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Stochastic activity networks
Sůva, Pavel ; Dupačová, Jitka (advisor) ; Kaňková, Vlasta (referee)
In the present work, stochastic network models representing a project as a set of activities are studied, as well as different approaches to these models. The critical path method, stochastic network models with probability constraints, finding a reference project duration, worst-case analysis in stochastic networks and optimization of the parameters of the probability distributions of the activity durations are studied. Use of stochastic network models in telecommunications networks is also briefly presented. In a numerical study, some of these models are implemented and the~related numerical results are analyzed.
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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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Scenario generation for multidimensional distributions
Olos, Marek ; Dupačová, Jitka (advisor) ; Kaňková, Vlasta (referee)
Some methods for generating scenarios from multidimensional distribution assume we are able to generate scenarios from the one-dimensional distribution. We dedicate chapter 3 to this problem. At the end of the chapter, we provide references for applicable algorithms. Chapter 4 is focused on selected methods for generating scenarios from multidimensional distributions. In chapter 4.3, we introduce an algorithm for generating scenarios, which do not use any assumption about the distribution, except the first four moments and correlations to be specified. A method of generating scenarios based on approximation of multivariate normal distribution by the binomial distribution is described in chapter 4.5. Dimension reduction technique using principal components is presented in chapter 4.4. The algorithm is presented under the assumption of normal distribution. In chapter 4.6, we introduce the basics of the copula theory and a method for generating scenarios by C-vine copula. In chapter 5, we implement selected methods for generating scenarios for the estimation of daily value at risk for selected indexes and we discuss the results. Powered by TCPDF (www.tcpdf.org)
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A Note on Stochastic Optimization Problems with Nonlinear Dependence on a Probability Measure
Kaňková, Vlasta
Nonlinear dependence on a probability measure begins to appear (last time) in a stochastic optimization rather often. Namely, the corresponding type of problems corresponds to many situations in applications. The nonlinear dependence can appear as in the objective functions so in a constraints set. We plan to consider the case of static (one-objective) problems in which nonlinear dependence appears in the objective function with a few types of constraints sets. In details we consider constraints sets “deterministic”, depending nonlinearly on the probability measure, constraints set determined by second order stochastic dominance and the sets given by mean-risk problems. The last case means that the constraints set corresponds to solutions those guarantee an acceptable value in both criteria. To introduce corresponding assertions we employ the stability results based on the Wasserstein metric and L1 norm. Moreover, we try to deal also with the case when all results have to be obtained (estimated) on the data base.
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Mean-Risk Optimization Problem via Scalarization, Stochastic Dominance, Empirical Estimates
Kaňková, Vlasta
Many economic and financial situations depend simultaneously on a random element and on a decision parameter. Mostly it is possible to influence the above mentioned situation by an optimization model depending on a probability measure. We focus on a special case of one-stage two objective stochastic “Mean-Risk problem”. Of course to determine optimal solution simultaneously with respect to the both criteria is mostly impossible. Consequently, it is necessary to employ some approaches. A few of them are known (from the literature), however two of them are very important: first of them is based on a scalarizing technique and the second one is based on the stochastic dominance. First approach has been suggested (in special case) by Markowitz, the second approach is based on the second order stochastic dominance. The last approach corresponds (under some assumptions) to partial order in the set of the utility functions.\nThe aim of the contribution is to deal with the both main above mentioned approaches. First, we repeat their properties and further we try to suggest possibility to improve the both values simultaneously with respect to the both criteria. However, we focus mainly on the case when probability characteristics has to be estimated on the data base.
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A Note on Optimal Value of Loans
Kaňková, Vlasta
People try to gain (in the last decades) own residence (a flat or a little house). Since young people do not posses necessary financial resources, bank sector offers them a mortgage. Of course, the aim of any bank is to profit from such a transaction. Therefore, according to their possibilities, the banks employ excellent experts to analyze the financial situation of potenitial clients. Consequently, the banks know what could be a maximal size of the loan (in dependence on the debtor's position, salary and age) and what is reasonable size of installments. The aim of this contribution is to analyze the situation from the second size. In particular, the aim is to investgate the possibilities of the debtors not only on the dependence on their present - day situation, but also on their future private and subjective decisions and on possible “unpleasant” events. Moreover, consequently according to these indexes, the aim of this contribution is to suggest a method for a recognition of a “safe” loan and simultaneously to offer tactics to state a suitable environment for future time.The stochastic programming theory will be employed to it.
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Multi-Objective Optimization Problems with Random Elements - Survey of Approaches
Kaňková, Vlasta
Many economic and financial situations depend simultaneously on a random element and a decision parameter. Mostly, it is possible to influence the above mentioned situation only by an optimization model depending on a probability measure. This optimization problem can be static (one-stage), dynamic with finite or infinite horizon, single-objective or multi-objective. We focus on one-stage multi-objective problems corresponding to applications those are suitable to evaluate simultaneously by a few objectives. The aim of the contribution is to give a survey of different approaches (as they are known from the literature) of the above mentioned applications. To this end we start with well-known mean-risk model and continue with other known approaches. Moreover, we try to complete every model by a suitable application. Except an analysis of a choice of the objective functions type we try to discuss suitable constraints set with respect to the problem base, possible investigation and relaxation. At the end we mention properties of the problem in the case when the theoretical „underlying“ probability measure is replaced by its „deterministic“ or „stochastic“ estimate.
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Multicriteria and robust extension of news-boy problem
Šedina, Jaroslav ; Kopa, Miloš (advisor) ; Kaňková, Vlasta (referee)
This thesis studies a classic single-period stochastic optimization problem called the newsvendor problem. A news-boy 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 price-dependent 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
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Optimal Value of Loans via Stochastic Programming
Kaňková, Vlasta
A question of mortgage leads to serious and complicated problems of financial mathematics. On one side is a bank with an aim to have a “good” profit, on the other side is the client trying to invest money safely, with possible “small” risk.Let us suppose that a young married couple is in a position of client. Young people know that an expected and also unexpected unpleasant financial situation can happen. Many unpleasant financial situation can be caused by a random factor. Consequently stochastic methods are suitable to secure against them. The aim of the suggested model is not only to state a maximal reasonable value of loans, but also to endure unpleasant financial period. To this end we employ stochastic optimization theory. A few suitable models will be introduced. The choice of the model depends on environment of the young people. Models will be with “deterministic” constraints, probability constraints, but also with stochastic dominance constraints. The suggested models will be analyzed both from the numerical point of view and from possible method solution based on data. Except static one-objective problem we suggest also multi–objective models.
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