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Scenario trees in stochastic programming problems
Malá, Alena ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
This thesis deals with multi-stage stochastic linear programming and its ap- plictions in the portfolio selection problem. It presents several models of invest- ment planning, the emphasis is on the basic model with transaction costs and risk adjusted model for every investment level. Random returns entering the above models are modelled by the scenario trees which are generated using the moment- matching method. The thesis presents the optimal investment strategy for each model. It then examines distance of optimal values of objective functions in de- pendence on the nested distance of these generated trees. All calculations were performed using Mathematica software version 9. 1
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Empiciral Estimates in Stochastic Programming; Dependent Data
Kolafa, Ondřej ; Kaňková, Vlasta (advisor) ; Dupačová, Jitka (referee)
This thesis concentrates on stochastic programming problems based on empirical and theoretical distributions and their relationship. Firstly, it focuses on the case where the empirical distribution is an independent random sample. The basic properties are shown followed by the convergence between the problem based on the empirical distribution and the same problem applied to the theoretical distribution. The thesis continues with an overview of some types of dependence - m-dependence, mixing, and also more general weak dependence. For sequences with some of these types of dependence, properties are shown to be similar to those holding for independent sequences. In the last section, the theory is demonstrated using numerical examples, and dependent and independent sequences, including sequences with different types of dependence, are compared.
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Stochastic models in consumer theory
Vlčková, Ivona ; Kopa, Miloš (advisor) ; Dvořák, Marek (referee)
The goal of this bachelor's thesis is the stochastic extension of deterministic models with a special attention to finding the consumer's optimum. The first chapter is devoted to utility theory, defining basic notions and it also studies characteristics of utility functions and indifference curves. Futhermore, the consumer's optimum is described from the perspective of marginal utility. At the end of this chapter, optimization problems are provided, including their solutions, and finally we examine the effect of price changes of particular goods and income. This effect is described and solved via the substitution and income effect. The second chapter shows the stochastic extension of the above mentioned models. Using the quantile function, we obtain the optimum in the case of random income and we use mean values for the case of random prices. Eventually, a considerable part of this work is devoted to scenario theory, which is also used in the final example.
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Stochastic programming problems with chance constraints
Harcek, Milan ; Branda, Martin (advisor) ; Kopa, Miloš (referee)
The thesis presents stochastic programming with chance contraints. We begin with the definition of convex set, convex and concave function and we study the convexity of programs with deterministic constraints. We continue with the definition of quasi-concave and quasi-convex function. After that, we put our mind to probabilistic constraints and the convexity of feasible set and show the formulation of joint and separate probabilistic constraints. We discuss properties of feasible set in general case, without any assumptions concerning the probability distribution of random variable. Finally, we apply our theory to random vectors with finite discrete distribution and multiva- riate normal distribution. 1
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Stochastic Programming Problems via Economic Problems
Kučera, Tomáš ; Kaňková, Vlasta (advisor) ; Dupačová, Jitka (referee)
This thesis' topic is stochastic programming, in particular with regard to portfolio optimization and heavy tailed data. The first part of the thesis mentions the most common types of problems associated with stochastic programming. The second part focuses on solving the stochastic programming problems via the SAA method, especially on the condition of data with heavy tailed distributions. In the final part, the theory is applied to the portfolio optimization problem and the thesis concludes with a numerical study programmed in R based on data collected from Google Finance.
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Engineering Process Optimization
Pluskal, Jaroslav ; Hrabec, Dušan (referee) ; Popela, Pavel (advisor)
This bachelor's thesis deals with optimization with emphasis on Newsvendor model and its usage. An overview of basic terms and theory related to probability, mathematical analysis and optimization is mentioned at the beginning. The main aim of this thesis is to formulate a Newsvendor problem in its basic form and then demonstrate the impact of various demand distributions. After that the gained knowledge is used to solve a project of factory, for which we want to set optimal parameters. Software GAMS is used to model and solve the project.
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Stochastic Optimization of Network Flows
Málek, Martin ; Holešovský, Jan (referee) ; Popela, Pavel (advisor)
Magisterská práce se zabývá stochastickou optimalizací síťových úloh. Teoretická část pokrývá tři témata - teorii grafů, optimalizaci a progressive hedging algoritmus. V rámci optimalizace je hlavní část věnována stochastickému programování a dvoustupňovému programování. Progressing hedging algoritmus zahrnuje také metodu přiřazování scénářů a modifikaci obecného algoritmu na dvou stupňové úlohy. Praktická část je věnována modelům na reálných datech z oblasti svozu odpadu v rámci České republiky. Data poskytl Ústav procesního inženýrství.
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Nonconvex stochastic programming problems-formulations, sample approximations and stability
Branda, Martin ; Lachout, Petr (advisor) ; Kaňková, Vlasta (referee) ; H.van der Vlerk, Maarten (referee)
Title: Nonconvex stochastic programming problems - formulations, sample approximations and stability Author: RNDr. Martin Branda Author's e-mail address: branda@karlin.mff.cuni.cz Supervisor: Doc. RNDr. Petr Lachout, CSc. Supervisor's e-mail address: lachout@karlin.mff.cuni.cz Abstract: We deal with problems where integer variables may appear, hence no assumptions on convexity are made throughout this thesis. The goal of Chapter 2 is to introduce stochastic programming problems and to outline the most important tasks connected with solving the problems. In Chapter 3, we compare basic formulations of static stochastic programming problems with chance constraints, with integrated chance constraints and with penalties in the objective function. We show that the problems are asymptotically equivalent under mild conditions. We discuss solving the problems using sample approximation techniques and extend some results on rates of convergence. All the formulations and corresponding sample approximations are compared on an investment problem with real features with Value at Risk constraint, integer allocations and transaction costs. Then, stability of financial decision models where two-stage mixed-integer value function appears as a loss variable is studied. In Chapter 4, we study qualitative properties of the...
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Transformations of optimization models with aplications
Rychtář, Adam ; Bednář, Josef (referee) ; Popela, Pavel (advisor)
The thesis deals with recent problems of waste management in the Czech Republic. In connection with the existing software implementation, the author focuses on the gradual development of advanced mathematical programming models, which generalize existing approaches. The author applies acquired knowledge in the areas of network flows, linear, integer, and stochastic programming. The important role is played by modifications and transformations of the discussed models. They are further used to obtain the experimental results for real-world input data by implementation in GAMS.
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Effective Investment Planning in Waste-to-Energy Systems
Šomplák, Radovan ; Klemeš,, Jiří (referee) ; Žaloudík, Petr (referee) ; Stehlík, Petr (advisor)
PhD thesis deals with the application of the simulation and optimization methods in the waste-to-energy field. An introduction describes the current state of the waste management in the EU with the focus on the Czech Republic. In the following chapter the evaluation criteria for investment intentions and the basic principles of stochastic programming are discussed. The core of the work lays in the mathematical models for the planning and operation of the process plants as well as in the mathematical models for the waste collection. The transportation problem involves all considered technological elements and therefore it is possible to simulate the waste streams between the producers and processors. This approach is demonstrated with five case studies. In the first three studies the calculations for the potential investor are presented. The main outcome of these case studies is the determination of the level of attractiveness of investment and the identification the greatest risks. Another case study is devoted to an analysis with the focus on perspective of government policies and in the last case study the issue of the waste management is analyzed in detail from the perspective of the waste producers. Developed computational tools are flexible and can be further developed and adapted based on the objectives of the specific tasks.
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