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Academic Commons Search Resultsen-usThe Theory of Systemic Risk
http://academiccommons.columbia.edu/catalog/ac:178176
Chen, Chenhttp://dx.doi.org/10.7916/D8W37TWCTue, 30 Sep 2014 00:00:00 +0000Systemic risk is an issue of great concern in modern financial markets as well as, more broadly, in the management of complex business and engineering systems. It refers to the risk of collapse of an entire complex system, as a result of the actions taken by the individual component entities or agents that comprise the system. We investigate the topic of systemic risk from the perspectives of measurement, structural sources, and risk factors. In particular, we propose an axiomatic framework for the measurement and management of systemic risk based on the simultaneous analysis of outcomes across agents in the system and over scenarios of nature. Our framework defines a broad class of systemic risk measures that accommodate a rich set of regulatory preferences. This general class of systemic risk measures captures many specific measures of systemic risk that have recently been proposed as special cases, and highlights their implicit assumptions. Moreover, the systemic risk measures that satisfy our conditions yield decentralized decompositions, i.e., the systemic risk can be decomposed into risk due to individual agents. Furthermore, one can associate a shadow price for systemic risk to each agent that correctly accounts for the externalities of the agent's individual decision-making on the entire system. Also, we provide a structural model for a financial network consisting of a set of firms holding common assets. In the model, endogenous asset prices are captured by the marketing clearing condition when the economy is in equilibrium. The key ingredients in the financial market that are captured in this model include the general portfolio choice flexibility of firms given posted asset prices and economic states, and the mark-to-market wealth of firms. The price sensitivity can be analyzed, where we characterize the key features of financial holding networks that minimize systemic risk, as a function of overall leverage. Finally, we propose a framework to estimate risk measures based on risk factors. By introducing a form of factor-separable risk measures, the acceptance set of the original risk measure connects to the acceptance sets of the factor-separable risk measures. We demonstrate that the tight bounds for factor-separable coherent risk measures can be explicitly constructed.Operations researchcc3136Industrial Engineering and Operations Research, BusinessDissertationsStochastic Approximation Algorithms in the Estimation of Quasi-Stationary Distribution of Finite and General State Space Markov Chains
http://academiccommons.columbia.edu/catalog/ac:177124
Zheng, Shuhenghttp://dx.doi.org/10.7916/D89C6VM9Tue, 12 Aug 2014 00:00:00 +0000This thesis studies stochastic approximation algorithms for estimating the quasi-stationary distribution of Markov chains. Existing numerical linear algebra methods and probabilistic methods might be computationally demanding and intractable in large state spaces. We take our motivation from a heuristic described in the physics literature and use the stochastic approximation framework to analyze and extend it. The thesis begins by looking at the finite dimensional setting. The finite dimensional quasi-stationary estimation algorithm was proposed in the Physics literature by [#latestoliveira, #oliveiradickman1, #dickman], however no proof was given there and it was not recognized as a stochastic approximation algorithm. This and related schemes were analyzed in the context of urn problems and the consistency of the estimator is shown there [#aldous1988two, #pemantle, #athreya]. The rate of convergence is studied by [#athreya] in special cases only. The first chapter provides a different proof of the algorithm's consistency and establishes a rate of convergence in more generality than [#athreya]. It is discovered that the rate of convergence is only fast when a certain restrictive eigenvalue condition is satisfied. Using the tool of iterate averaging, the algorithm can be modified and we can eliminate the eigenvalue condition. The thesis then moves onto the general state space discrete-time Markov chain setting. In this setting, the stochastic approximation framework does not have a strong theory in the current literature, so several of the convergence results have to be adapted because the iterates of our algorithm are measure-valued The chapter formulates the quasi-stationary estimation algorithm in this setting. Then, we extend the ODE method of [#kushner2003stochastic] and proves the consistency of algorithm. Through the proof, several non-restrictive conditions required for convergence of the algorithm are discovered. Finally, the thesis tests the algorithm by running some numerical experiments. The examples are designed to test the algorithm in various edge cases. The algorithm is also empirically compared against the Fleming-Viot method.Operations researchIndustrial Engineering and Operations ResearchDissertationsEssays in Financial Engineering
http://academiccommons.columbia.edu/catalog/ac:177072
Ahn, Andrewhttp://dx.doi.org/10.7916/D80K26R0Sat, 19 Jul 2014 00:00:00 +0000This thesis consists of three essays in financial engineering. In particular we study problems in option pricing, stochastic control and risk management. In the first essay, we develop an accurate and efficient pricing approach for options on leveraged ETFs (LETFs). Our approach allows us to price these options quickly and in a manner that is consistent with the underlying ETF price dynamics. The numerical results also demonstrate that LETF option prices have model-dependency particularly in high-volatility environments. In the second essay, we extend a linear programming (LP) technique for approximately solving high-dimensional control problems in a diffusion setting. The original LP technique applies to finite horizon problems with an exponentially-distributed horizon, T. We extend the approach to fixed horizon problems. We then apply these techniques to dynamic portfolio optimization problems and evaluate their performance using convex duality methods. The numerical results suggest that the LP approach is a very promising one for tackling high-dimensional control problems. In the final essay, we propose a factor model-based approach for performing scenario analysis in a risk management context. We argue that our approach addresses some important drawbacks to a standard scenario analysis and, in a preliminary numerical investigation with option portfolios, we show that it produces superior results as well.Operations researchaja2133Industrial Engineering and Operations ResearchDissertationsNew Quantitative Approaches to Asset Selection and Portfolio Construction
http://academiccommons.columbia.edu/catalog/ac:175867
Song, Irenehttp://dx.doi.org/10.7916/D83N21JVMon, 07 Jul 2014 00:00:00 +0000Since the publication of Markowitz's landmark paper "Portfolio Selection" in 1952, portfolio construction has evolved into a disciplined and personalized process. In this process, security selection and portfolio optimization constitute key steps for making investment decisions across a collection of assets. The use of quantitative algorithms and models in these steps has become a widely-accepted investment practice by modern investors. This dissertation is devoted to exploring and developing those quantitative algorithms and models. In the first part of the dissertation, we present two efficiency-based approaches to security selection: (i) a quantitative stock selection strategy based on operational efficiency and (ii) a quantitative currency selection strategy based on macroeconomic efficiency. In developing the efficiency-based stock selection strategy, we exploit a potential positive link between firm's operational efficiency and its stock performance. By means of data envelopment analysis (DEA), a non-parametric approach to productive efficiency analysis, we quantify firm's operational efficiency into a single score representing a consolidated measure of financial ratios. The financial ratios integrated into an efficiency score are selected on the basis of their predictive power for the firm's future operating performance using the LASSO (least absolute shrinkage and selection operator)-based variable selection method. The computed efficiency scores are directly used for identifying stocks worthy of investment. The basic idea behind the proposed stock selection strategy is that as efficient firms are presumed to be more profitable than inefficient firms, higher returns are expected from their stocks. This idea is tested in a contextual and empirical setting provided by the U.S. Information Technology (IT) sector. Our empirical findings confirm that there is a strong positive relationship between firm's operational efficiency and its stock performance, and further establish that firm's operational efficiency has significant explanatory power in describing the cross-sectional variations of stock returns. We moreover offer an economic argument that posits operational efficiency as a systematic risk factor and the most likely source of excess returns of investing in efficient firms. The efficiency-based currency selection strategy is developed in a similar way; i.e. currencies are selected based on a certain efficiency metric. An exchange rate has long been regarded as a reliable barometer of the state of the economy and the measure of international competitiveness of countries. While strong and appreciating currencies correspond to productive and efficient economies, weak and depreciating currencies correspond to slowing down and less efficient economies. This study hence develops a currency selection strategy that utilizes macroeconomic efficiency of countries measured based on a widely-accepted relationship between exchange rates and macroeconomic variables. For quantifying macroeconomic efficiency of countries, we first establish a multilateral framework using effective exchange rates and trade-weighted macroeconomic variables. This framework is used for transforming the three representative bilateral structural exchange rate models: the flexible price monetary model, the sticky price monetary model, and the sticky price asset model, into their multilateral counterparts. We then translate these multilateral models into DEA models, which yield an efficiency score representing an aggregate measure of macroeconomic variables. Consistent with the stock selection strategy, the resulting efficiency scores are used for identifying currencies worthy of investment. We evaluate our currency selection strategy against appropriate market and strategic benchmarks using historical data. Our empirical results confirm that currencies of efficient countries have stronger performance than those of inefficient countries, and further suggest that compared to the exchange rate models based on standard regression analysis, our models based on DEA improve on the predictability of the future performance of currencies. In the first part of the dissertation, we also develop a data-driven variable selection method for DEA based on the group LASSO. This method extends the LASSO-based variable selection method used for specifying a DEA model for estimating firm's operational efficiency. In our proposed method, we derive a special constrained version of the group LASSO with the loss function suited for variable selection in DEA models and solve it by a new tailored algorithm based on the alternating direction method of multipliers (ADMM). We conduct a thorough evaluation of the proposed method against two widely-used variable selection methods: the efficiency contribution measure (ECM) method and the regression-based (RB) test, in the DEA literature using Monte Carlo simulations. The simulation results show that our method provides more favorable performance compared with its benchmarks. In the second part of the dissertation, we propose a generalized risk budgeting (GRB) approach to portfolio construction. In a GRB portfolio, assets are grouped into possibly overlapping subsets, and each subset is allocated a risk budget that has been pre-specified by the investor. Minimum variance, risk parity and risk budgeting portfolios are all special instances of a GRB portfolio. The GRB portfolio optimization problem is to find a GRB portfolio with an optimal risk-return profile where risk is measured using any positively homogeneous risk measure. When the subsets form a partition, the assets all have identical returns and we restrict ourselves to long-only portfolios, then the GRB problem can in fact be solved as a convex optimization problem. In general, however, the GRB problem is a constrained non-convex problem, for which we propose two solution approaches. The first approach uses a semidefinite programming (SDP) relaxation to obtain an (upper) bound on the optimal objective function value. In the second approach we develop a numerical algorithm that integrates augmented Lagrangian and Markov chain Monte Carlo (MCMC) methods in order to find a point in the vicinity of a very good local optimum. This point is then supplied to a standard non-linear optimization routine with the goal of finding this local optimum. It should be emphasized that the merit of this second approach is in its generic nature: in particular, it provides a starting-point strategy for any non-linear optimization algorithms.Operations researchIndustrial Engineering and Operations ResearchDissertationsOn the Kidney Exchange Problem and Online Minimum Energy Scheduling
http://academiccommons.columbia.edu/catalog/ac:175610
Herrera Humphries, Tuliahttp://dx.doi.org/10.7916/D8125QSXMon, 07 Jul 2014 00:00:00 +0000The allocation and management of scarce resources are of central importance in the design of policies to improve social well-being. This dissertation consists of three essays; the first two deals with the problem of allocating kidneys and the third one on power management in computing devices. Kidney exchange programs are an attractive alternative for patients who need a kidney transplant and who have a willing, but medically incompatible, donor. A registry that keeps track of such patient-donor pairs can nd matches through exchanges amongst such pairs. This results in a quicker transplant for the patients involved, and equally importantly, keeps such patients from the long wait list of patients without an intended donor. As of March 2014, there were at least 99,000 candidates waiting for a kidney transplant in the U.S. However, in 2013 only 16,893 transplants were conducted. This imbalance between supply and demand among other factors, has driven the development of multiple kidney exchange programs in the U.S. and the subsequent development of matching mechanisms to run the programs. In the first essay we consider a matching problem arising in kidney exchanges between hospitals. Focusing on the case of two hospitals, we construct a strategy-proof matching mechanism that is guaranteed to return a matching that is at least 3/4 the size of a maximum cardinality matching. It is known that no better performance is possible if one focuses on mechanisms that return a maximal matching, and so our mechanism is best possible within this natural class of mechanisms. For path-cycle graphs we construct a mechanism that returns a matching that is at least 4/5 the size of max-cardinality matching. This mechanism does not necessarily return a maximal matching. Finally, we construct a mechanism that is universally truthful on path-cycle graphs and whose performance is within 2/3 of optimal. Again, it is known that no better ratio is possible. In most of the existing literature, mechanisms are typically evaluated by their overall performance on a large exchange pool, based on which conclusions and recommendations are drawn. In our second essay, we consider a dynamic framework to evaluate extensively used kidney exchange mechanisms. We conduct a simulation-based study of a dynamically evolving exchange pool during 9 years. Our results suggest that some of the features that are critical in a mechanism in the static setting have only a minor impact in its longrun performance when viewed in the dynamic setting. More importantly, features that are generally underestimated in the static setting turn to be relevant when we look at dynamically evolving exchange pool. For example, the pairs' arrival rates. In particular we provide insights into the eect on the waiting times and the probability to receive an oer of controllable features such as the frequency at which matching are run, the structures through which pairs could be matched (cycles or chains) as well as inherent features such as the pairs ABO-PRA characteristics, the availability of altruistic donors, and wether or not compatible pairs join the exchange etc. We evaluate the odds to receive an oer and the expected time to receive an oer for each ABO-PRA type of pairs in the model. Power management in computing devices aims to minimize energy consumption to perform tasks, meanwhile keeping acceptable performance levels. A widely used power management strategy for devices, is to transit the devices and/or components to lower power consumption states during inactivity periods. Transitions between power states consume energy, thus, depending on such costs, it may be advantageous to stay in high power state during some inactivity periods. In our third essay we consider the problem of minimizing the total energy consumed by a 2-power state device, to process jobs that are sent over time by a constrained adversary. Jobs can be preempted, but deadlines need to be met. In this problem, an algorithm must decide when to schedule the jobs, as well as a sequence of power states, and the discrete time thresholds at which these states will be reached. We provide an online algorithm to minimize the energy consumption when the cost of a transition to the low power state is small enough. In this case, the problem of minimizing the energy consumption is equivalent to minimizing the total number of inactivity periods. We also provide an algorithm to minimize the energy consumption when it may be advantageous to stay in high power state during some inactivity periods. In both cases we provide upper bounds on the competitive ratio of our algorithms, and lower bounds on the competitive ratio of all online algorithms.Operations researchIndustrial Engineering and Operations ResearchDissertationsData-driven Decisions in Service Systems
http://academiccommons.columbia.edu/catalog/ac:175604
Kim, Song-Heehttp://dx.doi.org/10.7916/D8D798KHMon, 07 Jul 2014 00:00:00 +0000This thesis makes contributions to help provide data-driven (or evidence-based) decision support to service systems, especially hospitals. Three selected topics are presented. First, we discuss how Little's Law, which relates average limits and expected values of stationary distributions, can be applied to service systems data that are collected over a finite time interval. To make inferences based on the indirect estimator of average waiting times, we propose methods for estimating confidence intervals and for adjusting estimates to reduce bias. We show our new methods are effective using simulations and data from a US bank call center. Second, we address important issues that need to be taken into account when testing whether real arrival data can be modeled by nonhomogeneous Poisson processes (NHPPs). We apply our method to data from a US bank call center and a hospital emergency department and demonstrate that their arrivals come from NHPPs. Lastly, we discuss an approach to standardize the Intensive Care Unit admission process, which currently lacks a well-defined criteria. Using data from nearly 200,000 hospitalizations, we discuss how we can quantify the impact of Intensive Care Unit admission on individual patient's clinical outcomes. We then use this quantified impact and a stylized model to discuss optimal admission policies. We use simulation to compare the performance of our proposed optimal policies to the current admission policy, and show that the gain can be significant.Operations researchsk3116Industrial Engineering and Operations ResearchDissertationsGraph Structure and Coloring
http://academiccommons.columbia.edu/catalog/ac:175631
Plumettaz, Matthieuhttp://dx.doi.org/10.7916/D87M0637Mon, 07 Jul 2014 00:00:00 +0000We denote by G=(V,E) a graph with vertex set V and edge set E. A graph G is claw-free if no vertex of G has three pairwise nonadjacent neighbours. Claw-free graphs are a natural generalization of line graphs. This thesis answers several questions about claw-free graphs and line graphs. In 1988, Chvatal and Sbihi proved a decomposition theorem for claw-free perfect graphs. They showed that claw-free perfect graphs either have a clique-cutset or come from two basic classes of graphs called elementary and peculiar graphs. In 1999, Maffray and Reed successfully described how elementary graphs can be built using line graphs of bipartite graphs and local augmentation. However gluing two claw-free perfect graphs on a clique does not necessarily produce claw-free graphs. The first result of this thesis is a complete structural description of claw-free perfect graphs. We also give a construction for all perfect circular interval graphs. This is joint work with Chudnovsky. Erdos and Lovasz conjectured in 1968 that for every graph G and all integers s,t≥ 2 such that s+t-1=χ(G) > ω(G), there exists a partition (S,T) of the vertex set of G such that ω(G|S)≥ s and χ(G|T)≥ t. This conjecture is known in the graph theory community as the Erdos-Lovasz Tihany Conjecture. For general graphs, the only settled cases of the conjecture are when s and t are small. Recently, the conjecture was proved for a few special classes of graphs: graphs with stability number 2, line graphs and quasi-line graphs. The second part of this thesis considers the conjecture for claw-free graphs and presents some progresses on it. This is joint work with Chudnovsky and Fradkin. Reed's ω, ∆, χ conjecture proposes that every graph satisfies χ≤ ⎡½ (Δ+1+ω)⎤ ; it is known to hold for all claw-free graphs. The third part of this thesis considers a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algorithms for constructing colorings that achieve our bounds: The complexity are O(n²) for line graphs and O(n³m²) for quasi-line graphs. For line graphs, this is faster than the best known algorithm for constructing a coloring that achieves the bound of Reed's original conjecture. This is joint work with Chudnovsky, King and Seymour.Operations researchmp2761Industrial Engineering and Operations ResearchDissertationsNetwork Resource Allocation Under Fairness Constraints
http://academiccommons.columbia.edu/catalog/ac:176038
Chandramouli, Shyam Sundarhttp://dx.doi.org/10.7916/D8S46Q3VMon, 07 Jul 2014 00:00:00 +0000This work considers the basic problem of allocating resources among a group of agents in a network, when the agents are equipped with single-peaked preferences over their assignments. This generalizes the classical claims problem, which concerns the division of an estate's liquidation value when the total claim on it exceeds this value. The claims problem also models the problem of rationing a single commodity, or the problem of dividing the cost of a public project among the people it serves, or the problem of apportioning taxes. A key consideration in this classical literature is equity: the good (or the ``bad,'' in the case of apportioning taxes or costs) should be distributed as fairly as possible. The main contribution of this dissertation is a comprehensive treatment of a generalization of this classical rationing problem to a network setting. Bochet et al. recently introduced a generalization of the classical rationing problem to the network setting. For this problem they designed an allocation mechanism---the egalitarian mechanism---that is Pareto optimal, envy free and strategyproof. In chapter 2, it is shown that the egalitarian mechanism is in fact group strategyproof, implying that no coalition of agents can collectively misreport their information to obtain a (weakly) better allocation for themselves. Further, a complete characterization of the set of all group strategyproof mechanisms is obtained. The egalitarian mechanism satisfies many attractive properties, but fails consistency, an important property in the literature on rationing problems. It is shown in chapter 3 that no Pareto optimal mechanism can be envy-free and consistent. Chapter 3 is devoted to the edge-fair mechanism that is Pareto optimal, group strategyproof, and consistent. In a related model where the agents are located on the edges of the graph rather than the nodes, the edge-fair rule is shown to be envy-free, group strategyproof, and consistent. Chapter 4 extends the egalitarian mechanism to the problem of finding an optimal exchange in non-bipartite networks. The results vary depending on whether the commodity being exchanged is divisible or indivisible. For the latter case, it is shown that no efficient mechanism can be strategyproof, and that the egalitarian mechanism is Pareto optimal and envy-free. Chapter 5 generalizes recent work on finding stable and balanced allocations in graphs with unit capacities and unit weights to more general networks. The existence of a stable and balanced allocation is established by a transformation to an equivalent unit capacity network.Operations researchIndustrial Engineering and Operations ResearchDissertationsData-driven System Design in Service Operations
http://academiccommons.columbia.edu/catalog/ac:163306
Lu, Yinahttp://hdl.handle.net/10022/AC:P:21080Tue, 16 Jul 2013 00:00:00 +0000The service industry has become an increasingly important component in the world's economy. Simultaneously, the data collected from service systems has grown rapidly in both size and complexity due to the rapid spread of information technology, providing new opportunities and challenges for operations management researchers. This dissertation aims to explore methodologies to extract information from data and provide powerful insights to guide the design of service delivery systems. To do this, we analyze three applications in the retail, healthcare, and IT service industries. In the first application, we conduct an empirical study to analyze how waiting in queue in the context of a retail store affects customers' purchasing behavior. The methodology combines a novel dataset collected via video recognition technology with traditional point-of-sales data. We find that waiting in queue has a nonlinear impact on purchase incidence and that customers appear to focus mostly on the length of the queue, without adjusting enough for the speed at which the line moves. We also find that customers' sensitivity to waiting is heterogeneous and negatively correlated with price sensitivity. These findings have important implications for queueing system design and pricing management under congestion. The second application focuses on disaster planning in healthcare. According to a U.S. government mandate, in a catastrophic event, the New York City metropolitan areas need to be capable of caring for 400 burn-injured patients during a catastrophe, which far exceeds the current burn bed capacity. We develop a new system for prioritizing patients for transfer to burn beds as they become available and demonstrate its superiority over several other triage methods. Based on data from previous burn catastrophes, we study the feasibility of being able to admit the required number of patients to burn beds within the critical three-to-five-day time frame. We find that this is unlikely and that the ability to do so is highly dependent on the type of event and the demographics of the patient population. This work has implications for how disaster plans in other metropolitan areas should be developed. In the third application, we study workers' productivity in a global IT service delivery system, where service requests from possibly globally distributed customers are managed centrally and served by agents. Based on a novel dataset which tracks the detailed time intervals an agent spends on all business related activities, we develop a methodology to study the variation of productivity over time motivated by econometric tools from survival analysis. This approach can be used to identify different mechanisms by which workload affects productivity. The findings provide important insights for the design of the workload allocation policies which account for agents' workload management behavior.Operations researchyl2494BusinessDissertationsThree Essays on Dynamic Pricing and Resource Allocation
http://academiccommons.columbia.edu/catalog/ac:151966
Nur, Cavdarogluhttp://hdl.handle.net/10022/AC:P:14492Thu, 23 Aug 2012 00:00:00 +0000This thesis consists of three essays that focus on different aspects of pricing and resource allocation. We use techniques from supply chain and revenue management, scenario-based robust optimization and game theory to study the behavior of firms in different competitive and non-competitive settings. We develop dynamic programming models that account for pricing and resource allocation decisions of firms in such settings. In Chapter 2, we focus on the resource allocation problem of a service firm, particularly a health-care facility. We formulate a general model that is applicable to various resource allocation problems of a hospital. To this end, we consider a system with multiple customer classes that display different reactions to delays in service. By adopting a dynamic-programming approach, we show that the optimal policy is not simple but exhibits desirable monotonicity properties. Furthermore, we propose a simple threshold heuristic policy that performs well in our experiments. In Chapter 3, we study a dynamic pricing problem for a monopolist seller that operates in a setting where buyers have market power, and where each potential sale takes the form of a bilateral negotiation. We review the dynamic programming formulation of the negotiation problem, and propose a simple and tractable deterministic "fluid" analogue for this problem. The main emphasis of the chapter is in expanding the formulation to the dynamic setting where both the buyer and seller have limited prior information on their counterparty valuation and their negotiation skill. In Chapter 4, we consider the revenue maximization problem of a seller who operates in a market where there are two types of customers; namely the "investors" and "regular-buyers". In a two-period setting, we model and solve the pricing game between the seller and the investors in the latter period, and based on the solution of this game, we analyze the revenue maximization problem of the seller in the former period. Moreover, we study the effects on the total system profits when the seller and the investors cooperate through a contracting mechanism rather than competing with each other; and explore the contracting opportunities that lead to higher profits for both agents.Operations researchIndustrial Engineering and Operations ResearchDissertationsEssays on Inventory Management and Object Allocation
http://academiccommons.columbia.edu/catalog/ac:144769
Lee, Thiam Huihttp://hdl.handle.net/10022/AC:P:12623Fri, 17 Feb 2012 00:00:00 +0000This dissertation consists of three essays. In the first, we establish a framework for proving equivalences between mechanisms that allocate indivisible objects to agents. In the second, we study a newsvendor model where the inventory manager has access to two experts that provide advice, and examine how and when an optimal algorithm can be efficiently computed. In the third, we study classical single-resource capacity allocation problem and investigate the relationship between data availability and performance guarantees. We first study mechanisms that solve the problem of allocating indivisible objects to agents. We consider the class of mechanisms that utilize the Top Trading Cycles (TTC) algorithm (these may differ based on how they prioritize agents), and show a general approach to proving equivalences between mechanisms from this class. This approach is used to show alternative and simpler proofs for two recent equivalence results for mechanisms with linear priority structures. We also use the same approach to show that these equivalence results can be generalized to mechanisms where the agent priority structure is described by a tree. Second, we study the newsvendor model where the manager has recourse to advice, or decision recommendations, from two experts, and where the objective is to minimize worst-case regret from not following the advice of the better of the two agents. We show the model can be reduced to the class machine-learning problem of predicting binary sequences but with an asymmetric cost function, allowing us to obtain an optimal algorithm by modifying a well-known existing one. However, the algorithm we modify, and consequently the optimal algorithm we describe, is not known to be efficiently computable, because it requires evaluations of a function v which is the objective value of recursively defined optimization problems. We analyze v and show that when the two cost parameters of the newsvendor model are small multiples of a common factor, its evaluation is computationally efficient. We also provide a novel and direct asymptotic analysis of v that differs from previous approaches. Our asymptotic analysis gives us insight into the transient structure of v as its parameters scale, enabling us to formulate a heuristic for evaluating v generally. This, in turn, defines a heuristic for the optimal algorithm whose decisions we find in a numerical study to be close to optimal. In our third essay, we study the classical single-resource capacity allocation problem. In particular, we analyze the relationship between data availability (in the form of demand samples) and performance guarantees for solutions derived from that data. This is done by describing a class of solutions called epsilon-backwards accurate policies and determining a suboptimality gap for this class of solutions. The suboptimality gap we find is in terms of epsilon and is also distribution-free. We then relate solutions generated by a Monte Carlo algorithm and epsilon-backwards accurate policies, showing a lower bound on the quantity of data necessary to ensure that the solution generated by the algorithm is epsilon-backwards accurate with a high probability. Combining the two results then allows us to give a lower bound on the data needed to generate an Î±-approximation with a given confidence probability 1-delta. We find that this lower bound is polynomial in the number of fares, M, and 1/Î±.Operations researchthl2102Industrial Engineering and Operations ResearchDissertationsAlgorithms for Sparse and Low-Rank Optimization: Convergence, Complexity and Applications
http://academiccommons.columbia.edu/catalog/ac:137539
Ma, ShiqianMon, 22 Aug 2011 00:00:00 +0000Solving optimization problems with sparse or low-rank optimal solutions has been an important topic since the recent emergence of compressed sensing and its matrix extensions such as the matrix rank minimization and robust principal component analysis problems. Compressed sensing enables one to recover a signal or image with fewer observations than the "length" of the signal or image, and thus provides potential breakthroughs in applications where data acquisition is costly. However, the potential impact of compressed sensing cannot be realized without efficient optimization algorithms that can handle extremely large-scale and dense data from real applications. Although the convex relaxations of these problems can be reformulated as either linear programming, second-order cone programming or semidefinite programming problems, the standard methods for solving these relaxations are not applicable because the problems are usually of huge size and contain dense data. In this dissertation, we give efficient algorithms for solving these "sparse" optimization problems and analyze the convergence and iteration complexity properties of these algorithms. Chapter 2 presents algorithms for solving the linearly constrained matrix rank minimization problem. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast and solved as a semidefinite programming problem, such an approach is computationally expensive when the matrices are large. In Chapter 2, we propose fixed-point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 × 1000 matrices of rank 50 with a relative error of 10-5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms. In Chapter 3, we study the convergence/recoverability properties of the fixed point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving linearly constrained matrix rank minimization problems are reported. Chapters 4 and 5 considers alternating direction type methods for solving composite convex optimization problems. We present in Chapter 4 alternating linearization algorithms that are based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most O(1/ε) iterations to obtain an ε-optimal solution, while our accelerated (i.e., fast) versions require at most O(1/√ε) iterations, with little change in the computational effort required at each iteration. For more general problem, i.e., minimizing the sum of K convex functions, we propose multiple-splitting algorithms for solving them. We propose both basic and accelerated algorithms with O(1/ε) and O(1/√ε) iteration complexity bounds for obtaining an ε-optimal solution. To the best of our knowledge, the complexity results presented in these two chapters are the first ones of this type that have been given for splitting and alternating direction type methods. Numerical results on various applications in sparse and low-rank optimization, including compressed sensing, matrix completion, image deblurring, robust principal component analysis, are reported to demonstrate the efficiency of our methods.Operations researchsm2756Industrial Engineering and Operations ResearchDissertationsMany-Server Queues with Time-Varying Arrivals, Customer Abandonment, and non-Exponential Distributions
http://academiccommons.columbia.edu/catalog/ac:136569
Liu, Yunanhttp://hdl.handle.net/10022/AC:P:10801Tue, 02 Aug 2011 00:00:00 +0000This thesis develops deterministic heavy-traffic fluid approximations for many-server stochastic queueing models. The queueing models, with many homogeneous servers working independently in parallel, are intended to model large-scale service systems such as call centers and health care systems. Such models also have been employed to study communication, computing and manufacturing systems. The heavy-traffic approximations yield relatively simple formulas for quantities describing system performance, such as the expected number of customers waiting in the queue. The new performance approximations are valuable because, in the generality considered, these complex systems are not amenable to exact mathematical analysis. Since the approximate performance measures can be computed quite rapidly, they usefully complement more cumbersome computer simulation. Thus these heavy-traffic approximations can be used to improve capacity planning and operational control. More specifically, the heavy-traffic approximations here are for large-scale service systems, having many servers and a high arrival rate. The main focus is on systems that have time-varying arrival rates and staffing functions. The system is considered under the assumption that there are alternating periods of overloading and underloading, which commonly occurs when service providers are unable to adjust the staffing frequently enough to economically meet demand at all times. The models also allow the realistic features of customer abandonment and non-exponential probability distributions for the service times and the times customers are willing to wait before abandoning. These features make the overall stochastic model non-Markovian and thus thus very difficult to analyze directly. This thesis provides effective algorithms to compute approximate performance descriptions for these complex systems. These algorithms are based on ordinary differential equations and fixed point equations associated with contraction operators. Simulation experiments are conducted to verify that the approximations are effective. This thesis consists of four pieces of work, each presented in one chapter. The first chapter (Chapter 2) develops the basic fluid approximation for a non-Markovian many-server queue with time-varying arrival rate and staffing. The second chapter (Chapter 3) extends the fluid approximation to systems with complex network structure and Markovian routing to other queues of customers after completing service from each queue. The extension to open networks of queues has important applications. For one example, in hospitals, patients usually move among different units such as emergency rooms, operating rooms, and intensive care units. For another example, in manufacturing systems, individual products visit different work stations one or more times. The open network fluid model has multiple queues each of which has a time-varying arrival rate and staffing function. The third chapter (Chapter 4) studies the large-time asymptotic dynamics of a single fluid queue. When the model parameters are constant, convergence to the steady state as time evolves is established. When the arrival rates are periodic functions, such as in service systems with daily or seasonal cycles, the existence of a periodic steady state and the convergence to that periodic steady state as time evolves are established. Conditions are provided under which this convergence is exponentially fast. The fourth chapter (Chapter 5) uses a fluid approximation to gain insight into nearly periodic behavior seen in overloaded stationary many-server queues with customer abandonment and nearly deterministic service times. Deterministic service times are of applied interest because computer-generated service times, such as automated messages, may well be deterministic, and computer-generated service is becoming more prevalent. With deterministic service times, if all the servers remain busy for a long interval of time, then the times customers enter service assumes a periodic behavior throughout that interval. In overloaded large-scale systems, these intervals tend to persist for a long time, producing nearly periodic behavior. To gain insight, a heavy-traffic limit theorem is established showing that the fluid model arises as the many-server heavy-traffic limit of a sequence of appropriately scaled queueing models, all having these deterministic service times. Simulation experiments confirm that the transient behavior of the limiting fluid model provides a useful description of the transient performance of the queueing system. However, unlike the asymptotic loss of memory results in the previous chapter for service times with densities, the stationary fluid model with deterministic service times does not approach steady state as time evolves independent of the initial conditions. Since the queueing model with deterministic service times approaches a proper steady state as time evolves, this model with deterministic service times provides an example where the limit interchange (limiting steady state as time evolves and heavy traffic as scale increases) is not valid.Operations researchyl2342Industrial Engineering and Operations ResearchDissertationsEssays in Consumer Choice Driven Assortment Planning
http://academiccommons.columbia.edu/catalog/ac:131420
Saure, Denis R.http://hdl.handle.net/10022/AC:P:10232Thu, 28 Apr 2011 00:00:00 +0000Product assortment selection is among the most critical decisions facing retailers: product variety and relevance is a fundamental driver of consumers' purchase decisions and ultimately of a retailer's profitability. In the last couple of decades an increasing number of firms have gained the ability to frequently revisit their assortment decisions during a selling season. In addition, the development and consolidation of online retailing have introduced new levels of operational flexibility, and cheap access to detailed transactional information. These new operational features present the retailer with both benefits and challenges. The ability to revisit the assortment decision frequently over time allows the retailer to introduce and test new products during the selling season, and adjust on the fly to unexpected changes in consumer preferences, and use customer profile information to customize (in real time) online shopping experience. Our main objective in this thesis is to formulate and solve assortment optimization models addressing the challenges present in modern retail environments. We begin by analyzing the role of the assortment decision in balancing information collection and revenue maximization, when consumer preferences are initially unknown. By considering utility maximizing consumers, we establish fundamental limits on the performance of any assortment policy whose aim is to maximize long run revenues. In addition, we propose adaptive assortment policies that attain such performance limits. Our results highlight salient features of this dynamic assortment problem that distinguish it from similar problems of sequential decision making under model uncertainty. Next, we extend the analysis to the case when additional consumer profile information is available; our primary motivation here is the emerging area of online advertisement. As in the previous setup, we identify fundamental performance limits and propose adaptive policies attaining these limits. Finally, we focus on the effects of competition and consumers' access to information on assortment strategies. In particular, we study competition among retailers when they have access to common products, i.e., products that are available to the competition, and where consumers have full information about the retailers' offerings. Our results shed light on equilibrium properties in such settings and the effect common products have on this behavior.Operations researchdrs2114BusinessDissertations