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Analysis of portfolio efficiency sets
Fehérová, Veronika ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
Pøedlo¾ená práce se zabývá dvìma pøístupy øe¹ení problému volby portfolia. Prvním jsou čmean-riskÿ modely, které minimalizují riziko pro pøedem zvolený výnos nebo maximalizují výnos pro pevnì stanovené riziko. Druhým je princip stochastické dominance, úzce související s teorií u¾itku. Cílem této diplomové práce je zkoumat vztah mezi mno¾inami e cientních portfolií, které jsou øe¹e- ním v obou pøístupech. Pro kvanti kaci rizika se kromì základních mìr jako jsou rozptyl, V aR nebo CV aR v práci uva¾ují i spektrální míry, zohledòující sub- jektivní postoj investora k riziku. Uká¾eme, za platnosti jakých podmínek jsou modely minimalizující spektrální míry konzistentní se stochastickou dominancí druhého øádu (SSD). Aplikujeme Kopa-Postùv test, který je jedním z více testù na SSD e cienci portfolia, na reálná data z americké burzy cenných papírù a SSD e cientní portfolia porovnáme s e cientními portfoliami získanými minimalizací CV aR-u uva¾ovaného na rùznych hladinách spolehlivosti. 1
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Bilevel optimization problems and their applications to portfolio selection
Goduľová, Lenka ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
Title: Bilevel optimization problems and their applications to portfolio selection Author: Lenka Godul'ová Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Ing. Miloš Kopa, Ph.D. Abstract: This work deals with the problem of bilevel tasks. First, it recalls the basic knowledge of mean-risk models, risk measure in singlelevel problems, and second degree stochastic dominance. Then it presents basic knowledge of bilevel tasks. bilevel problems have several advantages over singlelevel. In one process, it is possible to analyze two different or even conflicting situations. The bilevel role can better capture the relationship between the two objects. The main focus of the thesis is the formulation of various bilevel tasks and their reformulation into the simplest form. The numerical part deals with four types of formulated bilevel problems at selected risk measures. Keywords: Bilevel problems, Second degree stochastic dominance, Risk measures 1
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Algorithms for solving two-stage stochastic programs
Vlčková, Ivona ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
The thesis deals with the algorithms for two-stage stochastic programs. The first chapter considers the basic properties and theory. Specifically, we introduce the properites of the feasibility region and the objective function. Further, optimality conditions are discussed. In the second chapter we present algoritms which can be used to solve two-stage linear programs with fixed recourse. In the first section the basic L-shaped method is described in detail. The second section provides an explanation of the Stochastic Decomposition algorithm with the inclusion of a regularization term. The last chapter presents computational results. Three practical examples are provided both with a brief description of the problem and solutions by the studied algorithms.
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Fixed interval scheduling problems - stochastic extensions, formulations and algortihms
Leder, Ondřej ; Branda, Martin (advisor) ; Kopa, Miloš (referee)
Fixed interval scheduling problems have wide range of practical use in production planning, transportation, in hospitals or in schools when planning timetables. When solving these problems we often encounter requirement of integrality of solutions. Ignoring this condition is often not possible. In this thesis we propose some formulations of scheduling problems and their stochastic extensions. We also propone a new formulation of stochastic FIS problem, for which integrality of solution is byproduct of its definition. We present Gâteaux derivative and its relationship to stability of optimal value function of stochastic optimization problems under the influence of contamination. We propose a new theorem on the stability of such functions for fixed interval scheduling problems.
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Scheduling optimization problems in education
Puček, Samuel ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
This work deals with the theory of integer programming. After defining the ba- sic concepts, it presents two algorithms suitable for solving integer problems. Firstly, it talks about the branch and bound algorithm and secondly, it talks about the cutting plane algorithm. Next, it presents an assignment problem, which is a special case of integer programming. The work describes the hungarian method and explains its usage on exemplary examples. The last part of the work solves the real problem from the practice. The aim of this section is to find an optimal schedule for classes one to seven of the selected elementary school. It introduces input data processing, creating a model and the solution. Obtained results are accompanied by a brief discussion. 1
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Market of newsboys
Bureček, Tomáš ; Lachout, Petr (advisor) ; Kopa, Miloš (referee)
This thesis solves the task of a newspaper vendor, which fits into the classical tasks of stochastic programming. The work includes an extension to the market of newsboys and also considers various influences that vendors may suffer. A continuous version of the problem is considered. The task is to find the optimal amount of product the buyer can buy to maximize his yield. The thesis discusses the issue of solving this problem. First, the problem is demonstrated in a easier version, and later this problem is developed. Finally, we find an iterative algorithm that calculates the approximate solution and demonstrates its functionality on an example. 1
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Scenario reduction in Monte Carlo methods in optimization
Trégner, Tomáš ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
Tato práce se zabývá redukcí scénáøù pøi pou¾ití Monte Carlo metod. Hlavním cílem je posoudit, jaké výhody, èi zlep¹ení nám mù¾e redukce scénáøù poskytnout a zda nám mù¾e být v praxi u¾iteèná. V práci budeme prezentovat výsledky zís- kané pomocí vlastní implementace redukèního algoritmu v jazyku Python. Pro úèely posouzení efektivity redukce scénáøù byly vybrány dva konkrétní problémy. Prvním z nich je odhad konstanty π, který je pro tento úèel vhodný zejména proto, ¾e je znám pøesný výsledek. Druhým problém, na který se soustøedíme, je pak výbìr optimálního portfolia z daných akcií, který jsme vybrali proto, ¾e se jedná o pomìrnì nároèný a zajímavý problém umo¾òující posoudit èasovou efek- tivitu metody redukce scénáøù. Na základì na¹ich výpoètù docházíme k závìru, ¾e redukce scénáøù mù¾e být u¾iteèným nástrojem pro slo¾ité úlohy, je v¹ak tøeba si dávat pozor na vhodnou volbu pou¾ité metriky. 1
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Stochastic Programming Problems in Asset-Liability Management
Rusý, Tomáš ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
The main objective of this thesis is to build a multi-stage stochastic pro- gram within an asset-liability management problem of a leasing company. At the beginning, the business model of such a company is introduced and the stochastic programming formulation is derived. Thereafter, three various risk constraints, namely the chance constraint, the Value-at-Risk constraint and the conditional Value-at-Risk constraint along with the second-order stochastic dominance constraint are applied to the model to control for riski- ness of the optimal strategy. Their properties and their effects on the optimal decisions are thoroughly investigated, while various risk limits are considered. In order to obtain solutions of the problems, random elements in the model formulation had to be approximated by scenarios. The Hull - White model calibrated by a newly proposed method based on maximum likelihood esti- mation has been used to generate scenarios of future interest rates. In the end, the performances of the optimal solutions of the problems for unconsid- ered and unfavourable crisis scenarios were inspected. The used methodology of such a stress test has not yet been implemented in stochastic programming problems within an asset-liability management. 1
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SVM classifiers and heuristics for feature selection
Krupka, Tomáš ; Holub, Martin (advisor) ; Kopa, Miloš (referee)
In machine learning applications with a large number of computer-generated features, a selection of just a subset of features is often desirable. The Recursive Feature Elimination (SVM-RFE) algorithm proposed by Guyon et al. (2002) employs the mechanism of selecting the features based on their contribution to an SVM model decision rule, and has proven a state-of-the-art performance on the Gene Selection for Cancer Classification task (Tan et al. (2010)). This thesis expands on that work, and proposes a novel modification of the SVM-RFE feature selection method called Evaluation-Based RFE (EB-RFE). This heuristic significantly improves the performance of the SVM classifier in comparison to the original SVM-RFE on the studied machine learning task. In addition to the performance gain, the proposed algorithm has also, in experimental use, proven to have two other desirable properties. Firstly, EB-RFE produces much smaller feature subsets than SVM-RFE, which leads to more compact models. Secondly, unlike SVM-RFE, the EB-RFE heuristic is easily scalable with the computational time well beyond the possibilities of current high-end consumer CPUs. Powered by TCPDF (www.tcpdf.org)
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Stochastic dominance generated by a decreasing absolute risk aversion
Mrázková, Adéla ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
The aim of the thesis is to describe first order stochastic dominance, second order stochastic dominance and then to motivate and describe stochastic dominance generated by utility functions with a decreasing absolute risk aversion. A numerical application of described methods follows. Efficiency in the meaning of stochastic dominance generated by utility functions with a decreasing absolute risk aversion and second order stochastic dominance is tested. Connection between the results is clarified and used methods are compared in the meaning of computational demands. Powered by TCPDF (www.tcpdf.org)
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