This thesis investigates robust techniques for mean-variance (MV) portfolio optimization problems under the estimation risk in mean return. We evaluate the performance of the optimal portfolios generated by the min-max robust MV portfolio optimization model. With an ellipsoidal uncertainty set based on the statistics of the sample mean estimates, minmax robust portfolios equal to the ones from the standard MV model based on the nominal mean estimates but with larger risk aversion parameters. With an interval uncertainty set for mean return, min-max robust portfolios can vary significantly with the initial data used to generate the uncertainty set. In addition, by focusing on the worst-case scenario in the mean return uncertainty set, min-max robust portfolios can be too conservative and unable to achieve a high return. Adjusting the conservatism level of min-max robust portfolios can only be achieved by excluding poor mean return scenarios from the uncertainty set, which runs counter to the principle of min-max robustness. We propose a CVaR robust MV portfolio optimization model in which the estimation risk is measured by the Conditional Value-at-Risk (CVaR). We show that, using CVaR to quantify the estimation risk in mean return, the conservatism level of CVaR robust portfolios can be more naturally adjusted by gradually including better mean return scenarios. Moreover, we compare min-max robust portfolios (with an interval uncertainty set for mean return) and CVaR robust portfolios in terms of actual frontier variation, portfolio efficiency, and portfolio diversification. Finally, a computational method based on a smoothing technique is implemented to solve the optimization problem in the CVaR robust model. We numerically show that, compared with the quadratic programming (QP) approach, the smoothing approach is more computationally efficient for computing CVaR robust portfolios.

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A direction-finding system with multiple direction finders (DFs) is a military intelligence system designed to detect the positions of transmitters of radio frequencies. This dissertation studies three decision problems associated with the direction-finding system. The first part of this dissertation is to prescribe DF deployment to maximize the effectiveness with which transmitter positions are estimated in an area of interest (AOI). Three methods are presented to prescribe DF deployment. The first method uses Stansfield’s probability density function to compute objective function coefficients numerically. The second and the third employ surrogate measures of effectiveness as objective functions. The second method, like the first, involves complete enumerations; the third formulates the problem as an integer program and solves it with an efficient network-based label-setting algorithm. Our results show that the third method, which involved use of a surrogate measure as an objective function and an exact label-setting algorithm, is most effective. The second part of this dissertation is to minimize the number of DFs to cover an AOI effectively, considering obstacles between DFs and transmitters. We formulate this problem as a partial set multicover problem in which at least -fraction of the likely transmitter positions must be covered, each by at least direction finders. We present greedy heuristics with random selection rules for the partial set multicover problem, estimating statistical bounds on unknown optimal values. Our results show that the greedy heuristic with column selection rule, which gives priority for selecting a column that advances more rows to k-coverage, performs best on the partial set multicover problems. Results also show that the heuristic with random row and column selection rules is the best of the heuristics with respect to statistical bounds. The third part of this dissertation deals with the problem of deploying direction finders with the goal of maximizing the effectiveness with which transmitter positions can be estimated in an AOI while hedging against enemy threats. We present four formulations, considering the probability that a direction finder deployed at a location will survive enemy threats over the planning horizon (i.e., not be rendered inoperative by an attack). We formulate the first two as network flow problems and present an efficient label-setting algorithm. The third and the fourth use the well-known Conditional Value at Risk (CVaR) risk measure to deal with the risk of being rendered inoperative by the enemy. Computational results show that risk-averse decision models tend to deploy some or all DFs in locations that are not close to the enemy to reduce risk. Results also show that a direction-finding system with 5 DFs provides improved survivability under enemy threats.

01 Abstract: This thesis is concerned with the robust methods in portfolio theory. Different risk measures used in portfolio management are introduced and the corresponding robust portfolio optimization problems are formulated. The analytical solutions of the robust portfolio optimization problem with the lower partial moments (LPM), value-at-risk (VaR) or conditional value-at-risk (CVaR), as a risk measure, are presented. The application of the worst-case conditional value-at-risk (WCVaR) to robust portfolio management is proposed. This thesis considers WCVaR in the situation where only partial information on the underlying probability distribution is available. The minimization of WCVaR under mixture distribution uncertainty, box uncertainty, and ellipsoidal uncertainty are investigated. Several numerical examples based on real market data are presented to illustrate the proposed approaches and advantage of the robust formulation over the corresponding nominal approach.

In this thesis, probability theory and risk analysis are used to determine the riskof wind energy stocks. Three stocks of wind energy companies and three stocksof technology companies are gathered and risks are compared. Three difffferent riskmeasures: variance, value at risk, and conditional value at risk are used in this thesis.Conclusions which has been drawn, are that wind energy company stock risks arenot signifificantly lower than the stocks of other companies. Furthermore, optimalportfolios should include short positions of one or two of the energy companies forthe studied time period and under the difffferent risk measures.

This research presents a bi-level stochastic network interdiction model on an attack graph to enable a risk-averse resource constrained cyber network defender to optimally deploy security countermeasures to protect against attackers having an uncertain budget. This risk-averse conditional-value-at-risk model minimizes a weighted sum of the expected maximum loss over all scenarios and the expected maximum loss from the most damaging attack scenarios. We develop an exact algorithm to solve our model as well as several acceleration techniques to improve the computational efficiency. Computational experiments demonstrate that the application of all the acceleration techniques reduces the average computation time of the basic algorithm by 71% for 100-node graphs. Using metrics called mean-risk value of stochastic solution and value of risk-aversion, numerical results suggest that our stochastic risk-averse model significantly outperforms deterministic and risk-neutral models when 1) the distribution of attacker budget is heavy-right-tailed and 2) the defender is highly risk-averse.

A growing need for clean and sustainable energy is causing a significant shift in the electricity generation paradigm. In the electricity system of the future, integration of renewable energy sources with smart grid technologies can lead to potentially huge economical and environmental benefits ranging from lesser dependency on fossil fuels and improved efficiency to greater reliability and eventually reduced cost of electricity. In this context, microgrids serve as one of the main components of smart grids with high penetration of renewable resources and modern control strategies. This dissertation is concerned with developing optimal control strategies to manage an energy storage unit in a grid-connected microgrid under uncertainty of electricity demand and prices. Two methods are proposed based on the concept of rolling horizon control, where charge/discharge activities of the storage unit are determined by repeatedly solving an optimization problem over a moving control window. The predicted values of the microgrid net electricity demand and electricity prices over the control horizon are assumed uncertain. The first formulation of the control is based on the scenario-based stochastic conditional value at risk (CVaR) optimization, where the cost function includes electricity usage cost, battery operation costs, and grid signal smoothing objectives. Gaussian uncertainty is assumed in both net demand and electricity prices. The second formulation reduces the computations by taking a worst-case CVaR stochastic optimization approach. In this case, the uncertainty in demand is still stochastic but the problem constraints are made robust with respect to price changes in a given range. The optimization problems are initially formulated as mixed integer linear programs (MILP), which are non-convex. Later, reformulations of the optimization problems into convex linear programs are presented, which are easier and faster to solve. Simulation results under different operation scenarios are presented to demonstrate the effectiveness of the proposed methods. Finally, the energy management problem in network of grid-connected microgrids is investigated and a strategy is devised to allocate the resulting net savings/costs of operation of the microgrids to the individual microgrids. In the proposed approach, the energy management problem is formulated in a deterministic co-operative game theoretic framework for a group of connected microgrids as a single entity and the individual savings are distributed based on the Shapley value theory. Simulation results demonstrate that this co-operation leads to higher economical return for individual microgrids compared to the case where each of them is operating independently. Furthermore, this reduces the dependency of the microgrids on the utility grid by exchanging power locally. / Thesis / Master of Applied Science (MASc)

[pt] No seu planejamento plurianual de investimentos, as organizações do setor de Exploração e Produção (EeP) estruturam alternativas de projetos de produção de petróleo e gás natural, sujeitas a diversas restrições e a incertezas técnicas e econômicas. Como não há como assegurar que os resultados dos projetos ocorram conforme o previsto, é possível que seu retorno seja inferior ao esperado, o que, dependendo da relevância, pode provocar um efeito adverso no resultado operacional e nas condições financeiras da companhia. Nesse mérito, a dissertação apresenta e aplica um modelo de programação estocástica linear inteira mista para seleção de portfólio de projetos que permita a maximização dos resultados, com restrição de risco. A aplicação considerou dados realistas do segmento de upstream de uma empresa do setor. Para representar os cenários econômicos, optou-se pela utilização da simulação de Monte Carlo do modelo Movimento Geométrico Browniano. Com o Valor Presente Líquido como retorno e Conditional Value-at-Risk representando a medida de risco, foi possível estabelecer a fronteira eficiente do risco-retorno, com a qual o decisor pode definir uma solução de portfólio, conforme sua aversão ao risco. / [en] In their multi-annual investment planning, oil and gas companies consider alternatives of production projects, subject to a variety of constraints, and technical and economic uncertainties. Considering that it is not possible to guarantee that these projects will perform as predicted, the return can be less than expected and can lead to a significant adverse effect to the operational results and to financial conditions of a given organization. Therefore, this dissertation proposes a mixed integer linear stochastic programming model for project portfolio selection that maximizes the return with risk constraint. The application considered realistic data from the upstream segment of an oil and gas company. Monte Carlo simulation of the Geometric Brownian Motion model was considered to represent the economic scenarios. Using the Net Present Value as the function and Conditional Value-at-Risk as a risk measure, it was possible to establish the efficient frontier of risk-return, which can assist the decision-maker to define the project portfolio according to their risk aversion.

[pt] Nesta dissertação será proposta uma metodologia que contabiliza o risco de inadimplência no mercado, decorrentes de débitos não pagos à câmara de comercialização de energia elétrica (CCEE) nas estratégias de contratação de geradores renováveis. As incertezas relacionadas à geração e ao preço de curto prazo são consideradas através da simulação de cenários exógenos ao modelo como habitual em otimização estocástica. A otimização robusta é empregada através de conjuntos de incerteza poliédricos a fim de modelar a inadimplência do mercado. Dessa maneira, a metodologia proposta se baseia em um modelo matemático híbrido, robusto e estocástico. De forma mais objetiva, um modelo de dois níveis é proposto com tantos problemas de segundo nível quanto o número de cenários considerados para a produção renovável. No primeiro nível, as decisões de contratação são feitas. Em seguida, para cada cenário de geração, o problema de segundo nível encontra a pior inadimplência com base na carteira de contratos encontrados pelo primeiro nível. Para resolver o problema, o modelo de dois níveis é reescrito como um problema linear equivalente de um único nível. O perfil de risco do agente é definido por meio do conhecido valor condicional em risco (conditional value-a-risk), uma medida coerente de risco. Para ilustrar a eficácia do modelo de contratação, são realizados estudos de casos com dados realistas do sistema de energia brasileiro. / [en] In this dissertation we propose a new methodology to account for the market default risk, arising from debts not paid to the market clearing house, in the renewable generators contracting strategy. Renewable generation and spot price uncertainties are considered through exogenous simulated scenarios as customary in stochastic optimization. Robust optimization with polyhedral uncertainty sets is employed to account for the market default. Thus, the proposed methodology is based on a hybrid robust and stochastic mathematical program. More objectively, a bi-level model is proposed with as many second-level problems as the number of scenarios considered for the renewable production. In the first level, contracting decisions are made. Then, for each generation scenario, a second-level problem finds the worst-case default based on the portfolio of contracts found by the first level. To solve the problem, the bi-level model is rewritten as a single-level equivalent linear problem. The agent s risk profile is defined by means of the well-known conditional value-at-risk coherent risk measure. To illustrate the effectiveness of the contracting model, case studies are performed with realistic data from the Brazilian power system.

Orientador: Jose Mario Martinez Perez / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T15:09:35Z (GMT). No. of bitstreams: 1 Bueno_LuisFelipeCesardaRocha_M.pdf: 1111693 bytes, checksum: 531a933822f5dcf9cacad7dea6be5f53 (MD5) Previous issue date: 2008 / Resumo: Nesta dissertacao fazemos uma exposicao sobre alguns modelos matematicos com aplicacoes em economia. Dentre os modelos estudados destacamos a versao discreta das populares medidas de risco VaR (Value at Risk ) e C-VaR (Conditional Value at Risk ). Discutimos algumas propriedades de tais medidas, e, principalmente, expomos sobre algumas ideias para otimiza-las sob uma formulação do tipo OVO (Order Value Optimization) e propomos uma nova formulação para o problema de minimizar a VaR / Abstract: In this dissertation we make a presentation on some mathematical models with applications in economics. Among the studied models we highlight a discrete version of the popular risk measures VaR (Value at Risk) and C-VaR (Conditional Value at Risk). We discuss about some properties of such measures, and, above all, expose on some ideas for optimizing the VaR and CVaR under a OVO (Order Value Optimization) formulation and propose a new formulation to the problem of minimizing the VaR / Mestrado / Otimização / Mestre em Matemática Aplicada