
Stochastic dominance in portfolio optimization
Paulik, Marek ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
The main topic of this thesis is the application of stochastic dominance constrains to portfolio optimization problems. First, we recall Markowitz model. Then we present portfolio selection problems with stochastic dominance constraints. Finally, we compare performance of these two approaches in an empirical study presented in the last chapter.


Solving methods for bilevel optimization problems
Lžičař, Jiří ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
The presented thesis discusses bilevel programming problems with the focus on solution algorithms. Bilevel programming problem is a hierarchical programming problem, where constraints contain another programming problem. We formulate basic bilevel optimization theory and describe three types of so lution algorithms for bilevel programming problems: Algorithms based on KKT reformulation where the lower level is replaced by its KKT conditions, algorithms based on optimal value function where the bilevel programming problem is re duced to a single level problem using the optimal value function of the lower level problem, and algorithms solving linear bilevel programming problems. Using real data for portfolio optimization bilevel programming problems, we compare ability to solve the problems and computing time of some of the pre sented algorithms. 1


Multivariate stochastic dominance and its application in portfolio optimization problems
Petrová, Barbora ; Kopa, Miloš (advisor) ; Ortobelli, Sergio (referee) ; Branda, Martin (referee)
Title: Multivariate stochastic dominance and its application in portfolio optimization Problems Author: Barbora Petrová Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Ing. Miloš Kopa, Ph.D., Department of Probability and Mathematical Statistics Abstract: This thesis discusses the concept of multivariate stochastic dominance, which serves as a tool for ordering random vectors, and its possible usage in dynamic portfolio optimization problems. We strictly focus on different types of the firstorder multivariate stochastic dominance for which we describe their generators in the sense of von NeumannMorgenstern utility functions. The first one, called strong multivariate stochastic dominance, is generated by all nondecreasing multivariate utility functions. The second one, called weak multivariate stochastic dominance, is defined by relation between survival functions, and the last one, called the firstorder linear multivariate stochastic dominance, applies the firstorder univariate stochastic dominance notion to linear combinations of marginals. We focus on the main characteristics of these types of stochastic dominance, their relationships as well as their relation to the cumulative and marginal distribution functions of considered random vectors. Formulated...


Scenario structures in multistage stochastic programs
Harcek, Milan ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
This thesis deals with multistage stochastic programming in the context of random process representation. Basic structure for random process is a scenario tree. The thesis introduces general and stageindependent 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.


Portfolio optimization using risk premia
Novotná, Tereza ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
The main topic of this thesis is Portfolio Optimization Using Risk Premia. Basic terms are defined there such as utility function, investor's risk aversion, risk premia, absolute risk aversion measure and portfolio optimization. There are also stated important theorems about risk aversion. For better understanding, there can be found few examples. At the end of this thesis is shown empirical study. It presents how the restriction of risk premia affects optimal investment and other numerical results.


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 meanrisk 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


Assignment problem with application to heath service
Tlapák, Martin ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
This bachelor's thesis deals with the theory of the nurse scheduling problem using the theory of integer programming. That is why we define basic concepts and present basic theorem of integer programming. We present the algorithm made for solving integer programming. Next we define a basic concept of as sigment problem. We show how to solve assigment problem by the Hungarian method. Finally we solve the real nurse scheduling problem. The nurse's pref erences are included in our model. We are finding solution which is suitable for the labour code of the Czech Republic and the special requests of the stressful workplace. 1


A knapsack problem
Piskačová, Nikola ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
This work deals with the theory of integer programming. In the first part, there are defined the basic concepts and there are mentioned the two most used methods for solving integer problems. Namely, it is the Branch and Bound method and the Cutting Plane method. In the second chapter, there is described the Knapsack Problem and its various formulations. This problem is a special case of integer optimalization. Next, there is a practical part, where a real problem is solved. The problem is how to place the products in shelves in equipment in the most effectively way. In this chapter is described how to process input data, create a model and solve the problem. In the second part of the practical part, the basics of stochastic optimalization and solution of these problems by the Scenario method are presented. This method is used to solve the previously mentioned problem if the delivery days are random. The aim of this work is to show the applicability of formulations of Knapsack Problem and to compare the obtained results. 1


Comparison of statistical methods for the scoring models development
Mrázková, Adéla ; Vitali, Sebastiano (advisor) ; Kopa, Miloš (referee)
The aim of this thesis is to introduce and summarize the process of scoring model development in general and then basic statistical approaches used to resolve this problem, which are in particular logistic regression, neural networks and decision trees (random forests). Application of described methods on a real dataset provided by PROFI CREDIT Czech, a.s. follows, including discussion of some implementation issues and their resolution. Obtained results are discussed and compared.


Multicriteria and robust extension of newsboy problem
Šedina, Jaroslav ; Kopa, Miloš (advisor) ; Kaňková, Vlasta (referee)
This thesis studies a classic singleperiod stochastic optimization problem called the newsvendor problem. A newsboy must decide how many items to order un der the random demand. The simple model is extended in the following ways: endogenous demand in the additive and multiplicative manner, objective func tion composed of the expected value and Conditional Value at Risk (CVaR) of profit, multicriteria objective with pricedependent demand, multiproduct exten sion under dependent and independent demands, distributional robustness. In most cases, the optimal solution is provided. The thesis concludes with the nu merical study that compares results of two models after applying the Sample Average Approximation (SAA) method. This study is conducted on the real data. 1
