This dissertation considers graph theoretic generalizations of the maximum clique problem. Models that were originally proposed in social network analysis literature, are investigated from a mathematical programming perspective for the first time. A social network is usually represented by a graph, and cliques were the first models of "tightly knit groups" in social networks, referred to as cohesive subgroups. Cliques are idealized models and their overly restrictive nature motivated the development of clique relaxations that relax different aspects of a clique. Identifying large cohesive subgroups in social networks has traditionally been used in criminal network analysis to study organized crimes such as terrorism, narcotics and money laundering. More recent applications are in clustering and data mining wireless networks, biological networks as well as graph models of databases and the internet. This research has the potential to impact homeland security, bioinformatics, internet research and telecommunication industry among others. The focus of this dissertation is a degree-based relaxation called k-plex. A distance-based relaxation called k-clique and a diameter-based relaxation called k-club are also investigated in this dissertation. We present the first systematic study of the complexity aspects of these problems and application of mathematical programming techniques in solving them. Graph theoretic properties of the models are identified and used in the development of theory and algorithms. Optimization problems associated with the three models are formulated as binary integer programs and the properties of the associated polytopes are investigated. Facets and valid inequalities are identified based on combinatorial arguments. A branch-and-cut framework is designed and implemented to solve the optimization problems exactly. Specialized preprocessing techniques are developed that, in conjunction with the branch-and-cut algorithm, optimally solve the problems on real-life power law graphs, which is a general class of graphs that include social and biological networks. Computational experiments are performed to study the effectiveness of the proposed solution procedures on benchmark instances and real-life instances. The relationship of these models to the classical maximum clique problem is studied, leading to several interesting observations including a new compact integer programming formulation. We also prove new continuous non-linear formulations for the classical maximum independent set problem which maximize continuous functions over the unit hypercube, and characterize its local and global maxima. Finally, clustering and network design extensions of the clique relaxation models are explored.

This dissertation considers a subclass of sum-of-ratios fractional combinatorial optimization problems (FCOPs) whose linear versions admit polynomial-time exact algorithms. This topic lies in the intersection of two scarcely researched areas of fractional programming (FP): sum-of-ratios FP and combinatorial FP. Although not extensively researched, the sum-of-ratios problems have a number of important practical applications in manufacturing, administration, transportation, data mining, etc. Since even in such a restricted research domain the problems are numerous, the main focus of this dissertation is a mathematical programming study of the three, probably, most classical FCOPs: Minimum Multiple Ratio Spanning Tree (MMRST), Minimum Multiple Ratio Path (MMRP) and Minimum Multiple Ratio Cycle (MMRC). The first two problems are studied in detail, while for the other one only the theoretical complexity issues are addressed. The dissertation emphasizes developing solution methodologies for the considered family of fractional programs. The main contributions include: (i) worst-case complexity results for the MMRP and MMRC problems; (ii) mixed 0-1 formulations for the MMRST and MMRC problems; (iii) a global optimization approach for the MMRST problem that extends an existing method for the special case of the sum of two ratios; (iv) new polynomially computable bounds on the optimal objective value of the considered class of FCOPs, as well as the feasible region reduction techniques based on these bounds; (v) an efficient heuristic approach; and, (vi) a generic global optimization approach for the considered class of FCOPs. Finally, extensive computational experiments are carried out to benchmark performance of the suggested solution techniques. The results confirm that the suggested global optimization algorithms generally outperform the conventional mixed 0{1 programming technique on larger problem instances. The developed heuristic approach shows the best run time, and delivers near-optimal solutions in most cases.

In this thesis, we develop an evolutionary algorithm for approximating the Pareto frontier of multi-objective continuous and combinatorial optimization problems. The algorithm tries to evolve the population of solutions towards the Pareto frontier and distribute it over the frontier in order to maintain a well-spread representation. The fitness score of each solution is computed with a Tchebycheff distance function and non-dominating sorting approach. Each solution chooses its own favorable weights according to the Tchebycheff distance function. Some seed solutions at initial population and a crowding measure also help to achieve satisfactory results. In order to test the performance of our evolutionary algorithm, we use some continuous and combinatorial problems. The continuous test problems taken from the literature have special difficulties that an evolutionary algorithm has to deal with. Experimental results of our algorithm on these problems are provided. One of the combinatorial problems we address is the multi-objective knapsack problem. We carry out experiments on test data for this problem given in the literature. We work on two bi-criteria p-hub location problems and propose an evolutionary algorithm to approximate the Pareto frontiers of these problems. We test the performance of our algorithm on Turkish Postal System (PTT) data set (TPDS), AP (Australian Post) and CAB (US Civil Aeronautics Board) data sets. The main contribution of this thesis is in the field of developing a multi-objective evolutionary algorithm and applying it to a number of multi-objective continuous and combinatorial optimization problems.

In this thesis, we develop two exact algorithms and a heuristic procedure for Multiobjective Combinatorial Optimization Problems (MOCO). Our exact algorithms guarantee to generate all nondominated solutions of any MOCO problem. We test the performance of the algorithms on randomly generated problems including the Multiobjective Knapsack Problem, Multi-objective Shortest Path Problem and Multi-objective Spanning Tree Problem. Although we showed the algorithms work much better than the previous ones, we also proposed a fast heuristic method to approximate efficient frontier since it will also be applicable for real-sized problems. Our heuristic approach is based on fitting a surface to approximate the efficient frontier. We experiment our heuristic on randomly generated problems to test how well the heuristic procedure approximates the efficient frontier. Our results showed the heuristic method works well.

In Traveling Salesman Problem (TSP) with profits, a profit is associated with each city and the requirement to visit all cities is removed. The purpose is to simultaneously minimize cost (excluding as many cities as possible) and maximize profit (including as many cities as possible). Although the reduced single-objective case of the problem has been well-studied, the true biobjective problem has been studied only by a few researchers. In this paper we study the true biobjective problem using the Multiobjective Genetic Algorithm NSGA II and the Lin-Kernighan Heuristic. We propose several improvements for NSGA II in solving the problem. Based on these improvements, we provide computational results of the approximated Pareto-optimal front for a set of practically large size TSP instances. Finally, we provide a framework and its computational results for a post-optimality analysis to guide the decision maker, using the data mining software Clementine.

In this thesis, we address the route selection problem for Unmanned Air Vehicles (UAV) under multiple objectives. We consider a general case for this problem where the UAV has to visit several targets and return to the base. For this case, there are multiple combinatorial problems to be considered. First, the paths to be followed between any pairs of targets should be determined. This part can be considered as a multi-objective shortest path problem. Additionally, we need to determine the order of the targets to be visited. This in turn, is a multi-objective traveling salesperson problem. The overall problem is a combination of these two combinatorial problems. The route selection for UAVs has been studied by several researchers, mainly in the military context. They considered a linear combination of the two objectives / minimizing distance traveled and minimizing radar detection threat / and proposed heuristics for the minimization of the composite single objective problem. We treat these two objectives separately. We develop an evolutionary algorithm to determine the efficient tours. We also consider an exact interactive approach to identify the best paths and tours of a decision maker. We tested the two solution approaches on both small-sized and large-sized problem instances.

In recent years, evolving technology has provided a wide range of resources for Military forces. However, that wideness also caused resource management difficulties in combat missions. Air Tasking Order (ATO) is prepared for various missions of air combats in order to reach objectives by an optimized resource management. Considering combinatorial military aspects with dynamic objectives and various constraints / computer support became inevitable for optimizing the resource management in air force operations. In this thesis, we study different optimization approaches for resource allocation problem of ATO preparation and analyze their performance. We proposed a genetic algorithm formulation with customized encoding, crossover and fitness calculation mechanisms by using the domain knowledge. A linear programming formulation of the problem is developed by integer decision variables and it is used for effectivity and efficiency analysis of genetic algorithm formulations.

No description available.

Electric propulsion has recently become a viable technology for spacecraft, enabling shorter flight times, fewer required planetary gravity assists, larger payloads, and/or smaller launch vehicles. With the maturation of this technology, however, comes a new set of challenges in the area of trajectory design. Because low-thrust trajectory optimization has historically required long run-times and significant user-manipulation, mission design has relied on expert-based knowledge for selecting departure and arrival dates, times of flight, and/or target bodies and gravitational swing-bys. These choices are generally based on known configurations that have worked well in previous analyses or simply on trial and error. At the conceptual design level, however, the ability to explore the full extent of the design space is imperative to locating the best solutions in terms of mass and/or flight times. Beginning in 2005, the Global Trajectory Optimization Competition posed a series of difficult mission design problems, all requiring low-thrust propulsion and visiting one or more asteroids. These problems all had large ranges on the continuous variables - launch date, time of flight, and asteroid stay times (when applicable) - as well as being characterized by millions or even billions of possible asteroid sequences. Even with recent advances in low-thrust trajectory optimization, full enumeration of these problems was not possible within the stringent time limits of the competition. This investigation develops a systematic methodology for determining a broad suite of good solutions to the combinatorial, low-thrust, asteroid tour problem. The target application is for conceptual design, where broad exploration of the design space is critical, with the goal being to rapidly identify a reasonable number of promising solutions for future analysis. The proposed methodology has two steps. The first step applies a three-level heuristic sequence developed from the physics of the problem, which allows for efficient pruning of the design space. The second phase applies a global optimization scheme to locate a broad suite of good solutions to the reduced problem. The global optimization scheme developed combines a novel branch-and-bound algorithm with a genetic algorithm and an industry-standard low-thrust trajectory optimization program to solve for the following design variables: asteroid sequence, launch date, times of flight, and asteroid stay times. The methodology is developed based on a small sample problem, which is enumerated and solved so that all possible discretized solutions are known. The methodology is then validated by applying it to a larger intermediate sample problem, which also has a known solution. Next, the methodology is applied to several larger combinatorial asteroid rendezvous problems, using previously identified good solutions as validation benchmarks. These problems include the 2nd and 3rd Global Trajectory Optimization Competition problems. The methodology is shown to be capable of achieving a reduction in the number of asteroid sequences of 6-7 orders of magnitude, in terms of the number of sequences that require low-thrust optimization as compared to the number of sequences in the original problem. More than 70% of the previously known good solutions are identified, along with several new solutions that were not previously reported by any of the competitors. Overall, the methodology developed in this investigation provides an organized search technique for the low-thrust mission design of asteroid rendezvous problems.

This thesis makes a number of contributions to the theory of approximation algorithm design for multi-agent systems. In particular, we focus on two research directions. The first direction is to generalize the classical framework of combinatorial optimization to the submodular setting, where we assume that each agent has a submodular cost function. We show hardness results from both the information-theoretic and computational aspects for several fundamental optimization problems in the submodular setting, and provide matching approximation algorithms for most of them. The second direction is to introduce game-theoretic issues to approximation algorithm design. Towards this direction, we study the application of approximation algorithms in the theory of truthful mechanism design. We study both the standard objectives of revenue and social welfare, by designing efficient algorithms that satisfy the requirement of truthfulness and guarantee approximate optimality.