Dissertations / Theses on the topic 'Multistage stochastic optimization'

In this dissertation, we focus on developing sampling-based algorithms for solving stochastic linear programs. The work covers both two stage and multistage versions of stochastic linear programs. In particular, we first study the two stage stochastic decomposition (SD) algorithm and present some extensions associated with SD. Specifically, we study two issues: a) are there conditions under which the regularized version of SD generates a unique solution? and b) in cases where a user is willing to sacrifice optimality, is there a way to modify the SD algorithm so that a user can trade-off solution times with solution quality? Moreover, we present our preliminary approach to address these questions. Secondly, we investigate the multistage stochastic linear programs and propose a new approach to solving multistage stochastic decision models in the presence of constraints. The motivation for proposing the multistage stochastic decomposition algorithm is to handle large scale multistage stochastic linear programs. In our setting, the deterministic equivalent problems of the multistage stochastic linear program are too large to be solved exactly. Therefore, we seek an asymptotically optimum solution by simulating the SD algorithmic process, which was originally designed for two-stage stochastic linear programs (SLPs). More importantly, when SD is implemented in a time-staged manner, the algorithm begins to take the flavor of a simulation leading to what we refer to as optimization simulation. As for multistage stochastic decomposition, there are a couple of advantages that deserve mention. One of the benefits is that it can work directly with sample paths, and this feature makes the new algorithm much easier to be integrated within a simulation. Moreover, compared with other sampling-based algorithms for multistage stochastic programming, we also overcome certain limitations, such as a stage-wise independence assumption.

This dissertation focuses on extending solution methods in the area of stochastic optimization. Attention is focused to three specific problems in the field. First, a solution method for mixed integer programs subject to chance constraints is discussed. This class of problems serves as an effective modeling framework for a wide variety of applied problems. Unfortunately, chance constrained mixed integer programs tend to be very challenging to solve. Thus, the aim of this work is to address some of these challenges by exploiting the structure of the deterministic reformulation for the problem. Second, a stochastic program for integrating renewable energy sources into traditional energy systems is developed. As the global push for higher utilization of such green resources increases, such models will prove invaluable to energy system designers. Finally, a process for transforming clinical medical data into a model to assist decision making during the treatment planning phase for palliative chemotherapy is outlined. This work will likely provide decision support tools for oncologists. Moreover, given the new requirements for the usage electronic medical records, such techniques will have applicability to other treatment planning applications in the future.

Electric energy constitutes one of the most crucial elements to almost every aspect of life of people. The modern electric power systems face several challenges such as efficiency, economics, sustainability, and reliability. Increase in electrical energy demand, distributed generations, integration of uncertain renewable energy resources, and demand side management are among the main underlying reasons of such growing complexity. Additionally, the elements of power systems are often vulnerable to failures because of many reasons, such as system limits, weak conditions, unexpected events, hidden failures, human errors, terrorist attacks, and natural disasters. One common factor complicating the operation of electrical power systems is the underlying uncertainties from the demands, supplies and failures of system components. Stochastic programming provides a mathematical framework for decision making under uncertainty. It enables a decision maker to incorporate some knowledge of the intrinsic uncertainty into the decision making process. In this dissertation, we focus on application of two-stage and multistage stochastic programming approaches to electric energy systems modeling and optimization. Particularly, we develop models and algorithms addressing the sustainability and reliability issues in power systems. First, we consider how to improve the reliability of power systems under severe failures or contingencies prone to cascading blackouts by so called islanding operations. We present a two-stage stochastic mixed-integer model to find optimal islanding operations as a powerful preventive action against cascading failures in case of extreme contingencies. Further, we study the properties of this problem and propose efficient solution methods to solve this problem for large-scale power systems. We present the numerical results showing the effectiveness of the model and investigate the performance of the solution methods. Next, we address the sustainability issue considering the integration of renewable energy resources into production planning of energy-intensive manufacturing industries. Recently, a growing number of manufacturing companies are considering renewable energies to meet their energy requirements to move towards green manufacturing as well as decreasing their energy costs. However, the intermittent nature of renewable energies imposes several difficulties in long term planning of how to efficiently exploit renewables. In this study, we propose a scheme for manufacturing companies to use onsite and grid renewable energies provided by their own investments and energy utilities as well as conventional grid energy to satisfy their energy requirements. We propose a multistage stochastic programming model and study an efficient solution method to solve this problem. We examine the proposed framework on a test case simulated based on a real-world semiconductor company. Moreover, we evaluate long-term profitability of such scheme via so called value of multistage stochastic programming.

Submitted by Guido Chagas (guido.chagas@fgv.br) on 2016-09-09T15:34:13Z No. of bitstreams: 1 Long-Term Asset Allocation Based on Stochastic Multistage Multi-Objective Portfolio Optimization.pdf: 6336618 bytes, checksum: 67d3dd1c3b982252c5012b3078278f95 (MD5)<br>Approved for entry into archive by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br) on 2016-09-09T17:20:03Z (GMT) No. of bitstreams: 1 Long-Term Asset Allocation Based on Stochastic Multistage Multi-Objective Portfolio Optimization.pdf: 6336618 bytes, checksum: 67d3dd1c3b982252c5012b3078278f95 (MD5)<br>Made available in DSpace on 2016-09-09T17:21:47Z (GMT). No. of bitstreams: 1 Long-Term Asset Allocation Based on Stochastic Multistage Multi-Objective Portfolio Optimization.pdf: 6336618 bytes, checksum: 67d3dd1c3b982252c5012b3078278f95 (MD5) Previous issue date: 2016-08-19<br>Multi-Period Stochastic Programming (MSP) offers an appealing approach to identity optimal portfolios, particularly over longer investment horizons, because it is inherently suited to handle uncertainty. Moreover, it provides flexibility to accommodate coherent risk measures, market frictions, and most importantly, major stylized facts as volatility clustering, heavy tails, leverage effects and tail co-dependence. However, to achieve satisfactory results a MSP model relies on representative and arbitrage-free scenarios of the pertaining multivariate financial series. Only after we have constructed such scenarios, we can exploit it using suitable risk measures to achieve robust portfolio allocations. In this thesis, we discuss a comprehensive framework to accomplish that. First, we construct joint scenarios based on a combined GJR-GARCH + EVT-GPD + t-Copula approach. Then, we reduce the original scenario tree and remove arbitrage opportunities using a method based on Optimal Discretization and Process Distances. Lastly, using the approximated scenario tree we perform a multi-period Mean-Variance-CVaR optimization taking into account market frictions such as transaction costs and regulatory restrictions. The proposed framework is particularly valuable to real applications because it handles various key features of real markets that are often dismissed by more common optimization approaches.<br>Programação Estocástica Multi-Período (MSP) oferece uma abordagem conveniente para identificar carteiras ótimas, particularmente para horizontes de investimento mais longos, pois incorpora adequadamente a incerteza no processo de otimização. Adicionalmente, ela proporciona flexibilidade para acomodar medidas coerentes de risco, fricções de mercado e fatos estilizados relevantes como agrupamento de volatilidade, caudas pesadas, efeitos de alavancagem e co-dependência nas caudas. No entanto, para alcançar resultados satisfatórios, um modelo MSP depende de cenários representativos e livres de arbitragem. Somente após construídos esses cenários, podemos explorá-los usando medidas de risco adequadas para alcançar alocações ótimas. Nessa tese, discutimos uma metodologia completa para alcançar esse objetivo. Em primeiro lugar, construímos cenários conjuntos baseados numa abordagem conjunta GJR-GARCH + EVT-GPD + t-Copula. Posteriormente, reduzimos a árvore original de cenários e removemos oportunidades de arbitragem utilizando um método de discretização ótima baseado nas distâncias de processos estocásticos. Por último, usando a árvore aproximada de cenários, realizamos uma otimização multi-período de média-variância-CVaR considerando fricções de mercado, custos de transação e restrições regulamentares. A metodologia proposta é particularmente útil para aplicações reais, porque considera várias características relevantes dos mercados reais que muitas vezes são ignorados por abordagens mais simples de otimização.

We consider two classes of stochastic programming models which are motivated by two applications related to the field of aviation. The first problem we consider is the network capacity planning problem, which arises in capacity planning of systems with network structures, such as transportation terminals, roadways and telecommunication networks. We study this problem in the context of airport terminal capacity planning. In this problem, the objective is to determine the optimal design and expansion capacities for different areas of the terminal in the presence of uncertainty in future demand levels and expansion costs, such that overall passenger delay is minimized. We model this problem as a nonlinear multistage stochastic integer program with a multicommodity network flow structure. The formulation requires the use of time functions for maximum delays in passageways and processing stations, for which we derive approximations that account for the transient behavior of flow. The deterministic equivalent of the developed model is solved via a branch and bound procedure, in which a bounding heuristic is used at the nodes of the branch and bound tree to obtain integer solutions. In the second study, we consider the project portfolio optimization problem. This problem falls in the class of stochastic programs in which times of uncertainty realizations are dependent on the decisions made. The project portfolio optimization problem deals with the selection of research and development (R&D) projects and determination of optimal resource allocations for the current planning period such that the expected total discounted return or a function of this expectation for all projects over an infinite time horizon is maximized, given the uncertainties and resource limitations over a planning horizon. Accounting for endogeneity in some parameters, we propose efficient modeling and solution approaches for the resulting multistage stochastic integer programming model. We first develop a formulation that is amenable to scenario decomposition, and is applicable to the general class of stochastic problems with endogenous uncertainty. We then demonstrate the use of the sample average approximation method in solving large scale problems of this class, where the sample problems are solved through Lagrangian relaxation and lower bounding heuristics.

In this thesis, we have developed a strategic optimization model of investments in infrastructure in the LNG value chain. The focus is on floating LNG production units: when they are a viable solution and what value they add to the LNG value chain. First a deterministic model is presented with focus on describing the value chain, before it is expanded to a multistage stochastic model with uncertain field sizes and gas prices. The objective is to maximize expected discounted profits through optimal investments in infrastructure. A dataset based on a set of potential fields on the Norwegian continental shelf, with shipping of LNG to three markets in the Atlantic basin, is used to solve the model. The results illustrate when FLNG units can add value to the value chain. They are used as a supplement to onshore processing plants; for example expanding peak capacity or to react to the resolution of uncertain parameters. The floating liquefaction option is especially attractive for fields located far from shore. We also find that the main reason for using FLNG units is their lower liquefaction costs, not the ability to move between fields. The stochastic version of the model results in solutions very similar to the solutions of the deterministic model, even though it is significantly harder to solve. Dantzig-Wolfe decomposition is implemented to reduce run times, but does not converge.

The main objective of this thesis is to investigate risk neutral and risk averse approaches to multistage stochastic programming with applications to hydrothermal operation planning problems. The purpose of hydrothermal system operation planning is to define an operation strategy which, for each stage of the planning period, given the system state at the beginning of the stage, produces generation targets for each plant. This problem can be formulated as a large scale multistage stochastic linear programming problem. The energy rationing that took place in Brazil in the period 2001/2002 raised the question of whether a policy that is based on a criterion of minimizing the expected cost (i.e. risk neutral approach) is a valid one when it comes to meet the day-to-day supply requirements and taking into account severe weather conditions that may occur. The risk averse methodology provides a suitable framework to remedy these deficiencies. This thesis attempts to provide a better understanding of the risk averse methodology from the practice perspective and suggests further possible alternatives using robust optimization techniques. The questions investigated and the contributions of this thesis are as follows. First, we suggest a multiplicative autoregressive time series model for the energy inflows that can be embedded into the optimization problem that we investigate. Then, computational aspects related to the stochastic dual dynamic programming (SDDP) algorithm are discussed. We investigate the stopping criteria of the algorithm and provide a framework for assessing the quality of the policy. The SDDP method works reasonably well when the number of state variables is relatively small while the number of stages can be large. However, as the number of state variables increases the convergence of the SDDP algorithm can become very slow. Afterwards, performance improvement techniques of the algorithm are discussed. We suggest a subroutine to eliminate the redundant cutting planes in the future cost functions description which allows a considerable speed up factor. Also, a design using high performance computing techniques is discussed. Moreover, an analysis of the obtained policy is outlined with focus on specific aspects of the long term operation planning problem. In the risk neutral framework, extreme events can occur and might cause considerable social costs. These costs can translate into blackouts or forced rationing similarly to what happened in 2001/2002 crisis. Finally, issues related to variability of the SAA problems and sensitivity to initial conditions are studied. No significant variability of the SAA problems is observed. Second, we analyze the risk averse approach and its application to the hydrothermal operation planning problem. A review of the methodology is suggested and a generic description of the SDDP method for coherent risk measures is presented. A detailed study of the risk averse policy is outlined for the hydrothermal operation planning problem using different risk measures. The adaptive risk averse approach is discussed under two different perspectives: one through the mean-$avr$ and the other through the mean-upper-semideviation risk measures. Computational aspects for the hydrothermal system operation planning problem of the Brazilian interconnected power system are discussed and the contributions of the risk averse methodology when compared to the risk neutral approach are presented. We have seen that the risk averse approach ensures a reduction in the high quantile values of the individual stage costs. This protection comes with an increase of the average policy value - the price of risk aversion. Furthermore, both of the risk averse approaches come with practically no extra computational effort and, similarly to the risk neutral method, there was no significant variability of the SAA problems. Finally, a methodology that combines robust and stochastic programming approaches is investigated. In many situations, such as the operation planning problem, the involved uncertain parameters can be naturally divided into two groups, for one group the robust approach makes sense while for the other the stochastic programming approach is more appropriate. The basic ideas are discussed in the multistage setting and a formulation with the corresponding dynamic programming equations is presented. A variant of the SDDP algorithm for solving this class of problems is suggested. The contributions of this methodology are illustrated with computational experiments of the hydrothermal operation planning problem and a comparison with the risk neutral and risk averse approaches is presented. The worst-case-expectation approach constructs a policy that is less sensitive to unexpected demand increase with a reasonable loss on average when compared to the risk neutral method. Also, we comp are the suggested method with a risk averse approach based on coherent risk measures. On the one hand, the idea behind the risk averse method is to allow a trade off between loss on average and immunity against unexpected extreme scenarios. On the other hand, the worst-case-expectation approach consists in a trade off between a loss on average and immunity against unanticipated demand increase. In some sense, there is a certain equivalence between the policies constructed using each of these methods.

Diese Arbeit leistet Beiträge zu zwei Gebieten der mathematischen Programmierung: stochastische Optimierung und gemischt-ganzzahlige nichtlineare Optimierung (MINLP). Im ersten Teil erweitern wir quantitative Stetigkeitsresultate für zweistufige stochastische gemischt-ganzzahlige lineare Programme auf Situationen in denen Unsicherheit gleichzeitig in den Kosten und der rechten Seite auftritt, geben eine ausführliche Übersicht zu Dekompositionsverfahren für zwei- und mehrstufige stochastische lineare und gemischt-ganzzahlig lineare Programme, und diskutieren Erweiterungen und Kombinationen des Nested Benders Dekompositionsverfahrens und des Nested Column Generationsverfahrens für mehrstufige stochastische lineare Programme die es erlauben die Vorteile sogenannter rekombinierender Szenariobäume auszunutzen. Als eine Anwendung dieses Verfahrens betrachten wir die optimale Zeit- und Investitionsplanung für ein regionales Energiesystem unter Einbeziehung von Windenergie und Energiespeichern. Im zweiten Teil geben wir eine ausführliche Übersicht zum Stand der Technik bzgl. Algorithmen und Lösern für MINLPs und zeigen dass einige dieser Algorithmen innerhalb des constraint integer programming Softwaresystems SCIP angewendet werden können. Letzteres erlaubt uns die Verwendung schon existierender Technologien für gemischt-ganzzahlige linear Programme und constraint Programme für den linearen und diskreten Teil des Problems. Folglich konzentrieren wir uns hauptsächlich auf die Behandlung der konvexen und nichtkonvexen nichtlinearen Nebenbedingungen mittels Variablenschrankenpropagierung, äußerer Approximation und Reformulierung. In einer ausführlichen numerischen Studie untersuchen wir die Leistung unseres Ansatzes anhand von Anwendungen aus der Tagebauplanung und des Aufbaus eines Wasserverteilungssystems und mittels verschiedener Vergleichstests. Die Ergebnisse zeigen, dass SCIP ein konkurrenzfähiger Löser für MINLPs geworden ist.<br>This thesis contributes to two topics in mathematical programming: stochastic optimization and mixed-integer nonlinear programming (MINLP). In the first part, we extend quantitative continuity results for two-stage stochastic mixed-integer linear programs to include situations with simultaneous uncertainty in costs and right-hand side, give an extended review on decomposition algorithm for two- and multistage stochastic linear and mixed-integer linear programs, and discuss extensions and combinations of the Nested Benders Decomposition and Nested Column Generation methods for multistage stochastic linear programs to exploit the advantages of so-called recombining scenario trees. As an application of the latter, we consider the optimal scheduling and investment planning for a regional energy system including wind power and energy storages. In the second part, we give a comprehensive overview about the state-of-the-art in algorithms and solver technology for MINLPs and show that some of these algorithm can be applied within the constraint integer programming framework SCIP. The availability of the latter allows us to utilize the power of already existing mixed integer linear and constraint programming technologies to handle the linear and discrete parts of the problem. Thus, we focus mainly on the domain propagation, outer-approximation, and reformulation techniques to handle convex and nonconvex nonlinear constraints. In an extensive computational study, we investigate the performance of our approach on applications from open pit mine production scheduling and water distribution network design and on various benchmarks sets. The results show that SCIP has become a competitive solver for MINLPs.

Exploration and production (E&P) involves the upstream activities from looking for promising reservoirs to extracting oil and selling it to downstream companies. E&P is the most profitable business in the oil industry. However, it is also the most capital-intensive and risky. Hence, the proper assessment of E&P projects with effective management of uncertainties is crucial to the success of any upstream business. This dissertation is concentrated on developing portfolio optimization models to manage E&P projects. The idea is not new, but it has been mostly restricted to the conceptual level due to the inherent complications to capture interactions among projects. We disentangle the complications by modeling the project portfolio optimization problem as multistage stochastic programs with mixed integer programming (MIP) techniques. Due to the disparate nature of uncertainties, we separately consider explored and unexplored oil fields. We model portfolios of real options and portfolios of decision trees for the two cases, respectively. The resulting project portfolio models provide rigorous and consistent treatments to optimally balance the total rewards and the overall risk. For explored oil fields, oil price fluctuations dominate the geologic risk. The field development process hence can be modeled and assessed as sequentially compounded options with our optimization based option pricing models. We can further model the portfolio of real options to solve the dynamic capital budgeting problem for oil projects. For unexplored oil fields, the geologic risk plays the dominating role to determine how a field is optimally explored and developed. We can model the E&P process as a decision tree in the form of an optimization model with MIP techniques. By applying the inventory-style budget constraints, we can pool multiple project-specific decision trees to get the multistage E&P project portfolio optimization (MEPPO) model. The resulting large scale MILP is efficiently solved by a decomposition-based primal heuristic algorithm. The MEPPO model requires a scenario tree to approximate the stochastic process of the geologic parameters. We apply statistical learning, Monte Carlo simulation, and scenario reduction methods to generate the scenario tree, in which prior beliefs can be progressively refined with new information.<br>text

Multistage stochastic optimization is used to solve many real-life problems where decisions are taken at multiple times, e.g., portfolio selection problems. Such problems need the definition of stochastic processes, which are usually approxim- ated by scenario trees. The choice of the size of the scenario trees is the result of a compromise between the best approximation and the possibilities of the com- puter technology. Therefore, once a master scenario tree has been generated, it can be needed to reduce its dimension in order to make the problem computation- ally tractable. In this thesis, we introduce several scenario reduction algorithms and we compare them numerically for different types of master trees. A simple portfolio selection problem is also solved within the study. The distance from the initial scenario tree, the computational time, and the distance between the optimal objective values and solutions are compared for all the scenario reduction algorithms. In particular, we adopt the nested distance to measure the distance between two scenario trees. 1

We consider a hydrothermal scheduling problem with a mid-term horizon(HTSPM) modeled as a large-scale multistage stochastic program with stochastic monthly inflows of water to each hydro generator. In the HTSPM we seek an operating policy to minimize the sum of present and expected future costs, which include thermal generation costs and load curtailment costs. In addition to various simple bounds, problem constraints involve water balance, demand satisfaction and power interchanges. Sampling-based decomposition algorithms (SBDAs) have been used in the literature to solve HTSPM. SBDAs can be used to approximately solve problem instances with many time stages and with inflows that exhibit interstage dependence. Such dependence requires care in computing valid cuts for the decomposition algorithm. In order to help maintain tractability, we employ an aggregate reservoir representation (ARR). In an ARR all the hydro generators inside a specific region are grouped to effectively form one hydro plant with reservoir storage and generation capacity proportional to the parameters of the hydro plants used to form that aggregate reservoir. The ARR has been used in the literature with energy balance constraints, rather than water balance constraints, coupled with time series forecasts of energy inflows. Instead, we prefer as a model primitive to have the time series model forecast water inflows. This, in turn, requires that we extend existing methods to compute valid cuts for the decomposition method under the resulting form of interstage dependence. We form a sample average approximation of the original problem and then solve this problem by these special-purpose algorithms. And, we assess the quality of the resulting policy for operating the system. In our analysis, we compute a confidence interval on the optimality gap of a policy generated by solving an approximation on a sampled scenario tree. We present computational results on test problems with 24 monthly stages in which the inter-stage dependency of hydro inflows is modeled using a dynamic linear model. We further develop a parallel implementation of an SBDA. We apply SBDA to solve the HTSPM for the Brazilian power system that has 150 hydro generators, 151 thermal generators and 4 regions that each characterize an aggregate reservoir. We create and solve four different HTSPM instances where we change the input parameters with respect to generation capacity, transmission capacity and load in order to analyze the difference in the total expected cost.<br>text