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Robust approaches in portfolio optimization with stochastic dominance
Kozmík, Karel ; Kopa, Miloš (advisor)
We use modern approach of stochastic dominance in portfolio optimization, where we want the portfolio to dominate a benchmark. Since the distribution of returns is often just estimated from data, we look for the worst distribution that differs from empirical distribution at maximum by a predefined value. First, we define in what sense the distribution is the worst for the first and second order stochastic dominance. For the second order stochastic dominance, we use two different formulations for the worst case. We derive the robust stochastic dominance test for all the mentioned approaches and find the worst case distribution as the optimal solution of a non-linear maximization problem. Then we derive programs to maximize an objective function over the weights of the portfolio with robust stochastic dominance in constraints. We consider robustness either in returns or in probabilities for both the first and the second order stochastic dominance. To the best of our knowledge nobody was able to derive such program before. We apply all the derived optimization programs to real life data, specifically to returns of assets captured by Dow Jones Industrial Average, and we analyze the problems in detail using optimal solutions of the optimization programs with multiple setups. The portfolios calculated using...
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Robust approaches in portfolio optimization with stochastic dominance
Kozmík, Karel ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
We use modern approach of stochastic dominance in portfolio optimization, where we want the portfolio to dominate a benchmark. Since the distribution of returns is often just estimated from data, we look for the worst distribution that differs from empirical distribution at maximum by a predefined value. First, we define in what sense the distribution is the worst for the first and second order stochastic dominance. For the second order stochastic dominance, we use two different formulations for the worst case. We derive the robust stochastic dominance test for all the mentioned approaches and find the worst case distribution as the optimal solution of a non-linear maximization problem. Then we derive programs to maximize an objective function over the weights of the portfolio with robust stochastic dominance in constraints. We consider robustness either in returns or in probabilities for both the first and the second order stochastic dominance. To the best of our knowledge nobody was able to derive such program before. We apply all the derived optimization programs to real life data, specifically to returns of assets captured by Dow Jones Industrial Average, and we analyze the problems in detail using optimal solutions of the optimization programs with multiple setups. The portfolios calculated using...
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Robust optimization in classification and regression problems
Semela, Ondřej ; Kalina, Jan (advisor) ; Lachout, Petr (referee)
In this thesis, we present selected methods of regression and classification analysis in terms of robust optimization which aim to compensate for data imprecisions and measurement errors. In the first part, ordinary least squares method and its generalizations derived within the context of robust optimization - ridge regression and Lasso method are introduced. The connection between robust least squares and stated generalizations is also shown. Theoretical results are accompanied with simulation study investigating from a different perspective the robustness of stated methods. In the second part, we define a modern classification method - Support Vector Machines (SVM). Using the obtained knowledge, we formulate a robust SVM method, which can be applied in robust classification. The final part is devoted to the biometric identification of a style of typing and an individual based on keystroke dynamics using the formulated theory. Powered by TCPDF (www.tcpdf.org)
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Robust optimization for solution of uncertain optimization programs
Komora, Antonín ; Dupačová, Jitka (advisor) ; Kopa, Miloš (referee)
Robust optimization is a valuable alternative to stochastic programming, where all underlying probabilistic structures are replaced by the so-called uncertainty sets and all related conditions must be satisfied under all circumstances. This thesis reviews the fundamental aspects of robust optimization and discusses the most common types of problems as well as different choices of uncertainty sets. It focuses mainly on polyhedral and elliptical uncertainty and for the latter, in the case of linear, quadratic, semidefinite or discrete programming, computationally tractable equivalents are formulated. The final part of this thesis then deals with the well-known Flower-girl problem. First, using the principles of robust methodology, a basis for the construction of the robust counterpart is provided, then multiple versions of computationally tractable equivalents are formulated, tested and compared. Powered by TCPDF (www.tcpdf.org)
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