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Optimization in energy problems
Fürst, Matouš ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
Title: Optimization in energy problems Author: Matouš Fürst Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Ing. Miloš Kopa, Ph.D., Department of Probability and Mathematical Statistics Abstract: In this thesis we present an optimization model of a semi-autonomous household, which aims to make energy management more efficient. The household is equipped with solar panels and an electric vehicle with a high-capacity battery. In the first part we summarize the basic properties of linear programming and two- stage stochastic linear programming. Subsequently, a two-stage stochastic linear program is formulated and solved in order to optimize the purchase, sale and storage of energy in the household during a single day. The program is formulated in two versions - with present and with departing vehicle. The final solution represents optimal decisions of the household and we discuss it with respect to the input data. In both versions the solution leads to a substantial reduction in costs compared to a household without a battery. Keywords: stochastic optimization, linear programming, domestic microgrid 1
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High-order stochastic dominance
Mikulka, Jakub ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
The thesis deals with high-order stochastic dominance of random variables and portfolios. The summary of findings about high-order stochastic dominance and portfolio efficiency is presented. As a main part of the thesis it is proven that under assumption of both normal and gamma distribution the infinite-order stochastic dominance is equivalent to the second-order stochastic dominance. The necessary and sufficient condition for the infinite-order stochastic dominance portfolio efficiency is derived under the assumption of normality. The condition is used in the empirical part of the thesis where parametrical approach to the portfolio efficiency is compared to the nonparametric scenario approach. The derived necessary and sufficient condition is based on the assumption of normality; therefore we use two sets of data, one with fulfilled assumption of normality and the other for which the assumption of normality was unambigously rejected. Consequently, the influence of fulfillment of the normality assumption on the results of the necessary and sufficient condition for portfolio efficiency is estimated.
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Multidimensional risk measures
Chromíková, Dana ; Kopa, Miloš (advisor) ; Cipra, Tomáš (referee)
This thesis deals with multiperiod risk measures and multiperiod models with these risk measures in the objective are formulated. Multiperiod models consider the possibility of an intermediate actions within the investment horizont and represent the real situation in a better way than one-period models. First the basic properties for one-period risk measures are summarized. Then multiperiod risk measures are de ned and several ways of construction concrete risk measures are discussed as extension of one-period risk measures. Multiperiod portfolio selection mean-risk models with di erent risk measures are formulated, transaction costs are included and short sales are not allowed. Using scenario approach the analysis on real data is performed and optimal strategies for one-period and multiperiod models are compared. A transaction costs e ect on optimal strategy is examined.
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Convexity in chance constraints programming
Olos, Marek ; Kopa, Miloš (advisor) ; Adam, Lukáš (referee)
This thesis deals with chance constrained stochastic programming pro- blems. The first chapter is an introduction. We formulate several stochastic pro- gramming problems in the second chapter. In chapter 3 we present the theory of α-concave functions and measures as a basic tool for proving convexity of the problems formulated in chapter 2 for the continuous distributions of the random vectors. We use the results of the theory to characterize a large class of the conti- nuous distributions, that satisfy the sufficient conditions for the convexity and to prove convexity of concrete sets. In chapter 4 we present sufficient conditions for the convexity of the problems and we briefly discuss the method of the p-level ef- ficient points. In chapter 5 we solve a portfolio selection problem using Kataoka's model. 1
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Stochastic dominance portfolio efficiency measures
Jakubcová, Monika ; Kopa, Miloš (advisor) ; Dupačová, Jitka (referee)
In the present work we study the stochastic dominance portfolio e ciency measures. The investor's risk attitude is given by the type of an utility function. If this information is unknown or a general investor is assumed, it is possible to use the stochastic dominance principle, in which the portfolio is only classi ed as e cient or ine cient. We build on the works of Post, Kuosmanen and Kopa, who formulated the criteria of portfolio e ciency for nonsatiate and risk averse investors. On the basis of these criteria, we de ne the second-order stochastic dominance (SSD) portfolio e ciency measures. We examine the properties of SSD ine ciency measures, which allow to compare SSD ine cient portfolios. We prove mutual relationships for the de ned SSD ine ciency measures. Eventually, we test the SSD e ciency of a US market portfolio on real-world US Stock Exchange data.
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Robust portfolio selection problem
Zákutná, Tatiana ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
In this thesis, a portfolio optimization with integer variables which influ- ence optimal assets allocation, is studied. Measures of risk are defined and the cor- responding mean-risk models are derived. Two methods are used to develop robust models involving uncertainty in probability distribution: the worst-case analyses and contamination. The uncertainty in values of scenarios and in their probabili- ties of the discrete probability distribution is assumed separately followed by their combination. These models are applied to stock market data with using optimization software GAMS.
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Optimal investment problems solvable using linear programming
Jančařík, Joel ; Branda, Martin (advisor) ; Kopa, Miloš (referee)
Portfolio optimization problem is a classical optimization problem, where the expected return of the portfolio is maximized and the risk is minimized. In this bachelor thesis some LP solvable portfolio optimization models are studied. Application on real life financial data is also included. Model with Conditional Value at Risk, MAD-model and Minimax model are described. In numerical analysis data from Frankfurt Stock Exchange are used and optimization has been made by Wolfram Mathematica 9.0 function LinearProgramming. As a result we got optimal portfolios for eleven different models for each of six minimal expected return constraints. The portfolios have been then evaluated according to the data from next year period. Powered by TCPDF (www.tcpdf.org)
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Scenario structures in multistage stochastic programs
Harcek, Milan ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
This thesis deals with multi-stage stochastic programming in the context of random process representation. Basic structure for random process is a scenario tree. The thesis introduces general and stage-independent scenario tree and their properties. Scenario trees can be also combined with Markov chains which describe the state of the system and determine which scenario tree should be used. Another structure which enables reduce the complexity of the problem is a scenario lattice. Scenario generation is performed using moment method. Scenario trees are used for representation of random returns as the input to the investment problem.
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