Minmax regret combinatorial optimization problems: an Algorithmic Perspective

Candia-Vejar, A (reprint author), Univ Talca, Modeling & Ind Management Dept, Curico, Chile.Uncertainty in optimization is not a new ingredient. Diverse models considering uncertainty have been developed over the last 40 years. In our paper we essentially discuss a particular uncertainty model associated with combinatorial optimization problems, developed in the 90's and broadly studied in the past years. This approach named minmax regret (in particular our emphasis is on the robust deviation criteria) is different from the classical approach for handling uncertainty, stochastic approach, where uncertainty is modeled by assumed probability distributions over the space of all possible scenarios and the objective is to find a solution with good probabilistic performance. In the minmax regret (MMR) approach, the set of all possible scenarios is described deterministically, and the search is for a solution that performs reasonably well for all scenarios, i.e., that has the best worst-case performance. In this paper we discuss the computational complexity of some classic combinatorial optimization problems using MMR. approach, analyze the design of several algorithms for these problems, suggest the study of some specific research problems in this attractive area, and also discuss some applications using this model

Contingency-Constrained Unit Commitment with Post-Contingency Corrective Recourse

We consider the problem of minimizing costs in the generation unit commitment problem, a cornerstone in electric power system operations, while enforcing an N-k-e reliability criterion. This reliability criterion is a generalization of the well-known $N$-$k$ criterion, and dictates that at least $(1-e_ j)$ fraction of the total system demand must be met following the failures of $k$ or fewer system components. We refer to this problem as the Contingency-Constrained Unit Commitment problem, or CCUC. We present a mixed-integer programming formulation of the CCUC that accounts for both transmission and generation element failures. We propose novel cutting plane algorithms that avoid the need to explicitly consider an exponential number of contingencies. Computational studies are performed on several IEEE test systems and a simplified model of the Western US interconnection network, which demonstrate the effectiveness of our proposed methods relative to current state-of-the-art

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

Enhancing network robustness via shielding

We consider shielding critical links to guarantee network connectivity under geographical and general failure models. We develop a mixed integer linear program (MILP) to obtain the minimum cost shielding to guarantee the connectivity of a single SD pair under a general failure model, and exploit geometric properties to decompose the shielding problem under a geographical failure model. We extend our MILP formulation to guarantee the connectivity of the entire network, and use Benders decomposition to significantly reduce the running time by exploiting its partial separable structure. We also apply simulated annealing to solve larger network problems to obtain near-optimal solutions in much shorter time. Finally, we extend the algorithms to guarantee partial network connectivity, and observe significant reduction in shielding cost, especially when the failure region is small. For example, when the failure region radius is 60 miles, we observe as much as 75% reduction in shielding cost by relaxing the connectivity requirement to 95% on a major US infrastructure network

Contributions to robust and bilevel optimization models for decision-making

Los problemas de optimización combinatorios han sido ampliamente estudiados en la literatura especializada desde mediados del siglo pasado. No obstante, en las últimas décadas ha habido un cambio de paradigma en el tratamiento de problemas cada vez más realistas, en los que se incluyen fuentes de aleatoriedad e incertidumbre en los datos, múltiples criterios de optimización y múltiples niveles de decisión. Esta tesis se desarrolla en este contexto. El objetivo principal de la misma es el de construir modelos de optimización que incorporen aspectos inciertos en los parámetros que de nen el problema así como el desarrollo de modelos que incluyan múltiples niveles de decisión. Para dar respuesta a problemas con incertidumbre usaremos los modelos Minmax Regret de Optimización Robusta, mientras que las situaciones con múltiples decisiones secuenciales serán analizadas usando Optimización Binivel. En los Capítulos 2, 3 y 4 se estudian diferentes problemas de decisión bajo incertidumbre a los que se dará una solución robusta que proteja al decisor minimizando el máximo regret en el que puede incurrir. El criterio minmax regret analiza el comportamiento del modelo bajo distintos escenarios posibles, comparando su e ciencia con la e ciencia óptima bajo cada escenario factible. El resultado es una solución con una eviciencia lo más próxima posible a la óptima en el conjunto de las posibles realizaciones de los parámetros desconocidos. En el Capítulo 2 se estudia un problema de diseño de redes en el que los costes, los pares proveedor-cliente y las demandas pueden ser inciertos, y además se utilizan poliedros para modelar la incertidumbre, permitiendo de este modo relaciones de dependencia entre los parámetros. En el Capítulo 3 se proponen, en el contexto de la secuenciación de tareas o la computación grid, versiones del problema del camino más corto y del problema del viajante de comercio en el que el coste de recorrer un arco depende de la posición que este ocupa en el camino, y además algunos de los parámetros que de nen esta función de costes son inciertos. La combinación de la dependencia en los costes y la incertidumbre en los parámetros da lugar a dependencias entre los parámetros desconocidos, que obliga a modelar los posibles escenarios usando conjuntos más generales que los hipercubos, habitualmente utilizados en este contexto. En este capítulo, usaremos poliedros generales para este cometido. Para analizar este primer bloque de aplicaciones, en el Capítulo 4, se analiza un modelo de optimización en el que el conjunto de posibles escenarios puede ser alterado mediante la realización de inversiones en el sistema. En los problemas estudiados en este primer bloque, cada decisión factible es evaluada en base a la reacción más desfavorable que pueda darse en el sistema. En los Capítulos 5 y 6 seguiremos usando esta idea pero ahora se supondrá que esa reacción a la decisión factible inicial está en manos de un adversario o follower. Estos dos capítulos se centran en el estudio de diferentes modelos binivel. La Optimización Binivel aborda problemas en los que existen dos niveles de decisión, con diferentes decisores en cada uno ellos y la decisión se toma de manera jerárquica. En concreto, en el Capítulo 5 se estudian distintos modelos de jación de precios en el contexto de selección de carteras de valores, en los que el intermediario nanciero, que se convierte en decisor, debe jar los costes de invertir en determinados activos y el inversor debe seleccionar su cartera de acuerdo a distintos criterios. Finalmente, en el Capítulo 6 se estudia un problema de localización en el que hay distintos decisores, con intereses contrapuestos, que deben determinar secuencialmente la ubicación de distintas localizaciones. Este modelo de localización binivel se puede aplicar en contextos como la localización de servicios no deseados o peligrosos (plantas de reciclaje, centrales térmicas, etcétera) o en problemas de ataque-defensa. Todos estos modelos se abordan mediante el uso de técnicas de Programación Matemática. De cada uno de ellos se analizan algunas de sus propiedades y se desarrollan formulaciones y algoritmos, que son examinados también desde el punto de vista computacional. Además, se justica la validez de los modelos desde un enfoque de las aplicaciones prácticas. Los modelos presentados en esta tesis comparten la peculiaridad de requerir resolver distintos problemas de optimización encajados.Combinatorial optimization problems have been extensively studied in the specialized literature since the mid-twentieth century. However, in recent decades, there has been a paradigm shift to the treatment of ever more realistic problems, which include sources of randomness and uncertainty in the data, multiple optimization criteria and multiple levels of decision. This thesis concerns the development of such concepts. Our objective is to study optimization models that incorporate uncertainty elements in the parameters de ning the model, as well as the development of optimization models integrating multiple decision levels. In order to consider problems under uncertainty, we use Minmax Regret models from Robust Optimization; whereas the multiplicity and hierarchy in the decision levels is addressed using Bilevel Optimization. In Chapters 2, 3 and 4, we study di erent decision problems under uncertainty to which we give a robust solution that protects the decision-maker minimizing the maximum regret that may occur. This robust criterion analyzes the performance of the system under multiple possible scenarios, comparing its e ciency with the optimum one under each feasible scenario. We obtain, as a result, a solution whose e ciency is as close as possible to the optimal one in the set of feasible realizations of the uncertain parameters. In Chapter 2, we study a network design problem in which the costs, the pairs supplier-customer, and the demands can take uncertain values. Furthermore, the uncertainty in the parameters is modeled via polyhedral sets, thereby allowing relationships among the uncertain parameters. In Chapter 3, we propose time-dependent versions of the shortest path and traveling salesman problems in which the costs of traversing an arc depends on the relative position that the arc occupies in the path. Moreover, we assume that some of the parameters de ning these costs can be uncertain. These models can be applied in the context of task sequencing or grid computing. The incorporation of time-dependencies together with uncertainties in the parameters gives rise to dependencies among the uncertain parameters, which require modeling the possible scenarios using more general sets than hypercubes, normally used in this context. In this chapter, we use general polyhedral sets with this purpose. To nalize this rst block of applications, in Chapter 4, we analyze an optimization model in which the set of possible scenarios can be modi ed by making some investments in the system. In the problems studied in this rst block, each feasible decision is evaluated based on the most unfavorable possible reaction of the system. In Chapters 5 and 6, we will still follow this idea, but assuming that the reaction to the initial feasible decision will be held by a follower or an adversary, instead of assuming the most unfavorable one. These two chapters are focused on the study of some bilevel models. Bilevel Optimization addresses optimization problems with multiple decision levels, di erent decision-makers in each level and a hierarchical decision order. In particular, in Chapter 5, we study some price setting problems in the context of portfolio selection. In these problems, the nancial intermediary becomes a decisionmaker and sets the transaction costs for investing in some securities, and the investor chooses her portfolio according to di erent criteria. Finally, in Chapter 6, we study a location problem with several decision-makers and opposite interests, that must set, sequentially, some location points. This bilevel location model can be applied in practical applications such as the location of semi-obnoxious facilities (power or electricity plants, waste dumps, etc.) or interdiction problems. All these models are stated from a Mathematical Programming perspective, analyzing their properties and developing formulations and algorithms, that are tested from a computational point of view. Furthermore, we pay special attention to justifying the validity of the models from the practical applications point of view. The models presented in this thesis share the characteristic of involving the resolution of nested optimization problems.Premio Extraordinario de Doctorado U

Optimizing dynamic investment decisions for railway systems protection

Past and recent events have shown that railway infrastructure systems are particularly vulnerable to natural catastrophes, unintentional accidents and terrorist attacks. Protection investments are instrumental in reducing economic losses and preserving public safety. A systematic approach to plan security investments is paramount to guarantee that limited protection resources are utilized in the most efficient manner. In this article, we present an optimization model to identify the railway assets which should be protected to minimize the impact of worst case disruptions on passenger flows. We consider a dynamic investment problem where protection resources become available over a planning horizon. The problem is formulated as a bilevel mixed-integer model and solved using two different decomposition approaches. Random instances of different sizes are generated to compare the solution algorithms. The model is then tested on the Kent railway network to demonstrate how the results can be used to support efficient protection decisions

An Analysis of Multiple Layered Networks

Current infrastructure network models of single functionality do not typically account for the interdependent nature of infrastructure networks. Infrastructure networks are generally modeled individually, as an isolated network or with minimal recognition of interactions. This research develops a methodology to model the individual infrastructure network types while explicitly modeling their interconnected effects. The result is a formulation built with two sets of variables (the original set to model infrastructure characteristics and an additional set representing cuts of interdependent elements). This formulation is decomposed by variable type using Benders Partitioning and solved to optimality using a Benders Partitioning algorithm. Current infrastructure network models of single functionality do not typically account for the interdependent nature of infrastructure networks, Infrastructure networks are generally modeled individually, as an isolated network or with minimal recognition of interactions, This research develops a methodology to model the individual infrastructure network types while explicitly modeling their interconnected effects, The result is a formulation built with two sets of variables (the original set to model infrastructure characteristics and an additional set representing cuts of interdependent elements) This formulation is decomposed by variable type using Benders\u27 Partitioning and solved to optimality using a Benders\u27 Partitioning algorithm

Computationally efficient solution of mixed integer model predictive control problems via machine learning aided Benders Decomposition

Mixed integer Model Predictive Control (MPC) problems arise in the operation of systems where discrete and continuous decisions must be taken simultaneously to compensate for disturbances. The efficient solution of mixed integer MPC problems requires the computationally efficient and robust online solution of mixed integer optimization problems, which are generally difficult to solve. In this paper, we propose a machine learning-based branch and check Generalized Benders Decomposition algorithm for the efficient solution of such problems. We use machine learning to approximate the effect of the complicating variables on the subproblem by approximating the Benders cuts without solving the subproblem, therefore, alleviating the need to solve the subproblem multiple times. The proposed approach is applied to a mixed integer economic MPC case study on the operation of chemical processes. We show that the proposed algorithm always finds feasible solutions to the optimization problem, given that the mixed integer MPC problem is feasible, and leads to a significant reduction in solution time (up to 97% or 50x) while incurring small error (in the order of 1%) compared to the application of standard and accelerated Generalized Benders Decomposition

OPTIMIZATION MODELS AND METHODOLOGIES TO SUPPORT EMERGENCY PREPAREDNESS AND POST-DISASTER RESPONSE

This dissertation addresses three important optimization problems arising during the phases of pre-disaster emergency preparedness and post-disaster response in time-dependent, stochastic and dynamic environments. The first problem studied is the building evacuation problem with shared information (BEPSI), which seeks a set of evacuation routes and the assignment of evacuees to these routes with the minimum total evacuation time. The BEPSI incorporates the constraints of shared information in providing on-line instructions to evacuees and ensures that evacuees departing from an intermediate or source location at a mutual point in time receive common instructions. A mixed-integer linear program is formulated for the BEPSI and an exact technique based on Benders decomposition is proposed for its solution. Numerical experiments conducted on a mid-sized real-world example demonstrate the effectiveness of the proposed algorithm. The second problem addressed is the network resilience problem (NRP), involving an indicator of network resilience proposed to quantify the ability of a network to recover from randomly arising disruptions resulting from a disaster event. A stochastic, mixed integer program is proposed for quantifying network resilience and identifying the optimal post-event course of action to take. A solution technique based on concepts of Benders decomposition, column generation and Monte Carlo simulation is proposed. Experiments were conducted to illustrate the resilience concept and procedure for its measurement, and to assess the role of network topology in its magnitude. The last problem addressed is the urban search and rescue team deployment problem (USAR-TDP). The USAR-TDP seeks an optimal deployment of USAR teams to disaster sites, including the order of site visits, with the ultimate goal of maximizing the expected number of saved lives over the search and rescue period. A multistage stochastic program is proposed to capture problem uncertainty and dynamics. The solution technique involves the solution of a sequence of interrelated two-stage stochastic programs with recourse. A column generation-based technique is proposed for the solution of each problem instance arising as the start of each decision epoch over a time horizon. Numerical experiments conducted on an example of the 2010 Haiti earthquake are presented to illustrate the effectiveness of the proposed approach

Synthesis, Interdiction, and Protection of Layered Networks

This research developed the foundation, theory, and framework for a set of analysis techniques to assist decision makers in analyzing questions regarding the synthesis, interdiction, and protection of infrastructure networks. This includes extension of traditional network interdiction to directly model nodal interdiction; new techniques to identify potential targets in social networks based on extensions of shortest path network interdiction; extension of traditional network interdiction to include layered network formulations; and develops models/techniques to design robust layered networks while considering trade-offs with cost. These approaches identify the maximum protection/disruption possible across layered networks with limited resources, find the most robust layered network design possible given the budget limitations while ensuring that the demands are met, include traditional social network analysis, and incorporate new techniques to model the interdiction of nodes and edges throughout the formulations. In addition, the importance and effects of multiple optimal solutions for these (and similar) models is investigated. All the models developed are demonstrated on notional examples and were tested on a range of sample problem sets