SOLVING TWO-LEVEL OPTIMIZATION PROBLEMS WITH APPLICATIONS TO ROBUST DESIGN AND ENERGY MARKETS

This dissertation provides efficient techniques to solve two-level optimization problems. Three specific types of problems are considered. The first problem is robust optimization, which has direct applications to engineering design. Traditionally robust optimization problems have been solved using an inner-outer structure, which can be computationally expensive. This dissertation provides a method to decompose and solve this two-level structure using a modified Benders decomposition. This gradient-based technique is applicable to robust optimization problems with quasiconvex constraints and provides approximate solutions to problems with nonlinear constraints. The second types of two-level problems considered are mathematical and equilibrium programs with equilibrium constraints. Their two-level structure is simplified using Schur's decomposition and reformulation schemes for absolute value functions. The resulting formulations are applicable to game theory problems in operations research and economics. The third type of two-level problem studied is discretely-constrained mixed linear complementarity problems. These are first formulated into a two-level mathematical program with equilibrium constraints and then solved using the aforementioned technique for mathematical and equilibrium programs with equilibrium constraints. The techniques for all three problems help simplify the two-level structure into one level, which helps gain numerical and application insights. The computational effort for solving these problems is greatly reduced using the techniques in this dissertation. Finally, a host of numerical examples are presented to verify the approaches. Diverse applications to economics, operations research, and engineering design motivate the relevance of the novel methods developed in this dissertation

Propriedades de convergência de um método PQS estabilizado para problemas matemáticos com condições de equilíbrio

Orientador: Prof. Dr. Ademir Alves RibeiroCoorientador: Prof. Dr. José Alberto Ramos FlorTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Matemática. Defesa : Curitiba, 29/01/2019Inclui referênciasResumo: Problemas de Otimização com Condições de Equilíbrio (MPEC) possuem a particularidade de não satisfazer as condições de qualificação usuais para problemas de otimização não linear. Isto representa uma dificuldade ao tentar resolver problemas MPEC com estes métodos. Recentemente, considerando a condição de qualificação MPEC-LICQ, uma adaptação para problemas MPEC da condição de qualificação de independência linear (LICQ) usual, Izmailov, Solodov e Uskov provaram que métodos baseados em Lagrangiano Aumentado de primeira ordem, convergem a pontos C-estacionários, que são mais fracos que pontos KKT. Posteriormente Andreani, Secchin e Silva melhoraram o resultado, mostrando que quando se consideram métodos baseados em Lagrangiano Aumentado de segunda ordem pode-se garantir convergência a pontos pelo menos M-estacionários, condição mais forte que C-estacionariedade, porém também mais fraca que KKT. Além disso mostraram também que considerando a condição MPECRCPLD, mais fraca que MPEC-LICQ, e que certa sequência dos multiplicadores é limitada tem-se convergência a pontos S-estacionários, que são equivalentes a KKT. Neste trabalho mostramos que estes resultados não são exclusivos do método de Lagrangiano Aumentado. Apresentamos um método baseado em PQS estabilizada de segunda ordem que pode ser aplicado a problemas MPEC, obtendo resultados equivalentes. Assim, mostramos que quando se considera MPEC-LICQ, o método PQS estabilizado também garante convergência a pontos M-estacionérios. Além disso mostramos que considerando MPEC-RCPLD e uma propriedade de limitação vinculada ao multiplicador da restrição de complementaridade também temos convergência a pontos S-estacionários. Testes numéricos são feitos para validar os resultados teóricos. Palavras-chave: Problemas de Otimização com Condições de Equilíbrio. Programação não linear. Otimização com restrições. Programação Quadrática Sequencial Estabilizada. M-estacionariedade.Abstract: Mathematical Programs with Equilibrium Constraints (MPEC) have the particularity do not satisfy the usual constraint qualifications for standard nonlinear optimization. This represents a difficulty in attempting to solve MPEC problems with the usual nonlinear optimization methods. Recently, considering the MPEC-LICQ constraint qualification, an adaptation to MPEC problems of the usual linear independence constraint qualification (LICQ), Izmailov, Solodov and Uskov proved that first order augmented Lagrangian methods converge to C-stationary points, which are weaker than KKT points. Later on Andreani, Secchin and Silva improved this result, showing that when considering second order augmented Lagrangian methods, it can be guaranteed convergence to at least M-stationary points, which are stronger than C-stationary, but still weaker than KKT. In addition, they also showed that considering the MPEC-RCPLD constraint qualification, which is weaker than MPEC-LICQ, and assuming that a certain multiplier sequence is bounded, it can be proved the convergence to S-stationary points, which are equivalent to KKT. In this work we show that these results are not exclusive of the Augmented Lagrangian method. We provide a method based on second order stabilized Sequential Quadratic Programming, which can be applied to MPEC problems, achieving equivalent results. Thus we show that when MPEC-LICQ is considered, the stabilized SQP method also guarantees convergence to M-stationary points. Moreover, we show that considering MPEC-RCPLD and a boundedness property related to the complementarity constraint multiplier, the method also has convergence to S-stationary points. Numerical tests were performed to validate the theoretical results. Keywords: Mathematical Programs with Equilibrium Constraints. Nonlinear programming. Constrained optimization. stabilized Sequentia

OPTIMIZATION AND EQUILIBRIUM MODELING FOR RENEWABLE ENERGY: FOCUS ON WASTEWATER-TO-ENERGY APPLICATIONS

This dissertation presents three novel optimization models for sustainable wastewater management. The Blue Plains Advance Wastewater treatment plant (AWTP) operated by the District of Columbia Water and Sewer Authority (DC Water) is used as a case study. The application to the Blue Plains AWTP is presented to illustrate the usefulness of the model and how wastewater treatment plants (WWTPs), solid waste disposal plants, community management groups can actively and positively participate in energy and agricultural markets. Besides the conversion of the solid end products into biogas and electricity via digesters, WWTP can also produce Class B biosolids for land application or Class A biosolids for use as fertilizer. Chapter 1 introduces the Blue Plains case study and other important aspects of wastewater management. The first problem, discussed in Chapter 2, is a multi-objective, mixed-integer optimization model with an application to wastewater-derived energy. The decisions involve converting the amount of solid end products into biogas, and/or electricity for internal or external purposes. Three objectives; maximizing total value, minimizing energy purchased from external sources and minimizing carbon dioxide equivalent (CDE) emissions were presented via an approximation to the Pareto optimal set of solutions. The second type of problem is a stochastic multi-objective, mixed-integer optimization model with an application to wastewater-derived energy and is presented in Chapter 3. This model considers operational and investment decisions under uncertainty. We also consider investments in solar power and processing waste from outside sources for revenue and other benefits. The tradeoff decision between operational and investment costs and CDE emissions are presented. The third type of optimization model is a stochastic mathematical program with equilibrium constraints (MPEC) for sustainable wastewater management and is presented in Chapter 4. This two-level optimization problem is a stochastic model with a strategic WWTP as the upper-level player. The lower-level players represent the fertilizer, natural gas, compressed natural gas (CNG) and electricity markets. All the lower-level players are price-takers. Chapter 5 considers a comparison of the three optimization models discussed above and highlights differences. Chapter 6 provides conclusions, contributions, and potential future directions

Network modeling and optimization for energy and sustainable transit

Energy and transportation systems are integral to our infrastructure. Along with other types of networks, critical challenges constantly arise, particularly with regard to accessibility, efficiency, optimality, and sustainability. In this dissertation, we use mixed integer programming, data mining and mixed complementarity techniques to address some of these challenges. We have developed an improved schematic mapping algorithm to facilitate the process of network representation for a variety of systems beyond transportation. We also discover fundamental patterns in bicycle ownership on a global scale with implications for sustainable urban planning and public health outcomes. Finally, we model the fast-growing crude oil market in North America, implementing scenarios that point to integrated approaches to exports, pipeline investments and targeted rail restrictions as most viable for addressing medium-term oil transportation concerns. The methods we employ are generalizable to other types of energy and transit systems, and beyond. Finally, we discuss the importance of these methods to newer applications

ONE-AND-TWO-LEVEL NATURAL GAS EQUILIBRIUM MODELS AND ALGORITHMS

This dissertation consists of three parts; Part 1 provides two applied studies for the current issue of the global natural gas market, Part 2 presents the World Gas Model (WGM) 2014 version-a significant extension of WGM 2012, and Part 3 develops a novel Benders decomposition procedure with SOS1 reformulation to solve mathematical programs with equilibrium constraints (MPECs) and then is applied to several applications in natural gas and additional test problems. Part 1 presents two applied studies related to the impacts of U.S. liquefied natural gas (LNG) exports on global gas markets as well as the influence of the Panama Canal tariff selection on global gas trade. The first study within Part 1 investigates the effect of the U.S. LNG exports on the global gas markets using the WGM 2012 (Gabriel et. al., 2012), a market equilibrium model for global LNG markets based on a mixed complementarity problem (MCP) format. The second study within Part 1 focuses on the influence of the Panama Canal tariffs on global trade using WGM 2012 as well. After a planned expansion, the Panama Canal waterway will accommodate more than eighty percent of LNG tankers, providing significant potential time and cost savings for LNG buyers and producers. The aim of the second applied study is to address how the Panama Canal tariffs affect global gas trades In Part 2, a significant extension of the World Gas Model 2012 is developed. This new version called WGM 2014, distinguishes itself from the previous version in the sense of more detail for LNG markets including more market participants e.g., liquefiers, regasifiers, LNG shipping operators, and a canal operator as new players with separate optimization problems and market-clearing conditions. Moreover, the LNG shipping costs and congestion tariffs for canal transit fees are endogenously determined inside the model as opposed to being exogenously determined before. Also, WGM 2014 has flexible LNG routes. In particular, there are three route options for each LNG shipping operator: 1. Sending LNG via the Panama Canal, 2. the Suez Canal, or using a regular route without a canal. Moreover, WGM 2014 takes into account the limitations of maritime transportation by limiting the size of the LNG tankers that can pass through the Panama and Suez canals which itself is a major improvement for natural gas policy study. In part 3, the method we develop uses an SOS1 approach based on (Siddiqui and Gabriel, 2012) to replace complementarity in the lower-level problem's optimality conditions. Then, Benders algorithm decomposes the MPECs into a master and a subproblem and solves the overall problem iteratively. This methodology is applied to small, illustrative examples and a large-scale MPEC version of the World Gas Model where the Panama Canal operator is a Stackelberg leader with a reduced version of the rest of the global gas markets considered as followers