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Academic Commons Search Resultsen-usExcluding Induced Paths: Graph Structure and Coloring
https://academiccommons.columbia.edu/catalog/ac:186521
Maceli, Peter Lawson10.7916/D8WW7GK4Mon, 12 Jun 2017 17:35:55 +0000An induced subgraph of a given graph is any graph which can be obtained by successively deleting vertices, possible none. In this thesis, we present several new structural and algorithmic results on a number of different classes of graphs which are closed under taking induced subgraphs.
The first result of this thesis is related to a conjecture of Hayward and Nastos on the structure of graphs with no induced four-edge path or four-edge antipath. They conjectured that every such graph which is both prime and perfect is either a split graph or contains a certain useful arrangement of simplicial and antisimplicial vertices. We give a counterexample to their conjecture, and prove a slightly weaker version. This is joint work with Maria Chudnovsky, and first appeared in Journal of Graph Theory.
The second result of this thesis is a decomposition theorem for the class of all graphs with no induced four-edge path or four-edge antipath. We show that every such graph can be obtained from pentagons and split graphs by repeated application of complementation, substitution, and split graph unification. Split graph unification is a new graph operation we introduced, which is a generalization of substitution and involves "gluing" two graphs along a common induced split graph. This is a combination of joint work with Maria Chudnovsky and Irena Penev, together with later work of Louis Esperet, Laetitia Lemoine and Frederic Maffray, and first appeared in.
The third result of this thesis is related to the problem of determining the complexity of coloring graphs which do not contain some fixed induced subgraph. We show that three-coloring graphs with no induced six-edge path or triangle can be done in polynomial-time. This is joint work with Maria Chudnovsky and Mingxian Zhong, and first appeared in. Working together with Flavia Bonomo, Oliver Schaudt, and Maya Stein, we have since simplified and extended this result.Operations research, Mathematics, Computer scienceplm2109Industrial Engineering and Operations ResearchThesesConvex Optimization Algorithms and Recovery Theories for Sparse Models in Machine Learning
https://academiccommons.columbia.edu/catalog/ac:175385
Huang, Bo10.7916/D8VM49DMThu, 08 Jun 2017 20:14:25 +0000Sparse modeling is a rapidly developing topic that arises frequently in areas such as machine learning, data analysis and signal processing. One important application of sparse modeling is the recovery of a high-dimensional object from relatively low number of noisy observations, which is the main focuses of the Compressed Sensing, Matrix Completion(MC) and Robust Principal Component Analysis (RPCA) . However, the power of sparse models is hampered by the unprecedented size of the data that has become more and more available in practice. Therefore, it has become increasingly important to better harnessing the convex optimization techniques to take advantage of any underlying "sparsity" structure in problems of extremely large size.
This thesis focuses on two main aspects of sparse modeling. From the modeling perspective, it extends convex programming formulations for matrix completion and robust principal component analysis problems to the case of tensors, and derives theoretical guarantees for exact tensor recovery under a framework of strongly convex programming. On the optimization side, an efficient first-order algorithm with the optimal convergence rate has been proposed and studied for a wide range of problems of linearly constraint sparse modeling problems.Mathematics, Statistics, Operations researchIndustrial Engineering and Operations ResearchThesesStudies in Stochastic Networks: Efficient Monte-Carlo Methods, Modeling and Asymptotic Analysis
https://academiccommons.columbia.edu/catalog/ac:177127
Dong, Jing10.7916/D8X63K4FThu, 08 Jun 2017 20:11:58 +0000This dissertation contains two parts. The first part develops a series of bias reduction techniques for: point processes on stable unbounded regions, steady-state distribution of infinite server queues, steady-state distribution of multi-server loss queues and loss networks and sample path of stochastic differential equations. These techniques can be applied for efficient performance evaluation and optimization of the corresponding stochastic models. We perform detailed running time analysis under heavy traffic of the perfect sampling algorithms for infinite server queues and multi-server loss queues and prove that the algorithms achieve nearly optimal order of complexity. The second part aims to model and analyze the load-dependent slowdown effect in service systems. One important phenomenon we observe in such systems is bi-stability, where the system alternates randomly between two performance regions. We conduct heavy traffic asymptotic analysis of system dynamics and provide operational solutions to avoid the bad performance region.Operations research, Mathematicsjd2736Industrial Engineering and Operations ResearchThesesApproximate dynamic programming for large scale systems
https://academiccommons.columbia.edu/catalog/ac:169790
Desai, Vijay V.10.7916/D82V2PH1Thu, 08 Jun 2017 16:12:40 +0000Sequential decision making under uncertainty is at the heart of a wide variety of practical problems. These problems can be cast as dynamic programs and the optimal value function can be computed by solving Bellman's equation. However, this approach is limited in its applicability. As the number of state variables increases, the state space size grows exponentially, a phenomenon known as the curse of dimensionality, rendering the standard dynamic programming approach impractical. An effective way of addressing curse of dimensionality is through parameterized value function approximation. Such an approximation is determined by relatively small number of parameters and serves as an estimate of the optimal value function. But in order for this approach to be effective, we need Approximate Dynamic Programming (ADP) algorithms that can deliver `good' approximation to the optimal value function and such an approximation can then be used to derive policies for effective decision-making. From a practical standpoint, in order to assess the effectiveness of such an approximation, there is also a need for methods that give a sense for the suboptimality of a policy. This thesis is an attempt to address both these issues. First, we introduce a new ADP algorithm based on linear programming, to compute value function approximations. LP approaches to approximate DP have typically relied on a natural `projection' of a well studied linear program for exact dynamic programming. Such programs restrict attention to approximations that are lower bounds to the optimal cost-to-go function. Our program -- the `smoothed approximate linear program' -- is distinct from such approaches and relaxes the restriction to lower bounding approximations in an appropriate fashion while remaining computationally tractable. The resulting program enjoys strong approximation guarantees and is shown to perform well in numerical experiments with the game of Tetris and queueing network control problem. Next, we consider optimal stopping problems with applications to pricing of high-dimensional American options. We introduce the pathwise optimization (PO) method: a new convex optimization procedure to produce upper and lower bounds on the optimal value (the `price') of high-dimensional optimal stopping problems. The PO method builds on a dual characterization of optimal stopping problems as optimization problems over the space of martingales, which we dub the martingale duality approach. We demonstrate via numerical experiments that the PO method produces upper bounds and lower bounds (via suboptimal exercise policies) of a quality comparable with state-of-the-art approaches. Further, we develop an approximation theory relevant to martingale duality approaches in general and the PO method in particular. Finally, we consider a broad class of MDPs and introduce a new tractable method for computing bounds by consider information relaxation and introducing penalty. The method delivers tight bounds by identifying the best penalty function among a parameterized class of penalty functions. We implement our method on a high-dimensional financial application, namely, optimal execution and demonstrate the practical value of the method vis-a-vis competing methods available in the literature. In addition, we provide theory to show that bounds generated by our method are provably tighter than some of the other available approaches.Operations research, Mathematicsvvd2101Industrial Engineering and Operations ResearchThesesResource Cost Aware Scheduling Problems
https://academiccommons.columbia.edu/catalog/ac:166566
Carrasco, Rodrigo10.7916/D8Z60WDXThu, 08 Jun 2017 16:12:25 +0000Managing the consumption of non-renewable and/or limited resources has become an important issue in many different settings. In this dissertation we explore the topic of resource cost aware scheduling. Unlike the purely scheduling problems, in the resource cost aware setting we are not only interested in a scheduling performance metric, but also the cost of the resources consumed to achieve a certain performance level. There are several ways in which the cost of non-renewal resources can be added into a scheduling problem. Throughout this dissertation we will focus in the case where the resource consumption cost is added, as part of the objective, to a scheduling performance metric such as weighted completion time and weighted tardiness among others. In our work we make several contributions to the problem of scheduling with non-renewable resources. For the specific setting in which only energy consumption is the important resource, our contributions are the following. We introduce a model that extends the previous energy cost models by allowing more general cost functions that can be job-dependent. We further generalize the problem by allowing arbitrary precedence constraints and release dates. We give approximation algorithms for minimizing an objective that is a combination of a scheduling metric, namely total weighted completion time and total weighted tardiness, and the total energy consumption cost. Our approximation algorithm is based on an interval-and-speed-indexed IP formulation. We solve the linear relaxation of this IP and we use this solution to compute a schedule. We show that these algorithms have small constant approximation ratios. Through experimental analysis we show that the empirical approximation ratios are much better than the theoretical ones and that in fact the solutions are close to optimal. We also show empirically that the algorithm can be used in additional settings not covered by the theoretical results, such as using flow time or an online setting, with good approximation and competitiveness ratios.Industrial engineering, MathematicsIndustrial Engineering and Operations ResearchThesesFinancial Portfolio Risk Management: Model Risk, Robustness and Rebalancing Error
https://academiccommons.columbia.edu/catalog/ac:161415
Xu, Xingbo10.7916/D8SX6MF1Thu, 08 Jun 2017 13:54:56 +0000Risk management has always been in key component of portfolio management. While more and more complicated models are proposed and implemented as research advances, they all inevitably rely on imperfect assumptions and estimates. This dissertation aims to investigate the gap between complicated theoretical modelling and practice. We mainly focus on two directions: model risk and reblancing error. In the first part of the thesis, we develop a framework for quantifying the impact of model error and for measuring and minimizing risk in a way that is robust to model error. This robust approach starts from a baseline model and finds the worst-case error in risk measurement that would be incurred through a deviation from the baseline model, given a precise constraint on the plausibility of the deviation. Using relative entropy to constrain model distance leads to an explicit characterization of worst-case model errors; this characterization lends itself to Monte Carlo simulation, allowing straightforward calculation of bounds on model error with very little computational effort beyond that required to evaluate performance under the baseline nominal model. This approach goes well beyond the effect of errors in parameter estimates to consider errors in the underlying stochastic assumptions of the model and to characterize the greatest vulnerabilities to error in a model. We apply this approach to problems of portfolio risk measurement, credit risk, delta hedging, and counterparty risk measured through credit valuation adjustment. In the second part, we apply this robust approach to a dynamic portfolio control problem. The sources of model error include the evolution of market factors and the influence of these factors on asset returns. We analyze both finite- and infinite-horizon problems in a model in which returns are driven by factors that evolve stochastically. The model incorporates transaction costs and leads to simple and tractable optimal robust controls for multiple assets. We illustrate the performance of the controls on historical data. Robustness does improve performance in out-of-sample tests in which the model is estimated on a rolling window of data and then applied over a subsequent time period. By acknowledging uncertainty in the estimated model, the robust rules lead to less aggressive trading and are less sensitive to sharp moves in underlying prices. In the last part, we analyze the error between a discretely rebalanced portfolio and its continuously rebalanced counterpart in the presence of jumps or mean-reversion in the underlying asset dynamics. With discrete rebalancing, the portfolio's composition is restored to a set of fixed target weights at discrete intervals; with continuous rebalancing, the target weights are maintained at all times. We examine the difference between the two portfolios as the number of discrete rebalancing dates increases. We derive the limiting variance of the relative error between the two portfolios for both the mean-reverting and jump-diffusion cases. For both cases, we derive ``volatility adjustments'' to improve the approximation of the discretely rebalanced portfolio by the continuously rebalanced portfolio, based on on the limiting covariance between the relative rebalancing error and the level of the continuously rebalanced portfolio. These results are based on strong approximation results for jump-diffusion processes.Operations research, Finance, Mathematicsxx2126Industrial Engineering and Operations ResearchThesesTournaments With Forbidden Substructures and the Erdos-Hajnal Conjecture
https://academiccommons.columbia.edu/catalog/ac:160247
Choromanski, Krzysztof10.7916/D84J0N9PThu, 08 Jun 2017 13:53:49 +0000A celebrated Conjecture of Erdos and Hajnal states that for every undirected graph H there exists ɛ(H)>0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n^{ɛ(H)}. In 2001 Alon, Pach and Solymosi proved that the conjecture has an equivalent directed version, where undirected graphs are replaced by tournaments and cliques and stable sets by transitive subtournaments. This dissertation addresses the directed version of the conjecture and some problems in the directed setting that are closely related to it. For a long time the conjecture was known to be true only for very specific small graphs and graphs obtained from them by the so-called substitution procedure proposed by Alon, Pach and Solymosi. All the graphs that are an outcome of this procedure have nontrivial homogeneous sets. Tournaments without nontrivial homogeneous sets are called prime. They play a central role here since if the conjecture is not true then the smallest counterexample is prime. We remark that for a long time the conjecture was known to be true only for some prime graphs of order at most 5. There exist 5-vertex graphs for which the conjecture is still open, however one of the corollaries of the results presented in the thesis states that all tournaments on at most 5 vertices satisfy the conjecture. In the first part of the thesis we will establish the conjecture for new infinite classes of tournaments containing infinitely many prime tournaments. We will first prove the conjecture for so-called constellations. It turns out that almost all tournaments on at most 5 vertices are either constellations or are obtained from constellations by substitutions. The only 5-vertex tournament for which this is not the case is a tournament in which every vertex has outdegree 2. We call this the tournament C_{5}. Another result of this thesis is the proof of the conjecture for this tournament. We also present here the structural characterization of the tournaments satisfying the conjecture in almost linear sense. In the second part of the thesis we focus on the upper bounds on coefficients epsilon(H) for several classes of tournaments. In particular we analyze how they depend on the structure of the tournament. We prove that for almost all h-vertex tournaments ɛ(H) ≤ 4/h(1+o(1)). As a byproduct of the methods we use here, we get upper bounds for ɛ(H) of undirected graphs. We also present upper bounds on ɛ(H) of tournaments with small nontrivial homogeneous sets, in particular prime tournaments. Finally we analyze tournaments with big ɛ(H) and explore some of their structural properties.Mathematicskmc2178Industrial Engineering and Operations ResearchThesesRare Events in Stochastic Systems: Modeling, Simulation Design and Algorithm Analysis
https://academiccommons.columbia.edu/catalog/ac:156733
Shi, Yixi10.7916/D86H4QNSThu, 08 Jun 2017 13:53:34 +0000This dissertation explores a few topics in the study of rare events in stochastic systems, with a particular emphasis on the simulation aspect. This line of research has been receiving a substantial amount of interest in recent years, mainly motivated by scientific and industrial applications in which system performance is frequently measured in terms of events with very small probabilities.The topics mainly break down into the following themes: Algorithm Analysis: Chapters 2, 3, 4 and 5. Simulation Design: Chapters 3, 4 and 5. Modeling: Chapter 5. The titles of the main chapters are detailed as follows: Chapter 2: Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks Chapter 3: Splitting for Heavy-tailed Systems: An Exploration with Two Algorithms Chapter 4: State Dependent Importance Sampling with Cross Entropy for Heavy-tailed Systems Chapter 5: Stochastic Insurance-Reinsurance Networks: Modeling, Analysis and Efficient Monte CarloEngineering, Mathematicsys2347Industrial Engineering and Operations ResearchThesesQuantitative Modeling of Credit Derivatives
https://academiccommons.columbia.edu/catalog/ac:131549
Kan, Yu Hang10.7916/D8MP598QWed, 07 Jun 2017 02:46:06 +0000The recent financial crisis has revealed major shortcomings in the existing approaches for modeling credit derivatives. This dissertation studies various issues related to the modeling of credit derivatives: hedging of portfolio credit derivatives, calibration of dynamic credit models, and modeling of credit default swap portfolios. In the first part, we compare the performance of various hedging strategies for index collateralized debt obligation (CDO) tranches during the recent financial crisis. Our empirical analysis shows evidence for market incompleteness: a large proportion of risk in the CDO tranches appears to be unhedgeable. We also show that, unlike what is commonly assumed, dynamic models do not necessarily perform better than static models, nor do high-dimensional bottom-up models perform better than simpler top-down models. On the other hand, model-free regression-based hedging appears to be surprisingly effective when compared to other hedging strategies. The second part is devoted to computational methods for constructing an arbitrage-free CDO pricing model compatible with observed CDO prices. This method makes use of an inversion formula for computing the aggregate default rate in a portfolio from expected tranche notionals, and a quadratic programming method for recovering expected tranche notionals from CDO spreads. Comparing this approach to other calibration methods, we find that model-dependent quantities such as the forward starting tranche spreads and jump-to-default ratios are quite sensitive to the calibration method used, even within the same model class. The last chapter of this dissertation focuses on statistical modeling of credit default swaps (CDSs). We undertake a systematic study of the univariate and multivariate properties of CDS spreads, using time series of the CDX Investment Grade index constituents from 2005 to 2009. We then propose a heavy-tailed multivariate time series model for CDS spreads that captures these properties. Our model can be used as a framework for measuring and managing the risk of CDS portfolios, and is shown to have better performance than the affine jump-diffusion or random walk models for predicting loss quantiles of various CDS portfolios.Finance, Mathematicsyk2246Industrial Engineering and Operations ResearchThesesFirst Order Methods for Large-Scale Sparse Optimization
https://academiccommons.columbia.edu/catalog/ac:135750
Aybat, Necdet Serhat10.7916/D8V69RJJWed, 07 Jun 2017 02:46:03 +0000In today's digital world, improvements in acquisition and storage technology are allowing us to acquire more accurate and finer application-specific data, whether it be tick-by-tick price data from the stock market or frame-by-frame high resolution images and videos from surveillance systems, remote sensing satellites and biomedical imaging systems. Many important large-scale applications can be modeled as optimization problems with millions of decision variables. Very often, the desired solution is sparse in some form, either because the optimal solution is indeed sparse, or because a sparse solution has some desirable properties. Sparse and low-rank solutions to large scale optimization problems are typically obtained by regularizing the objective function with L1 and nuclear norms, respectively. Practical instances of these problems are very high dimensional (~ million variables) and typically have dense and ill-conditioned data matrices. Therefore, interior point based methods are ill-suited for solving these problems. The large scale of these problems forces one to use the so-called first-order methods that only use gradient information at each iterate. These methods are efficient for problems with a "simple" feasible set such that Euclidean projections onto the set can be computed very efficiently, e.g. the positive orthant, the n-dimensional hypercube, the simplex, and the Euclidean ball. When the feasible set is "simple", the subproblems used to compute the iterates can be solved efficiently. Unfortunately, most applications do not have "simple" feasible sets. A commonly used technique to handle general constraints is to relax them so that the resulting problem has only "simple" constraints, and then to solve a single penalty or Lagrangian problem. However, these methods generally do not guarantee convergence to feasibility. The focus of this thesis is on developing new fast first-order iterative algorithms for computing sparse and low-rank solutions to large-scale optimization problems with very mild restrictions on the feasible set - we allow linear equalities, norm-ball and conic inequalities, and also certain non-smooth convex inequalities to define the constraint set. The proposed algorithms guarantee that the sequence of iterates converges to an optimal feasible solution of the original problem, and each subproblem is an optimization problem with a "simple" feasible set. In addition, for any eps > 0, by relaxing the feasibility requirement of each iteration, the proposed algorithms can compute an eps-optimal and eps-feasible solution within O(log(1/eps)) iterations which requires O(1/eps) basic operations in the worst case. Algorithm parameters do not depend on eps > 0. Thus, these new methods compute iterates arbitrarily close to feasibility and optimality as they continue to run. Moreover, the computational complexity of each basic operation for these new algorithms is the same as that of existing first-order algorithms running on "simple" feasible sets. Our numerical studies showed that only O(log(1/eps)) basic operations, as opposed to O(1/eps) worst case theoretical bound, are needed for obtaining eps-feasible and eps-optimal solutions. We have implemented these new first-order methods for the following problem classes: Basis Pursuit (BP) in compressed sensing, Matrix Rank Minimization, Principal Component Pursuit (PCP) and Stable Principal Component Pursuit (SPCP) in principal component analysis. These problems have applications in signal and image processing, video surveillance, face recognition, latent semantic indexing, and ranking and collaborative filtering. To best of our knowledge, an algorithm for the SPCP problem that has O(1/eps) iteration complexity and has a per iteration complexity equal to that of a singular value decomposition is given for the first time.Operations research, Mathematicsnsa2106Industrial Engineering and Operations ResearchTheses