available Optimization of the Multi-Threaded Interval Algorithm for the Pareto-Set Computation
Kubica B. J., Wodniak A.
Journal of Telecommunications and Information Technology
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2010
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nr 1
70-75
EN
Previous investigations of the authors surveyed the possibility of applying interval methods to seek the Paretofront of a multicriterial nonlinear problem. An efficient algorithm has been proposed and its implementation in a multicore environment has been done and tested. This paper has two goals. First one is to tune the developed algorithm to increase the speedup of the multi-threaded variant. The second one is to extend the algorithm to compute not only the Paretofront (in the criteria space), but also the Pareto-set (in the decision space). Numerical results for suitable test problems are presented.
4
Shared-Memory Parallelization of an Interval Equations Systems Solver - Comparison of Tools
Kubica B.J.
Prace Naukowe Politechniki Warszawskiej. Elektronika
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2009
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z. 169
121-128
EN
This paper considers the shared-memory parallelization of an interval solver of underdetermined systems of nonlinear equations. Four threading libraries are investigated: OpenMP, POSIX threads, Boost threads and TBB. Directions for further investigations on multi-threaded interval algorithms are outlined.
5
Performance inversion of interval Newton narrowing operators
Kubica B.J.
Prace Naukowe Politechniki Warszawskiej. Elektronika
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2009
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z. 169
111-119
EN
This paper describes a phenomenon of performance inversion of Newton operators - more precise operators might result in longer computation of a branch-and-prune method. Examples are presented and possible reasons of this behavior are discussed.
6
Breakthrough in Interval Data Fitting II. From Ranges to Means and Standard Deviations
Gutowski M.W.
Prace Naukowe Politechniki Warszawskiej. Elektronika
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2009
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z. 169
73-78
EN
Interval analysis, when applied to the so called problem of experimental data fitting, appears to be still in its infancy. Sometimes, partly because of the unrivaled reliability of interval methods, we do not obtain any results at all. Worse yet, if this happens, then we are left in the state of complete ignorance concerning the unknown parameters of interest. This is in sharp contrast with widespread statistical methods of data analysis. In this paper I show the connections between those two approaches: how to process experimental data rigorously, using interval methods, and present the final results either as intervals (guaranteed, rigorous results) or in a more familiar probabilistic form: as a mean value and its standard deviation.
7
Breakthrough in Interval Data Fitting I. The Role of Hausdorff Distance
Gutowski M.W.
Prace Naukowe Politechniki Warszawskiej. Elektronika
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2009
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z. 169
63-72
EN
This is the first of two papers describing the process of fitting experimental data under interval uncertainty. Probably the most often encountered application of global optimization methods is finding the so called best fitted values of various parameters, as well as their uncertainties, based on experimental data. Here I present the methodology, designed from the very beginning as an interval-oriented tool, meant to replace to the large extent the famous Least Squares (LSQ) and other slightly less popular methods. Contrary to its classical counterparts, the presented method does not require any poorly justified prior assumptions, like smallness of experimental uncertainties or their normal (Gaussian) distribution. Using interval approach, we are able to fit rigorously and reliably not only the simple functional dependencies, with no extra effort when both variables are uncertain, but also the cases when the constitutive equation exists in implicit rather than explicit functional form. The magic word and a key to success of interval approach appears the Hausdorff distance.
8
Exploring the search space with intervals
Gutowski M.
Prace Naukowe Politechniki Warszawskiej. Elektronika
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2008
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z. 165
77-84
EN
The term global optimization is used in several contexts. Most often we are interested in finding such a point (or points) in many-dimensional search space at which the objective function's value is optimal, i.e. maximal or minimal. Sometimes, however, we are also interested in stability of the solution, that is in its robustness against small perturbations. Here I present the original, interval-analysis-based family of methods designed for exhaustive exploration of the search space. The power of interval methods makes it possible to reach all mentioned goals within a single, unified framework.
9
Can interval computations be applied over spaces of non-numbers?
Kubica B.
Prace Naukowe Politechniki Warszawskiej. Elektronika
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2008
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z. 165
103-112
EN
Interval methods proved to be a useful tool for solving global optimization and nonlinear equations systems problems over Rn. But an interval may be defined not only over the set of real numbers or real vectors, but over any partially ordered set. The paper shows how basic ideas of interval computations can be generalized for such spaces. Some specific applications are proposed and preliminary computational results are presented.
10
Content available Kontrola dokładności numerycznego przetwarzania danych pomiarowych
Górka P.
Pomiary Automatyka Kontrola
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2008
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R. 54, nr 12
854-856
PL
W artykule przedstawiono wybrane metody automatycznej kontroli dokładności obliczeń w procesie przetwarzania danych pomiarowych. Metody te powinny - w założeniu - uwzględniać dokładność wyników pomiarów, jak i błędy numeryczne. Najwięcej uwagi poświęcono omówieniu możliwości zastosowania arytmetyki przedziałowej jako najbardziej uniwersalnej metody kontroli dokładności obliczeń. Przedstawiono zasady jej stosowania, zalety jak i uwagi dotyczące ominięcia jej mankamentów.
EN
The paper presents some methods of the automatic accuracy check of calculations performed during computer processing of measurement data. The mentioned methods should take into account the measurement data accuracy and numerical errors. The paper discusses mainly the interval arithmetic method which appears as the most universal one. The basis of the method, its advantages and possibility to avoid some problems which can be connected with the use of interval arithmetic are presented as well.
11
CuikSLAM with unknown correspondence - preliminary results
Kubica B., Szynkiewicz W.
Prace Naukowe Politechniki Warszawskiej. Elektronika
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2007
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z. 160
143-151
EN
This paper considers the problem of simultaneous localization and mapping of a mobile robot. The kinematic approach of CuikSLAM is adopted applying constrained satisfaction and interval methods. The novelty is that we do not assume the landmark identification problem to br solved.
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