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Esta Tesis esta motivada por el interés en aplicar procedimientos de optimización global a problemas del mundo real. Para ello, nos hemos centrado en el problema de Coordinación Hidrotérmica de la Generación Eléctrica a Corto Plazo (llamado Problema de Generación en esta Tesis) donde la función objetivo y las restricciones no lineales son polinomios de grado como máximo cuatro. En el Problema de Generación no tenemos disponible una representación en diferencia convexa de las funciones involucradas ni tampoco es posible utilizar la estructura del problema para simplificarlo. No obstante, cuando disponemos de una función continua f(x) definida en un conjunto cerrado y no vacío S el problema puede transformarse en otro equivalente expresado mediante minimize l(z) subject to z 2 D n int. (programa d.c. canónico), donde l(z) es una función convexa (en general suele ser una función lineal) con D y C conjuntos convexos y cerrados. Una estructura matemática tal como Dnint C no resulta siempre aparente y aunque lo fuera siempre queda por realizar una gran cantidad de cálculos para expresarla de manera que se pueda resolver el problema de una manera eficiente desde un punto de vista computacional.La característica más importante de esta estructura es que aparecen conjuntos convexos y complementarios de conjuntos convexos. Por este motivo en tales problemas se pueden usar herramientas analíticas tales como subdifernciales y hiperplanos soporte. Por otro lado, como aparecen conjuntos complementarios de conjuntos convexos, estas herramientas analíticas se deben usar de una manera determinada y combinándolas con herramientas combinatorias tales como cortes por planos, Branco and bound y aproximación interior.En esta tesis se pone de manifiesto la estructura matemática subyacente en el Problema de Generación utilizando el hecho de que los polinomios son expresables como diferencia de funciones convexas. Utilizando esta propiedad describimos el problema como un programa d.c. canónico equivalente. Pero aun mas, partiendo de la estructura de las funciones del Problema de Generación es posible rescribirlo de una manera mas conveniente y obtener de este modo ventajas numéricas desde elpunto de vista de la implementación.Basándonos en la propiedad de que los polinomios homogéneos de grado 1 son un conjunto de generadores del espacio vectorial de los polinomios homogéneos de grado m hemos desarrollamos los conceptos y propiedades necesarios que nos permiten expresar un polinomio cualquiera como diferencia de polinomios convexos, También, se ha desarrollado y demostrado la convergencia de un nuevo algoritmo de optimización global (llamado Algoritmo Adaptado) que permite resolver el Problema de Generación. Como el programa equivalente no esta acotado se ha introducido una técnica de subdivisión mediante prismas en lugar de la habitual subdivisión mediante conos.Para obtener una descomposición óptima de un polinomio en diferencia de polinomios convexos, se ha enunciado el Problema de Norma Mínima mediante la introducción del concepto de Descomposición con Mínima Desviación, con lo cual obtenemos implementaciones m´as eficientes, al reducir el n´umero de iteraciones del Algoritmo Adaptado. Para resolver el problema de Norma Mínima hemos implementado un algoritmo de programación cuadrática semi-infinita utilizando una estrategia de build-up and build-down, introducida por Den Hertog (1997) para resolver programas lineales semi-infinitos, la cual usa un procedimiento de barrera logarítmica.Finalmente, se describen los resultados obtenidos por la implementación de los algoritmos anteriormente mencionados y se dan las conclusiones. / This Thesis has been motivated by the interest in applying deterministic global optimization procedures to problems in the real world with no special structures. We have focused on the Short-Term Hydrothermal Coordination of Electricity Generation Problem (also named Generation Problem in this Thesis) where the objective function and the nonlinear constraints are polynomials of degree up to four. In the Generation Problem there is no available d.c. representation of the involved functions and we cannot take advantage of any special structure of the problem either. Hence, a very general problem, such as the above-mentioned, does not seem to have any mathematical structure conducive to computational implementations. Nevertheless, when f(x) is a continuous function and S is a nonempty closed set the problem can be transformed into an equivalent problem expressed by minimize l(z) subject to z 2 D n intC (canonical d.c. program), where l(z) is a convex function (which is usually a linear function) and D and C are closed convex sets. A mathematical complementary convex structure such as D n int C is not always apparent and even when it is explicit, a lot of work still remains to be done to bring it into a form amenable to efficient computational implementations. The attractive feature of the mathematicalcomplementary convex structure is that it involves convexity. Thus, we can use analytical tools from convex analysis like sub differential and supporting hyper plane.On the other hand, since convexity is involved in a reverse sense, these tools must be used in some specific way and combined with combinatorial tools like cutting planes, branch and bound and outer approximation.We introduce the common general mathematical complementary convex structure underlying in global optimization problems and describe the Generation Problem, whose functions are d.c. functions because they are polynomials. Thus, by using the properties of the d.c. functions, we describe the Generation Problem as an equivalent canonical d.c. programming problem. From the structure of its functions the Generation Problem can be rewritten as a more suitable equivalent reverse convex program in order to obtain an adaptation for advantageous numerical implementations.Concepts and properties are introduced which allow us to obtain an explicit representation of a polynomial as a deference of convex polynomials, based on the fact that the set of mth powers of homogeneous polynomials of degree 1 is a generating set for the vector space of homogeneous polynomials of degree m.We also describe a new global optimization algorithm (adapted algorithm) in order to solve the Generation Problem. Since the equivalent reverse convex program is unbounded we use prismatical subdivisions instead of conical ones. Moreover, we prove the convergence of the adapted algorithm by using a prismatical subdivision process together with an outer approximation procedure.We enounce the Minimal Norm Problem by using the concept of Least Deviation Decomposition in order to obtain the optimal d.c. representation of a polynomial function, which allows a more efficient implementation, by reducing the number of iterations of the adapted algorithm.A quadratic semi-infinite algorithm is described. We propose a build-up and down strategy, introduced by Den Hertog (1997) for standard linear programs that uses a logarithmic barrier method.Finally, computational results are given and conclusions are explained.

This dissertation addresses five fundamental resource allocation problems on networks, all of which have applications to support Homeland Security or industry challenges. In the first application, we model and solve the strategic problem of minimizing the expected loss inflicted by a hostile terrorist organization. An appropriate allocation of certain capability-related, intent-related, vulnerability-related, and consequence-related resources is used to reduce the probabilities of success in the respective attack-related actions, and to ameliorate losses in case of a successful attack. Given the disparate nature of prioritizing capital and material investments by federal, state, local, and private agencies to combat terrorism, our model and accompanying solution procedure represent an innovative, comprehensive, and quantitative approach to coordinate resource allocations from various agencies across the breadth of domains that deal with preventing attacks and mitigating their consequences. Adopting a nested event tree optimization framework, we present a novel formulation for the problem as a specially structured nonconvex factorable program, and develop two branch-and-bound schemes based respectively on utilizing a convex nonlinear relaxation and a linear outer-approximation, both of which are proven to converge to a global optimal solution. We also investigate a fundamental special-case variant for each of these schemes, and design an alternative direct mixed-integer programming model representation for this scenario. Several range reduction, partitioning, and branching strategies are proposed, and extensive computational results are presented to study the efficacy of different compositions of these algorithmic ingredients, including comparisons with the commercial software BARON. The developed set of algorithmic implementation strategies and enhancements are shown to outperform BARON over a set of simulated test instances, where the best proposed methodology produces an average optimality gap of 0.35% (compared to 4.29% for BARON) and reduces the required computational effort by a factor of 33. A sensitivity analysis is also conducted to explore the effect of certain key model parameters, whereupon we demonstrate that the prescribed algorithm can attain significantly tighter optimality gaps with only a near-linear corresponding increase in computational effort. In addition to enabling effective comprehensive resource allocations, this research permits coordinating agencies to conduct quantitative what-if studies on the impact of alternative resourcing priorities. The second application is motivated by the author's experience with the U.S. Army during a tour in Iraq, during which combined operations involving U.S. Army, Iraqi Army, and Iraqi Police forces sought to interdict the transport of selected materials used for the manufacture of specialized types of Improvised Explosive Devices, as well as to interdict the distribution of assembled devices to operatives in the field. In this application, we model and solve the problem of minimizing the maximum flow through a network from a given source node to a terminus node, integrating different forms of superadditive synergy with respect to the effect of resources applied to the arcs in the network. Herein, the superadditive synergy reflects the additional effectiveness of forces conducting combined operations, vis-à-vis unilateral efforts. We examine linear, concave, and general nonconcave superadditive synergistic relationships between resources, and accordingly develop and test effective solution procedures for the underlying nonlinear programs. For the linear case, we formulate an alternative model representation via Fourier-Motzkin elimination that reduces average computational effort by over 40% on a set of randomly generated test instances. This test is followed by extensive analyses of instance parameters to determine their effect on the levels of synergy attained using different specified metrics. For the case of concave synergy relationships, which yields a convex program, we design an inner-linearization procedure that attains solutions on average within 3% of optimality with a reduction in computational effort by a factor of 18 in comparison with the commercial codes SBB and BARON for small- and medium-sized problems; and outperforms these softwares on large-sized problems, where both solvers failed to attain an optimal solution (and often failed to detect a feasible solution) within 1800 CPU seconds. Examining a general nonlinear synergy relationship, we develop solution methods based on outer-linearizations, inner-linearizations, and mixed-integer approximations, and compare these against the commercial software BARON. Considering increased granularities for the outer-linearization and mixed-integer approximations, as well as different implementation variants for both these approaches, we conduct extensive computational experiments to reveal that, whereas both these techniques perform comparably with respect to BARON on small-sized problems, they significantly improve upon the performance for medium- and large-sized problems. Our superlative procedure reduces the computational effort by a factor of 461 for the subset of test problems for which the commercial global optimization software BARON could identify a feasible solution, while also achieving solutions of objective value 0.20% better than BARON. The third application is likewise motivated by the author's military experience in Iraq, both from several instances involving coalition forces attempting to interdict the transport of a kidnapping victim by a sectarian militia as well as, from the opposite perspective, instances involving coalition forces transporting detainees between interment facilities. For this application, we examine the network interdiction problem of minimizing the maximum probability of evasion by an entity traversing a network from a given source to a designated terminus, while incorporating novel forms of superadditive synergy between resources applied to arcs in the network. Our formulations examine either linear or concave (nonlinear) synergy relationships. Conformant with military strategies that frequently involve a combination of overt and covert operations to achieve an operational objective, we also propose an alternative model for sequential overt and covert deployment of subsets of interdiction resources, and conduct theoretical as well as empirical comparative analyses between models for purely overt (with or without synergy) and composite overt-covert strategies to provide insights into absolute and relative threshold criteria for recommended resource utilization. In contrast to existing static models, in a fourth application, we present a novel dynamic network interdiction model that improves realism by accounting for interactions between an interdictor deploying resources on arcs in a digraph and an evader traversing the network from a designated source to a known terminus, wherein the agents may modify strategies in selected subsequent periods according to respective decision and implementation cycles. We further enhance the realism of our model by considering a multi-component objective function, wherein the interdictor seeks to minimize the maximum value of a regret function that consists of the evader's net flow from the source to the terminus; the interdictor's procurement, deployment, and redeployment costs; and penalties incurred by the evader for misperceptions as to the interdicted state of the network. For the resulting minimax model, we use duality to develop a reformulation that facilitates a direct solution procedure using the commercial software BARON, and examine certain related stability and convergence issues. We demonstrate cases for convergence to a stable equilibrium of strategies for problem structures having a unique solution to minimize the maximum evader flow, as well as convergence to a region of bounded oscillation for structures yielding alternative interdictor strategies that minimize the maximum evader flow. We also provide insights into the computational performance of BARON for these two problem structures, yielding useful guidelines for other research involving similar non-convex optimization problems. For the fifth application, we examine the problem of apportioning railcars to car manufacturers and railroads participating in a pooling agreement for shipping automobiles, given a dynamically determined total fleet size. This study is motivated by the existence of such a consortium of automobile manufacturers and railroads, for which the collaborative fleet sizing and efforts to equitably allocate railcars amongst the participants are currently orchestrated by the \textit{TTX Company} in Chicago, Illinois. In our study, we first demonstrate potential inequities in the industry standard resulting either from failing to address disconnected transportation network components separately, or from utilizing the current manufacturer allocation technique that is based on average nodal empty transit time estimates. We next propose and illustrate four alternative schemes to apportion railcars to manufacturers, respectively based on total transit time that accounts for queuing; two marginal cost-induced methods; and a Shapley value approach. We also provide a game-theoretic insight into the existing procedure for apportioning railcars to railroads, and develop an alternative railroad allocation scheme based on capital plus operating costs. Extensive computational results are presented for the ten combinations of current and proposed allocation techniques for automobile manufacturers and railroads, using realistic instances derived from representative data of the current business environment. We conclude with recommendations for adopting an appropriate apportionment methodology for implementation by the industry. / Ph. D.

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. / 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.