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Metody aproximace plně pravděpodobnostního návrhu rozhodování za neúplné znalosti
Pištěk, Miroslav ; Kárný, Miroslav (advisor) ; Andrýsek, Josef (referee)
In this thesis, we introduce an efficient algorithm for an optimal decision strategy approximation. It approximates the optimal equations of dynamic programming without omitting the principal uncertainty stemming from an uncomplete knowledge of a controlled system. Thus, the algorithm retains the ability to constantly verify the actual knowledge, which is the essence of dual control. An integral part of solution proposed is a reduction of memory demands using HDMR approximation. We have developed a general method for numerical solution of linear integral equations based on this approximation, and applied it to solve a linearized variant of optimal equations. To achieve such a variant, it was necessary to apply a different control design called fully probabilistic design which allows easier finding of a linearized approximation. The result of this method is a pair of linear algebraic systems for the upper and lower bound on the central function describing the optimal strategy. One illustrative example has been completely resolved.
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Theory of SSB Representation of Preferences Revised
Pištěk, Miroslav
A continuous skew-symmetric bilinear (SSB) representation of preferences has recently been proposed in a topological vector space, assuming a weaker notion of convexity of preferences than in the classical (algebraic) case. Equipping a linear vector space with the so-called inductive linear topology, we derive the algebraic SSB representation on a topological basis, thus weakening\nthe convexity assumption. Such a unifying approach to SSB representation permits also to fully discuss the relationship of topological and algebraic axioms of continuity, and leads to a stronger existence result for a maximal element. By applying this theory to probability measures we show the existence of a maximal preferred measure for an infinite set of pure outcomes, thus generalizing all available existence theorems in this context.
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Alternative Formulation of Pay-as-clear Auction in Electricity Markets
Aussel, D. ; Červinka, Michal ; Henrion, R. ; Pištěk, Miroslav
In widely used formulation of pay-as-clear electricity market the clearing price is given by the Lagrange multiplier of the demand sat- isfaction constraint in the problem of the Independent System Operator (ISO). Following this idea, one may usually calculate the market clearing\nprice analytically even for problems of higher dimensions. However, the economic interpretation of such a market setting is in question, since the minimized criterion does not correspond neither to the cost of production nor to the overall payment of consumers. This observation motivated us\nto propose an alternative clearing mechanism where the total payment of consumers is explicitly minimized. We show existence and uniqueness of the clearing price in such a setting.
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Metody aproximace plně pravděpodobnostního návrhu rozhodování za neúplné znalosti
Pištěk, Miroslav ; Andrýsek, Josef (referee) ; Kárný, Miroslav (advisor)
In this thesis, we introduce an efficient algorithm for an optimal decision strategy approximation. It approximates the optimal equations of dynamic programming without omitting the principal uncertainty stemming from an uncomplete knowledge of a controlled system. Thus, the algorithm retains the ability to constantly verify the actual knowledge, which is the essence of dual control. An integral part of solution proposed is a reduction of memory demands using HDMR approximation. We have developed a general method for numerical solution of linear integral equations based on this approximation, and applied it to solve a linearized variant of optimal equations. To achieve such a variant, it was necessary to apply a different control design called fully probabilistic design which allows easier finding of a linearized approximation. The result of this method is a pair of linear algebraic systems for the upper and lower bound on the central function describing the optimal strategy. One illustrative example has been completely resolved.
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Statistická fyzika frustrovaných evolučních her
Pištěk, Miroslav ; Janiš, Václav (referee) ; Slanina, František (advisor)
1 Title: Statistical Physics of Frustrated Evolutionary Games Author: Miroslav Pištěk Department: Institute of Theoretical Physics Supervisor: RNDr. František Slanina, CSc. Supervisor's e-mail address: slanina@fzu.cz Abstract: In last two decades, the effort devoted to interdisciplinary research of bounded sources allocation is growing, examining complex phenomena as stock markets or traffic jams. The Minority Game is a multiple-agent model of inevitable frus- tration arising in such situations. It is analytically tractable using the replica method originated in statistical physics of spin glasses. We generalised the Mi- nority Game introducing heterogenous agents. This heterogeneity causes a con- siderable decrease of an average agent's frustration. For many configurations, we achieve even a positive-sum game, which is not possible in the original game variant. This result is in accordance with real stock market data. Keywords: frustrated evolutionary games, Minority Game, Replica method
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Approximate Dynamic Programming based on High Dimensional Model Representation
Pištěk, Miroslav
In this article, an efficient algorithm for an optimal decision strategy approximation is introduced. The proposed approximation of the Bellman equation is based on HDMR technique. This non-parametric function approximation is used not only to reduce memory demands necessary to store Bellman function, but also to allow its fast approximate minimization. On that account, a clear connection between HDMR minimization and discrete optimization is newly established. In each time step of the backward evaluation of the Bellman function, we relax the parameterized discrete minimization subproblem to obtain parameterized trust region problem. We observe that the involved matrix is the same for all parameters owning to the structure of HDMR approximation. We find eigenvalue decomposition of this matrix to solve all trust region problems effectively.
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Implicitní aproximace Bellmanovy rovnice
Pištěk, Miroslav
In this article, an efficient algorithm for an optimal decision strategy approximation is introduced. It approximate the Bellman equation without omitting the principial uncertainty stemming from an uncomplete knowledge. An integral part of the proposed solution is a reduction of memory demands using HDMR approximation. The result of this method is a linear algebraic system for an approximated upper bound on the Bellman function. One illustrative example has been completely resolved.
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