
Market model with random inputs
Krch, Ivan ; Lachout, Petr (advisor) ; Branda, Martin (referee)
The thesis deals with market models with random inputs represented by the newsvendor problem for which the randomness is given through a random number of customers. Presented work is divided into three chapters. In the first chapter we present the elementar newsvendor problem as stochastic programming problem with a fixed recourse. In the second chapter we present the multiplayer game theory adapted to the newsvendors problem. Moreover, in the second chapter we extend the problem by the second newsvendor on the market and in the third chapter we generalize the problem for n newsvendors on the market. We deal with the situations that arise in the chapters two and three from the game theory point of view and we study characteristics of a Nash equilibrium. Presented theory is demonstrated on illustrative examples in the ends of the two last chapters. 1


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


A verification of an approximation of the continuous double auction by a sequence of call auctions.
Kubík, Petr ; Šmíd, Martin (advisor) ; Branda, Martin (referee)
The thesis deals with two kinds of double auction  with the continuous auction and a sequence of call auctions. We explain their rules and we define their models. We present results of simulations of the both kinds of double auction  the aim is to look for the call auction with such parameters that the prices and the traded volume of the continuous auction are approximated best. Finally, in a theoretical part, we characterize the dis tribution of the order book in the continuous auction and then we specify the joint distribution of the price and the traded volume in the call auction (the distribution of bid, ask and the traded volume given by the continuous auction may be immediately devised from the distribution of the order book).


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.


Network flows in scheduling problems
Rubín, Daniel ; Branda, Martin (advisor) ; Lachout, Petr (referee)
The goal of scheduling problems is to assign machines to a prespecified jobs which require processing. Standard approach leads to integer programming pro blems where machine assignment is represented by binary variables. However, the resulting problems are of high time complexity. Formulating the scheduling problems in terms of network flows shows to be a more effective approach. The aim of this thesis is to introduce basic scheduling tasks and methods used to formulate them in terms of network flows. By means of total unimodularity, we show that network flow algorithms are suitable for solving such problems. Finally, the results are demonstrated in a numerical study. 1


Benders decomposition in optimization
Minaříková, Michaela ; Branda, Martin (advisor) ; Rusý, Tomáš (referee)
The Bachelor thesis is dealing with Benders decomposition in optimization, especially in stochastic linear programming. In the begining the reader will be introduced to the important terms used in the decomposition algorithm. Con sequently it is demonstrated how to reformulate the problem of stochastic linear programming to a special structure suitable for Benders decomposition. In the third chapter, the decomposition algorithm, using the feasibility and optimality cuts, is explained including conditions of convergence of the algorithm. There follows modification of algorithm for two stage stochastic linear programming. Finally, we illustrate Benders algorithm on two smaller problems. 1


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
