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Characterization of convex sets
Lžičař, Jiří ; Lachout, Petr (advisor) ; Kozmík, Václav (referee)
The idea of convexity is very important especially for probability theory, optimization and stochastic optimization. Convexity is a unique set pro- perty in many ways, which is worth to be studied. Various properties of convex sets are generally known, such as the ones related to separability. It however becomes apparent that the definition of convexity is very interesting, since it is possible to replace the definition by various collections of properties which are equivalent to it. There also exist set operations preserving convexity and another ones which preserve it when supported by another requirements. 1
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Multi-Stage Stochastic Programming with CVaR: Modeling, Algorithms and Robustness
Kozmík, Václav ; Dupačová, Jitka (advisor) ; Morton, David (referee) ; Kaňková, Vlasta (referee)
Multi-Stage Stochastic Programming with CVaR: Modeling, Algorithms and Robustness RNDr. Václav Kozmík Abstract: We formulate a multi-stage stochastic linear program with three different risk measures based on CVaR and discuss their properties, such as time consistency. The stochastic dual dynamic programming algorithm is described and its draw- backs in the risk-averse setting are demonstrated. We present a new approach to evaluating policies in multi-stage risk-averse programs, which aims to elimi- nate the biggest drawback - lack of a reasonable upper bound estimator. Our approach is based on an importance sampling scheme, which is thoroughly ana- lyzed. A general variance reduction scheme for mean-risk sampling with CVaR is provided. In order to evaluate robustness of the presented models we extend con- tamination technique to the case of large-scale programs, where a precise solution cannot be obtained. Our computational results are based on a simple multi-stage asset allocation model and confirm usefulness of the presented procedures, as well as give additional insights into the behavior of more complex models. Keywords: Multi-stage stochastic programming, stochastic dual dynamic programming, im- portance sampling, contamination, CVaR
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Sample approximation technique in stochastic programming
Vörös, Eszter ; Branda, Martin (advisor) ; Kozmík, Václav (referee)
Title: Sample approximation technique in stochastic programming Author: Eszter V¨or¨os Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Martin Branda, Ph.D., Department of Probability and Mathematical Statistics Abstract: This thesis deals with the problem of stochastic programming. Sto- chastic problems are usually applied for optimalization problems involving uncer- tain parameters. The problem, which we are aimed to solve, is approximated with the so-called sample average approximation method. The sample used to estimate the true problem is generated by the Monte Carlo method. This technique allows us to use standard algorithms for the further treatment of the problem. The aim of this thesis is to discuss the convergence properites of the optimal value and the optimal solution of the approximed problem to the optimal value and the optimal solution of the real problem. The thesis ends with a practical demonstration of the theoretical results on a portfolio optimization problem. Keywords: stochastic programming, sample average approximation, Monte Carlo method, portfolio optimization 1
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Portfolio efficiency with continuous probability distribution of returns
Kozmík, Václav
Present work deals with the portfolio selection problem using mean-risk models. The main goal of this work is to investigate the convergence of approximate solutions using generated scenarios to the analytic solution and its sensitivity to chosen risk measure and probability distribution. The considered risk measures are: variance, VaR, cVaR, absolute deviation and semivariance. We present analytical solutions for all risk measures under the assumption of normal or Student distribution. For log-normal distribution, we use the approximate assumption that the sum of log-normal random variables has log-normal distribution. Optimization models for discrete scenarios are derived for all risk measures and compared with analytical solution. In case of approximate solution with scenarios, we repeat the procedure multiple times and present our own approach to finding the optimal solution using the cluster analysis. All optimization models are written in GAMS language. Testing and estimating are realized using an application developed in C++ language.
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Optimization of flow in graph
Popovič, Viktor ; Lachout, Petr (advisor) ; Kozmík, Václav (referee)
When it comes to maximization of effectively or minimizing of cost, optimization represents the key activity. There is a number of practical examples that can be implemented into Theory of Graphs and subsequently optimized. This thesis includes the introduction to transportation problem where the consumer demand is met by the lowest price. Also there is maximum flow problem which is to transfer maximum of commodity (petroleum, gas...) through the network where each edge has a capacity restriction. We will also look into the alternative situations where we will maximize the flow along with minimizing of cost. To resolve these problems we will establish numeric algorithms like distribute method, labeling algorithm, shortest augmented path algorithm, and Preflow-Push algorithms. We will also illustrate functionality on example which confirm appropriate application of algorithms and differences among them.
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Mean absolute deviation risk measure
Janouchová, Petra ; Kozmík, Václav (advisor) ; Branda, Martin (referee)
This bachelor thesis considers the mean absolute deviation as a risk me- asure. It deals with its properties and its application in the case of the asset allocation problem. The Markowitz model is described and we demonstrated the relation between our model with mean absolute deviation and the Mar- kowitz model. We study the influence of changes in the input data for the linear model with mean absolute deviation. The primary data used in this thesis are historical relative rates of profit of shares in the Prague Stock Ex- change. The testing is done on the selected subsets of scenarios from primary data and the stability is discussed in conclusion.
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Investment problems with stochastic dominance constraints
Dorová, Bianka ; Kopa, Miloš (advisor) ; Kozmík, Václav (referee)
This thesis focuses on stochastic dominance in portfolio selection problems. The thesis recalls basic knowledge from the area of portfolio optimization with utility functions and first, second, $N$-th and infinite order of stochastic dominance. It sumarizes Post's, Kuosmanen's and Kopa's criteria for portfolio efficiency and necessary and sufficient conditions of stochastic dominance for discrete and continuous probability distributions. The thesis also contains formulations of optimization problems with second order stochastic dominance constraints derived for discrete and continuous probability distributions. A practical application is also a part of the thesis, where the optimization problems for monthly returns of Czech stocks are solved using optimization software GAMS.
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Optimization and stress tests
Fašungová, Diana ; Dupačová, Jitka (advisor) ; Kozmík, Václav (referee)
Title: Optimization and stress tests Author: Diana Fašungová Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Jitka Dupačová, DrSc., Department of Probability and Mathematical Statistics Abstract: In the thesis we apply contamination technique on a portfolio optimiza- tion problem using minimization of risk measure CVaR. The problem is considered from a risk manager point of view. We stress correlation structure of data and of revenues using appropriately chosen data for this kind of problem and for ge- nerated stress scenarios. From behaviour of CVaR with regard to contamination bounds, we formulate recommendations for the risk manager optimizing his port- folio. The recommendations are interpreted for both types of stress scenarios. In the end, limitations of the model and possible ways of improvement are discussed. Keywords: contamination bounds, stress tests, portfolio optimization, risk mana- gement
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