
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 nonlinear 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...


Minimax in scheduling problems under uncertainty
Jeliga, Jan ; Branda, Martin (advisor) ; Lachout, Petr (referee)
In this work, we deal with fixed interval scheduling problems with the possibility of random delay of the end of the tasks (FIS). First, we pre sent the basic deterministic FIS problems and ways to solve them. Next, we introduce the concept of minimax and present two wellknown and one new FIS problem under uncertainty, when random task delays are conside red to belong to a certain uncertainty set. Next, we deal with the solution of previously presented FIS problems for five chosen uncertainty sets. We present both previously achieved and original results. The work concludes with a summary of a numerical study of two problems. First, we explore the possibility of Lagrange relaxation application to the first presented problem. Next we explore the quality of approximation allowing to solve the later of problems as LP. 1


Optimization problems with decisiondependent uncertainty
Šípka, Stanislav ; Branda, Martin (advisor) ; Lachout, Petr (referee)
In practical optimization problems, uncertainty in parameter values is often present. This uncertainty needs to be taken in account when taking reallife de cisions. Such issues, where the parameters of the problem lie in the sets with a given shape, can be solved by a type of linear optimization called robust linear optimization. Special cases of these robust optimization are problems, where the sets depend on decisions. In this thesis we will focus on these special problems. The main aim of this thesis is to reformulate the classical form of the problems, leading to formulations which can be solved by standard computational software. We will then use these formulations in numerical study, focusing on behavior of robust shortest path in graphs. 1


Optimization in energy problems
Fürst, Matouš ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
Title: Optimization in energy problems Author: Matouš Fürst Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Ing. Miloš Kopa, Ph.D., Department of Probability and Mathematical Statistics Abstract: In this thesis we present an optimization model of a semiautonomous household, which aims to make energy management more efficient. The household is equipped with solar panels and an electric vehicle with a highcapacity battery. In the first part we summarize the basic properties of linear programming and two stage stochastic linear programming. Subsequently, a twostage stochastic linear program is formulated and solved in order to optimize the purchase, sale and storage of energy in the household during a single day. The program is formulated in two versions  with present and with departing vehicle. The final solution represents optimal decisions of the household and we discuss it with respect to the input data. In both versions the solution leads to a substantial reduction in costs compared to a household without a battery. Keywords: stochastic optimization, linear programming, domestic microgrid 1


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


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


Market model with random inputs
Krch, Ivan ; Lachout, Petr (advisor) ; Večeř, Jan (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


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


Optimization problems with chance constraints
Drobný, Miloslav ; Adam, Lukáš (advisor) ; Lachout, Petr (referee)
Autor se v diplomové práci zabývá optimalizačními úlohami s pravděpodob nostními omezeními. Konkrétně pak situacemi, kdy není známo pravděpo dobnostní rozdělení přítomného náhodného efektu. K řešení těchto problém· lze přistoupit metodami optimistických a pesimistických scénář·, kdy z dané rodiny možných pravděpodobnostních rozdělení vybíráme bu¤ nejpříznivější možnou variantu, nebo naopak tu nejméně výhodnou. Optimalizační úlohy s pravděpodobnostními omezeními formulovanými pomocí výše zmíněných přístup· byly za učinění jistých předpoklad· transformovány do jednoduš ších a řešitelných optimalizačních úloh. Dosažené výsledky byly aplikovány na reálná data z oblastí optimalizace portfolia a zpracování obrazu. 1
