47 research outputs found

    Algorithms for the continuous nonlinear resource allocation problem---new implementations and numerical studies

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    Patriksson (2008) provided a then up-to-date survey on the continuous,separable, differentiable and convex resource allocation problem with a single resource constraint. Since the publication of that paper the interest in the problem has grown: several new applications have arisen where the problem at hand constitutes a subproblem, and several new algorithms have been developed for its efficient solution. This paper therefore serves three purposes. First, it provides an up-to-date extension of the survey of the literature of the field, complementing the survey in Patriksson (2008) with more then 20 books and articles. Second, it contributes improvements of some of these algorithms, in particular with an improvement of the pegging (that is, variable fixing) process in the relaxation algorithm, and an improved means to evaluate subsolutions. Third, it numerically evaluates several relaxation (primal) and breakpoint (dual) algorithms, incorporating a variety of pegging strategies, as well as a quasi-Newton method. Our conclusion is that our modification of the relaxation algorithm performs the best. At least for problem sizes up to 30 million variables the practical time complexity for the breakpoint and relaxation algorithms is linear

    Auction algorithms for generalized nonlinear network flow problems

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    Thesis (Ph.D.)--Boston UniversityNetwork flow is an area of optimization theory concerned with optimization over networks with a range of applicability in fields such as computer networks, manufacturing, finance, scheduling and routing, telecommunications, and transportation. In both linear and nonlinear networks, a family of primal-dual algorithms based on "approximate" Complementary Slackness (ε-CS) is among the fastest in centralized and distributed environments. These include the auction algorithm for the linear assignment/transportation problems, ε-relaxation and Auction/Sequential Shortest Path (ASSP) for the min-cost flow and max-flow problems. Within this family, the auction algorithm is particularly fast, as it uses "second best" information, as compared to using the more generic ε-relaxation for linear assignment/transportation. Inspired by the success of auction algorithms, we extend them to two important classes of nonlinear network flow problems. We start with the nonlinear Resource Allocation Problem (RAP). This problem consists of optimally assigning N divisible resources to M competing missions/tasks each with its own utility function. This simple yet powerful framework has found applications in diverse fields such as finance, economics, logistics, sensor and wireless networks. RAP is an instance of generalized network (networks with arc gains) flow problem but it has significant special structure analogous to the assignment/transportation problem. We develop a class of auction algorithms for RAP: a finite-time auction algorithm for both synchronous and asynchronous environments followed by a combination of forward and reverse auction with ε-scaling to achieve pseudo polynomial complexity for any non-increasing generalized convex utilities including non-continuous and/ or non-differentiable functions. These techniques are then generalized to handle shipping costs on allocations. Lastly, we demonstrate how these techniques can be used for solving a dynamic RAP where nodes may appear or disappear over time. In later part of the thesis, we consider the convex nonlinear min-cost flow problem. Although E-relaxation and ASSP are among the fastest available techniques here, we illustrate how nonlinear costs, as opposed to linear, introduce a significant bottleneck on the progress that these algorithms make per iteration. We then extend the core idea of the auction algorithm, use of second best to make aggressive steps, to overcome this bottleneck and hence develop a faster version of ε-relaxation. This new algorithm shares the same theoretical complexity as the original but outperforms it in our numerical experiments based on random test problem suites

    Facility location, capacity acquisition and technology selection models for manufacturing strategy planning

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    Ankara : The Institute of Engineering and Science, Bilkent Univ., 1993.Thesis (Ph.D.) -- Bilkent University, 1993.Includes bibliographical references leaves 129-141.The primary aim of this dissertation research is to contribute to the manufacturing strategy planning process. The firm is perceived as a value chain which can be represented by a production-distribution network. Structural decisions regarding the value chain of a firm are the means to implement the firm’s manufacturing strategy. Thus, development of analytical methods to aid the design of production-distribution sytems constitutes the essence of this study. The differentiating features of the manufacturing strategy planning process within the multinational companies are especially taken into account due to the significance of the globalization in product, factor, and capital markets. A review of the state-of-the-art in production-distribution system design reveals that although the evaluation of strategy alternatives received much attention, the existing analytical methods are lacking the capability to produce manufacturing strategy options. Further, it is shown that the facility location, capacity acquisition, and technology selection decisions have been dealt with separately in the literature. Whereas, the interdependencies among these structural decisions are pronounced within the international context, and hence global manufacturing strategy planning requires their simultaneous optimization. Thus, an analytical method is developed for the integration of the facility location and sizing decisions in producing a single commodity. Then, presence of product-dedicated technology alternatives in acquiring the required production capacity at each facility is incorporated. The analytical method is further extended to the multicommodity problem where product- flexible technology is also available as a technology alternative. Not only the arising models facilitate analysis of the trade-offs associated with the scale and scope economies in capacity/technology acquisition on the basis of alternative facility locations, but they also provide valuable insights regarding the presence of some dominance properties in manufacturing strategy design.Verter, VedatPh.D

    Approximation Algorithms for Resource Allocation

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    This thesis is devoted to designing new techniques and algorithms for combinatorial optimization problems arising in various applications of resource allocation. Resource allocation refers to a class of problems where scarce resources must be distributed among competing agents maintaining certain optimization criteria. Examples include scheduling jobs on one/multiple machines maintaining system performance; assigning advertisements to bidders, or items to people maximizing profit/social fairness; allocating servers or channels satisfying networking requirements etc. Altogether they comprise a wide variety of combinatorial optimization problems. However, a majority of these problems are NP-hard in nature and therefore, the goal herein is to develop approximation algorithms that approximate the optimal solution as best as possible in polynomial time. The thesis addresses two main directions. First, we develop several new techniques, predominantly, a new linear programming rounding methodology and a constructive aspect of a well-known probabilistic method, the Lov\'{a}sz Local Lemma (LLL). Second, we employ these techniques to applications of resource allocation obtaining substantial improvements over known results. Our research also spurs new direction of study; we introduce new models for achieving energy efficiency in scheduling and a novel framework for assigning advertisements in cellular networks. Both of these lead to a variety of interesting questions. Our linear programming rounding methodology is a significant generalization of two major rounding approaches in the theory of approximation algorithms, namely the dependent rounding and the iterative relaxation procedure. Our constructive version of LLL leads to first algorithmic results for many combinatorial problems. In addition, it settles a major open question of obtaining a constant factor approximation algorithm for the Santa Claus problem. The Santa Claus problem is a NPNP-hard resource allocation problem that received much attention in the last several years. Through out this thesis, we study a number of applications related to scheduling jobs on unrelated parallel machines, such as provisionally shutting down machines to save energy, selectively dropping outliers to improve system performance, handling machines with hard capacity bounds on the number of jobs they can process etc. Hard capacity constraints arise naturally in many other applications and often render a hitherto simple combinatorial optimization problem difficult. In this thesis, we encounter many such instances of hard capacity constraints, namely in budgeted allocation of advertisements for cellular networks, overlay network design, and in classical problems like vertex cover, set cover and k-median

    Advances in electric power systems : robustness, adaptability, and fairness

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 151-157).The electricity industry has been experiencing fundamental changes over the past decade. Two of the arguably most significant driving forces are the integration of renewable energy resources into the electric power system and the creation of the deregulated electricity markets. Many new challenges arise. In this thesis, we focus on two important ones: How to reliably operate the power system under high penetration of intermittent and uncertain renewable resources and uncertain demand: and how to design an electricity market that considers both efficiency and fairness. We present some new advances in these directions. In the first part of the thesis, we focus on the first issue in the context of the unit commitment (UC) problem, one of the most critical daily operations of an electric power system. Unit commitment in large scale power systems faces new challenges of increasing uncertainty from both generation and load. We propose an adaptive robust model for the security constrained unit commitment problem in the presence of nodal net load uncertainty. We develop a practical solution methodology based on a combination of Benders decomposition type algorithm and outer approximation techniques. We present an extensive numerical study on the real-world large scale power system operated by the ISO New England (ISO-NE). Computational results demonstrate the advantages of the robust model over the traditional reserve adjustment approach in terms of economic efficiency, operational reliability, and robustness to uncertain distributions. In the second part of the thesis, we are concerned with a geometric characterization of the performance of adaptive robust solutions in a multi-stage stochastic optimization problem. We study the notion of finite adaptability in a general setting of multi-stage stochastic and adaptive optimization. We show a significant role that geometric properties of uncertainty sets, such as symmetry, play in determining the power of robust and finitely adaptable solutions. We show that a class of finitely adaptable solutions is a good approximation for both the multi-stage stochastic as well as the adaptive optimization problem. To the best of our knowledge, these are the first approximation results for multi-stage problems in such generality. Moreover, the results and the proof techniques are quite general and extend to include important constraints such as integrality and linear conic constraints. In the third part of the thesis, we focus on how to design an auction and pricing scheme for the day-ahead electricity market that achieves both economic efficiency and fairness. The work is motivated by two outstanding problems in the current practice - the uplift problem and equitable selection problem. The uplift problem is that the electricity payment determined by the electricity price cannot fully recover the production cost (especially the fixed cost) of some committed generators, and therefore the ISOs make side payments to such generators to make up the loss. The equitable selection problem is how to achieve fairness and integrity of the day-ahead auction in choosing from multiple (near) optimal solutions. We offer a new perspective and propose a family of fairness based auction and pricing schemes that resolve these two problems. We present numerical test result using ISO-NE's day-ahead market data. The proposed auction- pricing schemes produce a frontier plot of efficiency versus fairness, which can be used as a vaulable decision tool for the system operation.by Xu Andy Sun.Ph.D

    Indefinite Knapsack Separable Quadratic Programming: Methods and Applications

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    Quadratic programming (QP) has received significant consideration due to an extensive list of applications. Although polynomial time algorithms for the convex case have been developed, the solution of large scale QPs is challenging due to the computer memory and speed limitations. Moreover, if the QP is nonconvex or includes integer variables, the problem is NP-hard. Therefore, no known algorithm can solve such QPs efficiently. Alternatively, row-aggregation and diagonalization techniques have been developed to solve QP by a sub-problem, knapsack separable QP (KSQP), which has a separable objective function and is constrained by a single knapsack linear constraint and box constraints. KSQP can therefore be considered as a fundamental building-block to solve the general QP and is an important class of problems for research. For the convex KSQP, linear time algorithms are available. However, if some quadratic terms or even only one term is negative in KSQP, the problem is known to be NP-hard, i.e. it is notoriously difficult to solve. The main objective of this dissertation is to develop efficient algorithms to solve general KSQP. Thus, the contributions of this dissertation are five-fold. First, this dissertation includes comprehensive literature review for convex and nonconvex KSQP by considering their computational efficiencies and theoretical complexities. Second, a new algorithm with quadratic time worst-case complexity is developed to globally solve the nonconvex KSQP, having open box constraints. Third, the latter global solver is utilized to develop a new bounding algorithm for general KSQP. Fourth, another new algorithm is developed to find a bound for general KSQP in linear time complexity. Fifth, a list of comprehensive applications for convex KSQP is introduced, and direct applications of indefinite KSQP are described and tested with our newly developed methods. Experiments are conducted to compare the performance of the developed algorithms with that of local, global, and commercial solvers such as IBM CPLEX using randomly generated problems in the context of certain applications. The experimental results show that our proposed methods are superior in speed as well as in the quality of solutions
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