Essays on Multistage Stochastic Programming applied to Asset Liability Management

Uncertainty is a key element of reality. Thus, it becomes natural that the search for methods allows us to represent the unknown in mathematical terms. These problems originate a large class of probabilistic programs recognized as stochastic programming models. They are more realistic than deterministic ones, and their aim is to incorporate uncertainty into their definitions. This dissertation approaches the probabilistic problem class of multistage stochastic problems with chance constraints and joint-chance constraints. Initially, we propose a multistage stochastic asset liability management (ALM) model for a Brazilian pension fund industry. Our model is formalized in compliance with the Brazilian laws and policies. Next, given the relevance of the input parameters for these optimization models, we turn our attention to different sampling models, which compose the discretization process of these stochastic models. We check how these different sampling methodologies impact on the final solution and the portfolio allocation, outlining good options for ALM models. Finally, we propose a framework for the scenario-tree generation and optimization of multistage stochastic programming problems. Relying on the Knuth transform, we generate the scenario trees, taking advantage of the left-child, right-sibling representation, which makes the simulation more efficient in terms of time and the number of scenarios. We also formalize an ALM model reformulation based on implicit extensive form for the optimization model. This technique is designed by the definition of a filtration process with bundles, and coded with the support of an algebraic modeling language. The efficiency of this methodology is tested in a multistage stochastic ALM model with joint-chance constraints. Our framework makes it possible to reach the optimal solution for trees with a reasonable number of scenarios.A incerteza é um elemento fundamental da realidade. Então, torna-se natural a busca por métodos que nos permitam representar o desconhecido em termos matemáticos. Esses problemas originam uma grande classe de programas probabilísticos reconhecidos como modelos de programação estocástica. Eles são mais realísticos que os modelos determinísticos, e tem por objetivo incorporar a incerteza em suas definições. Essa tese aborda os problemas probabilísticos da classe de problemas de multi-estágio com incerteza e com restrições probabilísticas e com restrições probabilísticas conjuntas. Inicialmente, nós propomos um modelo de administração de ativos e passivos multi-estágio estocástico para a indústria de fundos de pensão brasileira. Nosso modelo é formalizado em conformidade com a leis e políticas brasileiras. A seguir, dada a relevância dos dados de entrada para esses modelos de otimização, tornamos nossa atenção às diferentes técnicas de amostragem. Elas compõem o processo de discretização desses modelos estocásticos Nós verificamos como as diferentes metodologias de amostragem impactam a solução final e a alocação do portfólio, destacando boas opções para modelos de administração de ativos e passivos. Finalmente, nós propomos um “framework” para a geração de árvores de cenário e otimização de modelos com incerteza multi-estágio. Baseados na tranformação de Knuth, nós geramos a árvore de cenários considerando a representação filho-esqueda, irmão-direita o que torna a simulação mais eficiente em termos de tempo e de número de cenários. Nós também formalizamos uma reformulação do modelo de administração de ativos e passivos baseada na abordagem extensiva implícita para o modelo de otimização. Essa técnica é projetada pela definição de um processo de filtragem com “bundles”; e codifciada com o auxílio de uma linguagem de modelagem algébrica. A eficiência dessa metodologia é testada em um modelo de administração de ativos e passivos com incerteza com restrições probabilísticas conjuntas. Nosso framework torna possível encontrar a solução ótima para árvores com um número razoável de cenários

A Node Formulation for Multistage Stochastic Programs with Endogenous Uncertainty

This paper introduces a node formulation for multistage stochastic programs with endogenous (i.e., decision-dependent) uncertainty. Problems with such structure arise when the choices of the decision maker determine a change in the likelihood of future random events. The node formulation avoids an explicit statement of non-anticipativity constraints, and as such keeps the dimension of the model sizeable. An exact solution algorithm for a special case is introduced and tested on a case study. Results show that the algorithm outperforms a commercial solver as the size of the instances increases

Multi-level, Multi-stage and Stochastic Optimization Models for Energy Conservation in Buildings for Federal, State and Local Agencies

Energy Conservation Measure (ECM) project selection is made difficult given real-world constraints, limited resources to implement savings retrofits, various suppliers in the market and project financing alternatives. Many of these energy efficient retrofit projects should be viewed as a series of investments with annual returns for these traditionally risk-averse agencies. Given a list of ECMs available, federal, state and local agencies must determine how to implement projects at lowest costs. The most common methods of implementation planning are suboptimal relative to cost. Federal, state and local agencies can obtain greater returns on their energy conservation investment over traditional methods, regardless of the implementing organization. This dissertation outlines several approaches to improve the traditional energy conservations models. Any public buildings in regions with similar energy conservation goals in the United States or internationally can also benefit greatly from this research. Additionally, many private owners of buildings are under mandates to conserve energy e.g., Local Law 85 of the New York City Energy Conservation Code requires any building, public or private, to meet the most current energy code for any alteration or renovation. Thus, both public and private stakeholders can benefit from this research. The research in this dissertation advances and presents models that decision-makers can use to optimize the selection of ECM projects with respect to the total cost of implementation. A practical application of a two-level mathematical program with equilibrium constraints (MPEC) improves the current best practice for agencies concerned with making the most cost-effective selection leveraging energy services companies or utilities. The two-level model maximizes savings to the agency and profit to the energy services companies (Chapter 2). An additional model presented leverages a single congressional appropriation to implement ECM projects (Chapter 3). Returns from implemented ECM projects are used to fund additional ECM projects. In these cases, fluctuations in energy costs and uncertainty in the estimated savings severely influence ECM project selection and the amount of the appropriation requested. A risk aversion method proposed imposes a minimum on the number of “of projects completed in each stage. A comparative method using Conditional Value at Risk is analyzed. Time consistency was addressed in this chapter. This work demonstrates how a risk-based, stochastic, multi-stage model with binary decision variables at each stage provides a much more accurate estimate for planning than the agency’s traditional approach and deterministic models. Finally, in Chapter 4, a rolling-horizon model allows for subadditivity and superadditivity of the energy savings to simulate interactive effects between ECM projects. The approach makes use of inequalities (McCormick, 1976) to re-express constraints that involve the product of binary variables with an exact linearization (related to the convex hull of those constraints). This model additionally shows the benefits of learning between stages while remaining consistent with the single congressional appropriations framework

Stochastic Dynamic Optimization Models in the Banking Sector

The thesis consists of an introduction and 3 essays presenting stochastic dynamic optimization models concerning decision making in the banking sector. The first two essays consider individual banks in an environment where financial crises may occur. The third essay considers the whole banking sector as one entity which is a part of the economy, and thereby the process of money creation in the banking system becomes a central issue. The first essay presents a model for analyzing the optimal dynamic decision making of a bank, which adjusts the size and composition of its balance sheet over time. The model considers the development of the bank's balance sheet in a situation involving the risk of a financial crisis which may or may not materialize, and the timing of which is uncertain. The crisis may involve defaulting of loans and a reduction in the availability of funding. The maturing of loans and deposits taking place in each period is explicitly modeled, assuming maturity mismatch. The outcomes of the model show e.g. a tendency of the bank to deleverage its balance sheet in preparation for an anticipated financial crisis, as well as a tendency to accumulate cash reserves in order to maintain sufficient liquidity. The second essay presents a portfolio model for analyzing a bank making decisions over time in a stochastic environment. The bank is assumed to make decisions regarding the amount of new loans given out in each period, thus affecting the allocation of its funds between liquid cash and non-liquid loans. The model involves maturity mismatch and the risk of a liquidity crisis during which the availability of new funding is restricted. Simulations of the model show that a positive amount is allocated to cash even though cash pays zero returns and no credit risk or investment risk is present in the model, as long as maturity mismatch and the risk of a liquidity crisis are both present. The third essay presents a model of an economy consisting of a central bank, a commercial banking sector, and a real economy experiencing stochastic productivity shocks. A stochastic dynamic programming model is formulated for modeling the policy decisions of the central bank, which dynamically adjusts the size of the monetary base, attempting to keep inflation close to a target. It is assumed that reserve requirements may or may not be binding at a given time. When reserve requirements are not binding, money creation is endogenous, i.e. determined by lending decisions of commercial banks. These lending decisions are affected by the condition of the real economy and, to some extent, by central bank policies acting through transmission channels such as the portfolio rebalance effect. Lending stimulates the real economy while also accelerating inflation as it causes the money supply to grow. The outcomes show that during a recession lending by commercial banks is reduced, deflation prevails, and the central bank carries out expansionary monetary policy. When the recession ends, lending increases and there is a period of increased inflation, while at the same time contractionary monetary policy is carried out.Väitöskirja koostuu johdannosta ja kolmesta esseestä. Esseissä esitellään pankkisektorin päätöksentekoa koskevia malleja, jotka sisältävät talouskehitykseen liittyvää epävarmuutta ja joissa päätöksenteko jakaantuu usealle ajanjaksolle (ns. stokastisia dynaamisia optimointimalleja). Ensimmäisessä ja toisessa esseessä tarkastellaan yksittäisiä pankkeja ympäristöissä, joihin sisältyy mahdollisten finanssikriisien riski. Kolmannessa esseessä tarkastellaan koko pankkisektoria osana taloutta, jolloin pankkijärjestelmän rahanluontiprosessi muodostuu keskeiseksi. Ensimmäisessä esseessä esitellään pankin optimaalista päätöksentekoa koskeva malli. Pankin oletetaan voivan säätää taseensa kokoa ja koostumusta ajan yli. Mallissa tarkastellaan pankin taseen kehittymistä tilanteessa, jossa on läsnä riski finanssikriisistä, jonka realisoituminen ja ajoitus ovat epävarmoja. Kriisiin voi kuulua lainan ottajien maksukyvyttömyyttä sekä pankin rahoituksen saatavuuden alentumista. Lainojen ja talletusten erääntymisessä on huomioitu pankin antolainauksen ja rahoituksen maturiteettien eriävyys (ns. maturity mismatch). Mallin tuloksista havaitaan pankin taipumus käyttää vähemmän velkavipua sen varautuessa mahdolliseen kriisiin sekä taipumus kerätä käteisreservejä likviditeetin varmistamiseksi. Toisessa esseessä esitellään portfoliomalli, joka koskee pankkia, joka tekee päätöksiä ajan yli epävarmuutta sisältävässä ympäristössä. Pankin oletetaan jokaisella ajanjaksolla tekevän lainojen myöntämistä koskevia päätöksiä, jotka vaikuttavat sen varallisuuden jakaantumiseen likvidin käteisen ja ei-likvidien lainojen välillä. Malliin sisältyy uuden rahoituksen saatavuutta alentavan likviditeettikriisin riski. Antolainauksen ja rahoituksen maturiteetit voivat erota toisistaan. Mallin simulaatioissa havaitaan, että pankki pitää taseessaan jonkin verran käteistä, vaikka tilanteeseen ei liittyisi luottoriskiä tai investointeja koskevaa riskiä siitä huolimatta, että käteisvarat eivät tuota voittoa. Tulosten perusteella käteisvaroja pidetään taseessa jos ja vain jos asetelmaan sisältyy sekä likviditeettikriisin uhka että antolainauksen ja rahoituksen maturiteettien eriävyys. Kolmannessa esseessä esitellään malli, joka sisältää keskuspankin, liikepankkisektorin sekä reaalitalouden, joka kokee satunnaisia tuottavuusshokkeja. Keskuspankin oletetaan säätävän ajan myötä perusrahan kokonaismäärää pyrkien pitämään inflaation tavoitetason lähellä. Kassareservirajoitteista oletetaan, että kullakin hetkellä ne voivat joko olla sitovia tai ei-sitovia. Silloin, kun kassareservirajoitteet eivät ole sitovia, rahanluonnin oletetaan olevan ns. endogeenista eli riippuvan liikepankkien lainanantopäätöksistä. Liikepankkien päätöksiin puolestaan vaikuttaa reaalitalous sekä jossakin määrin keskuspankkipolitiikka, joka toimii portfolion tasapainotusvaikutuksen kaltaisten välityskanavien kautta. Antolainaus piristää reaalitaloutta mutta lisää samalla inflaatiota kasvattamalla rahan tarjontaa. Mallin tuloksista havaitaan, että talouden ollessa lamassa liikepankkisektori rajoittaa lainanantoa, taloudessa ilmenee deflaatiota ja keskuspankki harjoittaa ns. ekspansiivista rahapolitiikka kasvattaen perusrahan kokonaismäärää. Laman päättyessä puolestaan liikepankkien lainananto lisääntyy, taloudessa ilmenee jonkin aikaa kohonnutta inflaatiota ja keskuspankki harjoittaa ns. kontraktiivista rahapolitiikkaa pienentäen perusrahan kokonaismäärää

Scheduling process operations under uncertainty and integration with long term planning

This thesis centers upon the application of mathematical modelling, optimization theory and uncertainty analysis to the problem of scheduling batch operations for large scale industries. Over the years, decision making strategies such as scheduling, that deals with allocation of plant resources, has been widely adopted by industries to efficiently carry out their operations and achieve the desired targets. In this thesis, the focus is on planning and scheduling under endogenous uncertainty in the context of multijob, multitasking batch plants. This class of scheduling problems are of practical importance, specially in the analytical services sector, where effective scheduling models could increase the efficiency in carrying out the plant operations and may lead to increased throughput, or reduced makespan, resulting in greater profits or customer satisfaction

On parallel computing for stochastic optimization models and algorithms

167 p.Esta tesis tiene como objetivo principal la resolución de problemas de optimización bajo incertidumbre a gran escala, mediante la interconexión entre las disciplinas de Optimización estocástica y Computación en paralelo. Se describen algoritmos de descomposición desde la perspectivas de programación matemática y del aprovechamiento de recursos computacionales con el fin de resolver problemas de manera más rápida, de mayores dimensiones o/y obtener mejores resultados que sus técnicas homónimas en serie. Se han desarrollado dos estrategias de paralelización, denotadas como inner y outer. La primera de las cuales, realiza tareas en paralelo dentro de un esquema algorítmico en serie, mientras que la segunda ejecuta de manera simultánea y coordinada varios algoritmos secuenciales. La mayor descomposición del problema original, compartiendo el área de factibilidad, creando fases de sincronización y comunicación entre ejecuciones paralelas o definiendo condiciones iniciales divergentes, han sido claves en la eficacia de los diseños de los algoritmos propuestos. Como resultado, se presentan tanto algoritmos exactos como matheurísticos, que combinan metodologías metaheurísticas y técnicas de programación matemática. Se analiza la escalabilidad de cada algoritmo propuesto, y se consideran varios bancos de problemas de diferentes dimensiones, hasta un máximo de 58 millones de restricciones y 54 millones de variables (de las cuales 15 millones son binarias). La experiencia computacional ha sido principalmente realizada en el cluster ARINA de SGI/IZO-SGIker de la UPV/EHU

Stochastic Optimization and Applications with Endogenous Uncertainties Via Discrete Choice Models

Stochastic optimization is an optimization method that solves stochastic problems for minimizing or maximizing an objective function when there is randomness in the optimization process. In this dissertation, various stochastic optimization problems from the areas of Manufacturing, Health care, and Information Cascade are investigated in networks systems. These stochastic optimization problems aim to make plan for using existing resources to improve production efficiency, customer satisfaction, and information influence within limitation. Since the strategies are made for future planning, there are environmental uncertainties in the network systems. Sometimes, the environment may be changed due to the action of the decision maker. To handle this decision-dependent situation, the discrete choice model is applied to estimate the dynamic environment in the stochastic programming model. In the manufacturing project, production planning of lot allocation is performed to maximize the expected output within a limited time horizon. In the health care project, physician is allocated to different local clinics to maximize the patient utilization. In the information cascade project, seed selection of the source user helps the information holder to diffuse the message to target users using the independent cascade model to reach influence maximization. The computation complexities of the three projects mentioned above grow exponentially by the network size. To solve the stochastic optimization problems of large-scale networks within a reasonable time, several problem-specific algorithms are designed for each project. In the manufacturing project, the sampling average approximation method is applied to reduce the scenario size. In the health care project, both the guided local search with gradient ascent and large neighborhood search with Tabu search are developed to approach the optimal solution. In the information cascade project, the myopic policy is used to separate stochastic programming by discrete time, and the Markov decision process is implemented in policy evaluation and updating