Asset and Liability Management (ALM) models have become well established decision tools for pension funds. ALMs are commonly modelled as multi-stage, in which a large terminal wealth is required, while at intermediate time periods, constraints on the funding ratio, that is, the ratio of assets to liabilities, are imposed. Underfunding occurs when the funding ratio is too low; a target value for funding ratios is pre-specified by the decision maker. The risk of underfunding has been usually modelled by employing established risk measures; this controls one single aspect of the funding ratio distributions. For example, controlling the expected shortfall below the target has limited power in controlling shortfall under worst-case scenarios. We propose ALM models in which the risk of underfunding is modelled based on the concept of Second Order Stochastic Dominance (SSD). This is a criterion of ranking random variables - in our case funding ratios - that takes the entire distributions of interest into account and works under the widely accepted assumptions of decision makers being rational and risk averse. In the proposed SSD models, investment decisions are taken such that the resulting short-term distribution of the funding ratio is non-dominated with respect to SSD, while a constraint is imposed on the expected terminal wealth. This is done by considering progressively larger tails of the funding ratio distribution and considering target levels for them; a target distribution is thus implied. Different target distributions lead to different SSD efficient solutions. Improved distributions of funding ratios may be thus achieved, compared to the existing risk models for ALM. This is the first contribution of this thesis. Interesting results are obtained in the special case when the target distribution is deterministic, specified by one single outcome. In this case, we can obtain equivalent risk minimisation models, with risk defined as expected shortfall or as worst case loss. This represents the second contribution. The third contribution is a framework for scenario generation based on the "Birth, Immigration, Death, Emigration" (BIDE) population model and the Empirical copula; the scenarios are used to evaluate the proposed models and their special cases both in-sample and out-of-sample. As an application, we consider the planning problem of a large DB pension fund in Saudi Arabia.

Submitted by Rosivalda Pereira (mrs.pereira@ufma.br) on 2017-08-14T18:59:08Z No. of bitstreams: 1 MauroSergioSilvaPinto.pdf: 1648622 bytes, checksum: 71f809c341318a8660df7cdd2182a4f0 (MD5) / Made available in DSpace on 2017-08-14T18:59:08Z (GMT). No. of bitstreams: 1 MauroSergioSilvaPinto.pdf: 1648622 bytes, checksum: 71f809c341318a8660df7cdd2182a4f0 (MD5) Previous issue date: 2016-07-01 / The Unit Commitment Problem (UC) in power generation is a difficult problem, traditionally modeled with a mixed-integer optimization formulation. What makes it especially difficult is the time-dependency of the generation decisions, caused by ramping limitation constraints applied mostly to thermal generation, as well as minimum shut down and start up times. The main types of uncertainty are usually taken in account: in the actual load values and the (un)reliability of the generators. The uncertainty in generator availability has been met with a specification of operational reserve policy. The uncertainty in load, taking in account that its magnitude is usually small, is in many cases simply ignored. With the significant inclusion of wind power in the portfolio of a county or region, it is no longer adequate to deal with the UC problem in the traditional way. The uncertainty in wind generation is at least one order of magnitude higher than the uncertainty in load. Moreover, the wind behavior includes the possibility of strong ramping, with important stressing effect on thermal generation. Dealing with such challenges in a business-as-usual manner is doomed to produce sub-optimal solutions and to put the system in jeopardy or cause substantial financial loss with costly emergency actions. The transition to models that take risk in account supposes a change in paradigm in the decisionmaking process in the UC process. Without clear guidelines, operators will tend to over-protect – while under commercial pressure, they may run excessive risks. To help in the transition to a UC decision-making process under uncertainty, this thesis contributes to the set of planning paradigms and makes an attempt to organize the comparative analysis and results and conclusions reached, from an illustrative case built around the IEEE RTS 30-bus system. The results show that the Pareto-optimal front, in a stochastic cost vs. risk space, may not be convex, which precludes the use of simplistic trade-off approaches. The conclusion, as a contribution from this thesis, is unmistakable: a stochastic programming approach is not adequately informative on the risks run as consequence of system operator decisions on unit commitment, in systems with a high penetration of wind power. Models that follow the Risk Analysis paradigm are necessary, in order to quantify the costs of hedging (protecting against adverse scenarios). Furthermore, by relying on an explicit multiple criteria representation, the thesis shows how this risk aversion perspective, in terms of undesired events, may be blended with a stochastic optimization perspective of average gain or expense On the planning matter, embedding the risk in the operating cost annually capitalized assists the decision-making in investments in system planning. / O problema do pré-despacho em sistemas de potência, conhecido na literatura como Unit Commitment -UC, é um problema não linear, tradicionalmente modelado como uma formulação de otimização inteira mista. Um dos pontos críticos deste processo é a sua interdependência temporal, além de restrições como tempos mínimos de parada e partida. Tradicionalmente, as fontes de incertezas no sistema são o valor atual da carga e a disponibilidade do fornecimento de potência por parte dos geradores, relacionando-se com a confiabilidade. A geração deve satisfazer a critérios que determinam um nível mínimo de reserva girante ou uma política de reserva operacional através de métricas determinísticas ou estocásticas. A incerteza da carga é menor e muitas vezes é desprezada no processo de pré-despacho. Em ambos os casos, o objetivo é transformar um problema com incerteza em um modelo determinístico. Devido à elevada integração de fontes eólicas na matriz energética, as abordagens tradicionais de pré-despacho se tornam inadequadas para lidar com as incertezas associadas a este tipo de fonte. O grau de incerteza das fontes eólicas é pelo menos maior em magnitude do que o grau de incertezas da carga. Além disso, o comportamento do vento inclui a possibilidade de fortes rajadas que podem se transformar em eventos de rampa não previstos. Diante disto, as formas usuais de tratar este problema podem produzir soluções sub-ótimas, colocando o sistema em risco ou causando perda financeira substancial por ações de correções técnicas dispendiosas. A transição para modelos que levam em conta o risco propõe uma mudança de paradigma no processo de tomada de decisão do problema do pré-desapcho. Desta forma, estes desafios exigem que novos modelos de decisão sejam elaborados levando em conta este novo quadro de incertezas e que forneçam soluções úteis para o planejamento da operação. Em particular, no que se refere à operação do sistema, são necessárias ferramentas que auxiliem os operadores na tomada de decisões levando em conta o risco decorrente do ambiente de incerteza presente no sistema. Esta tese vem contribuir no suporte ao processo de tomada de decisão, analisando um conjunto de paradigmas de planejamento quanto a sua capacidade/utilidade de fornecer soluções consistentes em sistemas com significativa integração eólica. Os resultados mostraram que a fronteira de Pareto das soluções ótimas não-dominadas em um espaço multicritério entre custo estocástico versus o risco pode não ser convexa, o que impede uma abordagem de análise simples de trade-off. Mostra-se, que o modelo tradicional estocástico pode não ser adequado para lidar com as incertezas geradas pelas fontes eólicas. Além disso, esta Tese mostra que eventos indesejados, sob uma perspectiva de risco em um espaço multicritério, podem ser negligenciados pela abordagem tradicional estocástica. No muito curto prazo, a abordagem de tomada de decisão com incertezas eólicas mostra que o simples despacho de mais reservas operacionais no sistema com alta penetração eólica pode ser insuficiente para lidar com as incertezas. Sob o aspecto do planejamento, a incorporação do risco nos custos de operação capitalizados anualmente auxilia a tomada de decisão de investimentos no planejamento do sistema.

Orientador: John Fredy Franco Baquero / Resumo: A presença de Geração Distribuída (GD) no Sistema de Distribuição de Energia Elétrica (SDEE) tem se incrementado nos últimos anos devido a mudanças na regulação e a incentivos governamentais, proporcionando benefícios técnicos e econômicos. Em particular, é esperado que a GD renovável (eólica ou solar) seja integrada adequadamente no SDEE, visando contribuir na redução de emissões de gases de efeito estufa. Entretanto, a presença da GD renovável, junto com suas inerentes incertezas, aumenta a complexidade no planejamento do SDEE. Diante do exposto, neste trabalho propõe-se um modelo de programação estocástica de dois estágios para o problema de planejamento da expansão do SDEE multi-período. As incertezas da geração renovável (associadas à irradiação solar e velocidade do vento) e demanda são representadas por meio de cenários. A função objetivo minimiza o valor presente líquido dos investimentos (subestações, circuitos, e alocação de GD), custo da energia, manutenção e operação, assim como o custo das emissões de CO2. A operação das unidades de GD é representada limitando a potência ativa/reativa que pode ser injetada segundo as curvas de capabilidade e restrições de fator de potência. O modelo proposto foi implementado na linguagem de modelamento AMPL e resolvido com o solver CPLEX. Testes utilizando um SDEE de 24 e 54 nós comprovam a eficiência do modelo. / Abstract: The presence of Distributed Generation (DG) in Electrical Distribution Systems (EDSs) has been increased in recent years due to changes in regulation and government incentives, leading to technical and economic benefits. In particular, renewable DG (wind or solar power) is expected to be properly integrated into the EDS, aiming to contribute to the reduction of greenhouse gas emissions. However, the presence of renewable DG, along with its inherent uncertainties, increases the complexity in the planning of the EDS. In this context, this work proposes a two-stage stochastic programming model for the problem of EDSs expansion planning. The uncertainties of renewable generation (associated with solar irradiation and wind speed) and demand, are represented through scenarios. The objective function minimizes the net present value of investments (substations, circuits, and DG allocation), energy cost, maintenance and operation, as well as the cost of CO2 emissions. The operation of the DG units is represented by limiting the active/reactive power that can be injected according to capability curves and power factor constraints. The proposed model was implemented in the modeling language AMPL and solved with the solver CPLEX. Tests using a 24 and 54-nodes EDS prove the efficiency of the proposed model. / Mestre

There is a lack of research in the area of hedging future contracts, especially in illiquid or very volatile market conditions. It is important to understand the volatility of the oil and currency markets because reduced fluctuations in these markets could lead to better hedging performance. This study compared different hedging methods by using a hedging error metric, supplementing the Receding Horizontal Control and Stochastic Programming (RHCSP) method by utilizing the London Interbank Offered Rate with the Levy process. The RHCSP hedging method was investigated to determine if improved hedging error was accomplished compared to the Black-Scholes, Leland, and Whalley and Wilmott methods when applied on simulated, oil, and currency futures markets. A modified RHCSP method was also investigated to determine if this method could significantly reduce hedging error under extreme market illiquidity conditions when applied on simulated, oil, and currency futures markets. This quantitative study used chaos theory and emergence for its theoretical foundation. An experimental research method was utilized for this study with a sample size of 506 hedging errors pertaining to historical and simulation data. The historical data were from January 1, 2005 through December 31, 2012. The modified RHCSP method was found to significantly reduce hedging error for the oil and currency market futures by the use of a 2-way ANOVA with a t test and post hoc Tukey test. This study promotes positive social change by identifying better risk controls for investment portfolios and illustrating how to benefit from high volatility in markets. Economists, professional investment managers, and independent investors could benefit from the findings of this study.

<p>We analyse several facets of bilevel decision problems under uncertainty. These problems can be interpreted as an extension of stochastic programming problems where part of the uncertainty is attributed to the behaviour of another actor.</p><p>The field of decision making under uncertainty with bilevel features is quite new and most approaches focus on the interactions and relations between the decision makers. In contrast to these studies, the approach of bilevel stochastic programming pursued here stresses the stochastic programming aspect of the problem formulation. The framework enables a direct application of stochastic programming concepts and solution methods to the bilevel relationship between the actors. Thus more complex problem structures can be studied and the aspect of uncertainty can be treated adequately.</p><p>Our analysis covers both theoretical and more practically oriented issues. We study different formulations of one and two stage bilevel stochastic programming problems and state necessary optimality conditions for each of the problem instances. Additionally we present a solution algorithm utilising a stochastic quasi-gradient method. A further study is concerned with the uniqueness of the minima of a convex stochastic programming problem with uncertainty about the decision variables. We state conditions on the distribution of the parameters representing the uncertainty such that the minima of the optimisation problem are unique. We formulate a model of competition and collaboration of two different types of telecom service providers, the owner of a bottleneck facility and a virtual network operator. This represents an application of a bilevel stochastic programming formulation to a liberalised telecommunications environment. Furthermore, the utilisation of the bilevel stochastic programming framework and the developed solution concepts for the analysis of principal agent models is demonstrated. Also here the background of a regulated telecom environment, more specific the relations between a regulator and a regulated telecommunications company, was chosen.</p>

We analyse several facets of bilevel decision problems under uncertainty. These problems can be interpreted as an extension of stochastic programming problems where part of the uncertainty is attributed to the behaviour of another actor. The field of decision making under uncertainty with bilevel features is quite new and most approaches focus on the interactions and relations between the decision makers. In contrast to these studies, the approach of bilevel stochastic programming pursued here stresses the stochastic programming aspect of the problem formulation. The framework enables a direct application of stochastic programming concepts and solution methods to the bilevel relationship between the actors. Thus more complex problem structures can be studied and the aspect of uncertainty can be treated adequately. Our analysis covers both theoretical and more practically oriented issues. We study different formulations of one and two stage bilevel stochastic programming problems and state necessary optimality conditions for each of the problem instances. Additionally we present a solution algorithm utilising a stochastic quasi-gradient method. A further study is concerned with the uniqueness of the minima of a convex stochastic programming problem with uncertainty about the decision variables. We state conditions on the distribution of the parameters representing the uncertainty such that the minima of the optimisation problem are unique. We formulate a model of competition and collaboration of two different types of telecom service providers, the owner of a bottleneck facility and a virtual network operator. This represents an application of a bilevel stochastic programming formulation to a liberalised telecommunications environment. Furthermore, the utilisation of the bilevel stochastic programming framework and the developed solution concepts for the analysis of principal agent models is demonstrated. Also here the background of a regulated telecom environment, more specific the relations between a regulator and a regulated telecommunications company, was chosen.

Stochastic dynamic programming is a recursive method for solving sequential or multistage decision problems. It helps economists and mathematicians construct and solve a huge variety of sequential decision making problems in stochastic cases. Research on stochastic dynamic programming is important and meaningful because stochastic dynamic programming reflects the behavior of the decision maker without risk aversion; i.e., decision making under uncertainty. In the solution process, it is extremely difficult to represent the existing or future state precisely since uncertainty is a state of having limited knowledge. Indeed, compared to the deterministic case, which is decision making under certainty, the stochastic case is more realistic and gives more accurate results because the majority of problems in reality inevitably have many unknown parameters. In addition, time scale calculus theory is applicable to any field in which a dynamic process can be described with discrete or continuous models. Many stochastic dynamic models are discrete or continuous, so the results of time scale calculus are directly applicable to them as well. The aim of this thesis is to introduce a general form of a stochastic dynamic sequence problem on complex discrete time domains and to find the optimal sequence which maximizes the sequence problem.

This dissertation studies optimization problems that involve probabilistic covering constraints. A probabilistic constraint evaluates and requires that the probability that a set of constraints involving random coefficients with known distributions hold satisfy a minimum requirement. A covering constraint involves a linear inequality on non-negative variables with a greater or equal to sign and non-negative coefficients. A variety of applications, such as set cover problems, node/edge cover problems, crew scheduling, production planning, facility location, and machine learning, in uncertain settings involve probabilistic covering constraints. In the first part of this dissertation we consider probabilistic covering linear programs. Using the sampling average approximation (SAA) framework, a probabilistic covering linear program can be approximated by a covering k-violation linear program (CKVLP), a deterministic covering linear program in which at most k constraints are allowed to be violated. We show that CKVLP is strongly NP-hard. Then, to improve the performance of standard mixed-integer programming (MIP) based schemes for CKVLP, we (i) introduce and analyze a coefficient strengthening scheme, (ii) adapt and analyze an existing cutting plane technique, and (iii) present a branching technique. Through computational experiments, we empirically verify that these techniques are significantly effective in improving solution times over the CPLEX MIP solver. In particular, we observe that the proposed schemes can cut down solution times from as much as six days to under four hours in some instances. We also developed valid inequalities arising from two subsets of the constraints in the original formulation. When incorporating them with a modified coefficient strengthening procedure, we are able to solve a difficult probabilistic portfolio optimization instance listed in MIPLIB 2010, which cannot be solved by existing approaches. In the second part of this dissertation we study a class of probabilistic 0-1 covering problems, namely probabilistic k-cover problems. A probabilistic k-cover problem is a stochastic version of a set k-cover problem, which is to seek a collection of subsets with a minimal cost whose union covers each element in the set at least k times. In a stochastic setting, the coefficients of the covering constraints are modeled as Bernoulli random variables, and the probabilistic constraint imposes a minimal requirement on the probability of k-coverage. To account for absence of full distributional information, we define a general ambiguous k-cover set, which is ``distributionally-robust." Using a classical linear program (called the Boolean LP) to compute the probability of events, we develop an exact deterministic reformulation to this ambiguous k-cover problem. However, since the boolean model consists of exponential number of auxiliary variables, and hence not useful in practice, we use two linear program based bounds on the probability that at least k events occur, which can be obtained by aggregating the variables and constraints of the Boolean model, to develop tractable deterministic approximations to the ambiguous k-cover set. We derive new valid inequalities that can be used to strengthen the linear programming based lower bounds. Numerical results show that these new inequalities significantly improve the probability bounds. To use standard MIP solvers, we linearize the multi-linear terms in the approximations and develop mixed-integer linear programming formulations. We conduct computational experiments to demonstrate the quality of the deterministic reformulations in terms of cost effectiveness and solution robustness. To demonstrate the usefulness of the modeling technique developed for probabilistic k-cover problems, we formulate a number of problems that have up till now only been studied under data independence assumption and we also introduce a new applications that can be modeled using the probabilistic k-cover model.

Monotonic optimization consists of minimizing or maximizing a monotonic objective function over a set of constraints defined by monotonic functions. Many optimization problems in economics and engineering often have monotonicity while lacking other useful properties, such as convexity. This thesis is concerned with the development and application of global optimization algorithms for monotonic optimization problems. First, we propose enhancements to an existing outer-approximation algorithm | called the Polyblock Algorithm | for monotonic optimization problems. The enhancements are shown to significantly improve the computational performance of the algorithm while retaining the convergence properties. Next, we develop a generic branch-and-bound algorithm for monotonic optimization problems. A computational study is carried out for comparing the performance of the Polyblock Algorithm and variants of the proposed branch-and-bound scheme on a family of separable polynomial programming problems. Finally, we study an important class of monotonic optimization problems | probabilistically constrained linear programs. We develop a branch-and-bound algorithm that searches for a global solution to the problem. The basic algorithm is enhanced by domain reduction and cutting plane strategies to reduce the size of the partitions and hence tighten bounds. The proposed branch-reduce-cut algorithm exploits the monotonicity properties inherent in the problem, and requires the solution of only linear programming subproblems. We provide convergence proofs for the algorithm. Some illustrative numerical results involving problems with discrete distributions are presented.

In this dissertation we study several non-convex and stochastic optimization problems. The common theme is the use of mixed-integer programming (MIP) techniques including valid inequalities and reformulation to solve these problems. We first study a strategic capacity planning model which captures the trade-off between the incentive to delay capacity installation to wait for improved technology and the need for some capacity to be installed to meet current demands. This problem is naturally formulated as a MIP with a bilinear objective. We develop several linear MIP formulations, along with classes of strong valid inequalities. We also present a specialized branch-and-cut algorithm to solve a compact concave formulation. Computational results indicate that these formulations can be used to solve large-scale instances. We next study methods for optimization with joint probabilistic constraints. These problems are challenging because evaluating solution feasibility requires multidimensional integration and the feasible region is not convex. We propose and analyze a Monte Carlo sampling scheme to simplify the probabilistic structure of such problems. Computational tests of the approach indicate that it can yield good feasible solutions and reasonable bounds on their quality. Next, we study a MIP formulation of the non-convex sample approximation problem. We obtain two strengthened formulations. As a byproduct of this analysis, we obtain new results for the previously studied mixing set, subject to an additional knapsack inequality. Computational results indicate that large-scale instances can be solved using the strengthened formulations. Finally, we study optimization problems with stochastic dominance constraints. A stochastic dominance constraint states that a random outcome which depends on the decision variables should stochastically dominate a given random variable. We present new formulations for both first and second order stochastic dominance which are significantly more compact than existing formulations. Computational tests illustrate the benefits of the new formulations.