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

Disjunctive Inequalities: Applications and Extensions

A general optimization problem can be expressed in the form min{cx: x ∈ S}, (1) where x ∈ R n is the vector of decision variables, c ∈ R n is a linear objective function and S ⊂ R n is the set of feasible solutions of (1). Because S is generall

A New Mathematical Programming Framework for Facility Layout Design

We present a new framework for efficiently finding competitive solutions for the facility layout problem. This framework is based on the combination of two new mathematical programming models. The first model is a relaxation of the layout problem and is intended to find good starting points for the iterative algorithm used to solve the second model. The second model is an exact formulation of the facility layout problem as a non-convex mathematical program with equilibrium constraints (MPEC). Aspect ratio constraints, which are frequently used in facility layout methods to restrict the occurrence of overly long and narrow departments in the computed layouts, are easily incorporated into this new framework. Finally, we present computational results showing that both models, and hence the complete framework, can be solved efficiently using widely available optimization software. This important feature of the new framework implies that it can be used to find competitive layouts with relatively little computational effort. This is advantageous for a user who wishes to consider several competitive layouts rather than simply using the mathematically optimal layout

An enumerative method for convex programs with linear complementarity constraints and application to the bilevel problem of a forecast model for high complexity products

The increasing variety of high complexity products presents a challenge in acquiring detailed demand forecasts. Against this backdrop, a convex quadratic parameter dependent forecast model is revisited, which calculates a prognosis for structural parts based on historical order data. The parameter dependency inspires a bilevel problem with convex objective function, that allows for the calculation of optimal parameter settings in the forecast model. The bilevel problem can be formulated as a mathematical problem with equilibrium constraints (MPEC), which has a convex objective function and linear constraints. Several new enumerative methods are presented, that find stationary points or global optima for this problem class. An algorithmic concept shows a recursive pattern, which finds global optima of a convex objective function on a general non-convex set defined by a union of polytopes. Inspired by these concepts the thesis investigates two implementations for MPECs, a search algorithm and a hybrid algorithm

Economic Engineering Modeling of Liberalized Electricity Markets: Approaches, Algorithms, and Applications in a European Context: Economic Engineering Modeling of Liberalized Electricity Markets: Approaches, Algorithms, and Applications in a European Context

This dissertation focuses on selected issues in regard to the mathematical modeling of electricity markets. In a first step the interrelations of electric power market modeling are highlighted a crossroad between operations research, applied economics, and engineering. In a second step the development of a large-scale continental European economic engineering model named ELMOD is described and the model is applied to the issue of wind integration. It is concluded that enabling the integration of low-carbon technologies appears feasible for wind energy. In a third step algorithmic work is carried out regarding a game theoretic model. Two approaches in order to solve a discretely-constrained mathematical program with equilibrium constraints using disjunctive constraints are presented. The first one reformulates the problem as a mixed-integer linear program and the second one applies the Benders decomposition technique. Selected numerical results are reported

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