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Academic Commons Search Resultsen-usApproximate dynamic programming for large scale systems
http://academiccommons.columbia.edu/catalog/ac:169790
Desai, Vijay V.http://hdl.handle.net/10022/AC:P:20875Fri, 28 Jun 2013 00:00:00 +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 Research, BusinessDissertationsContingent Capital: Valuation and Risk Implications Under Alternative Conversion Mechanisms
http://academiccommons.columbia.edu/catalog/ac:152933
Nouri, Behzadhttp://hdl.handle.net/10022/AC:P:14800Fri, 28 Sep 2012 00:00:00 +0000Several proposals for enhancing the stability of the financial system include requirements that banks hold some form of contingent capital, meaning equity that becomes available to a bank in the event of a crisis or financial distress. Specific proposals vary in their choice of conversion trigger and conversion mechanism, and have inspired extensive scrutiny regarding their effectivity in avoiding costly public rescues and bail-outs and potential adverse effects on market dynamics. While allowing banks to leverage and gain a higher return on their equity capital during the upturns in financial markets, contingent capital provides an automatic mechanism to reduce debt and raise the loss bearing capital cushion during the downturns and market crashes; therefore, making it possible to achieve stability and robustness in the financial sector, without reducing efficiency and competitiveness of the banking system with higher regulatory capital requirements. However, many researchers have raised concerns regarding unintended consequences and implications of such instruments for market dynamics. Death spirals in the stock price near the conversion, possibility of profitable stock or book manipulations by either the investors or the issuer, the marketability and demand for such hybrid instruments, contagion and systemic risks arising from the hedging strategies of the investors and higher risk taking incentives for issuers are among such concerns. Though substantial, many of such issues are addressed through a prudent design of the trigger and conversion mechanism. In the following chapters, we develop multiple models for pricing and analysis of contingent capital under different conversion mechanisms. In Chapter 2 we analyze the case of contingent capital with a capital-ratio trigger and partial and on-going conversion. The capital ratio we use is based on accounting or book value to approximate the regulatory ratios that determine capital requirements for banks. The conversion process is partial and on-going in the sense that each time a bank's capital ratio reaches the minimum threshold, just enough debt is converted to equity to meet the capital requirement, so long as the contingent capital has not been depleted. In Chapter 3 we simplify the design to all-at-once conversion however we perform the analysis through a much richer model which incorporates tail risk in terms of jumps, endogenous optimal default policy and debt rollover. We also investigate the case of bail-in debt, where at default the original shareholders are wiped out and the converted investors take control of the firm. In the case of contingent convertibles the conversion trigger is assumed as a contractual term specified by market value of assets. For bail-in debt the trigger is where the original shareholders optimally default. We study incentives of shareholders to change the capital structure and how CoCo's affect risk incentives. Several researchers have advocated use of a market based trigger which is forward looking, continuously updated and readily available, while some others have raised concerns regarding unintended consequences of a market based trigger. In Chapter 4 we investigate one of these issues, namely the existence and uniqueness of equilibrium when the conversion trigger is based on the stock price.Finance, Operations researchbn2164Industrial Engineering and Operations Research, BusinessDissertationsThree 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 ResearchDissertationsForbidden Substructures in Graphs and Trigraphs, and Related Coloring Problems
http://academiccommons.columbia.edu/catalog/ac:146465
Penev, Irenahttp://hdl.handle.net/10022/AC:P:13082Tue, 01 May 2012 00:00:00 +0000Given a graph G, χ(G) denotes the chromatic number of G, and ω(G) denotes the clique number of G (i.e. the maximum number of pairwise adjacent vertices in G). A graph G is perfect provided that for every induced subgraph H of G, χ(H) = ω(H). This thesis addresses several problems from the theory of perfect graphs and generalizations of perfect graphs. The bull is a five-vertex graph consisting of a triangle and two vertex-disjoint pendant edges; a graph is said to be bull-free provided that no induced subgraph of it is a bull. The first result of this thesis is a structure theorem for bull-free perfect graphs. This is joint work with Chudnovsky, and it first appeared in [12]. The second result of this thesis is a decomposition theorem for bull-free perfect graphs, which we then use to give a polynomial time combinatorial coloring algorithm for bull-free perfect graphs. We remark that de Figueiredo and Maffray [33] previously solved this same problem, however, the algorithm presented in this thesis is faster than the algorithm from [33]. We note that a decomposition theorem that is very similar (but slightly weaker) than the one from this thesis was originally proven in [52], however, the proof in this thesis is significantly different from the one in [52]. The algorithm from this thesis is very similar to the one from [52]. A class G of graphs is said to be χ-bounded provided that there exists a function f such that for all G in G, and all induced subgraphs H of G, we have that χ(H) ≤ f(ω(H)). χ-bounded classes were introduced by Gyarfas [41] as a generalization of the class of perfect graphs (clearly, the class of perfect graphs is χ-bounded by the identity function). Given a graph H, we denote by Forb*(H) the class of all graphs that do not contain any subdivision of H as an induced subgraph. In [57], Scott proved that Forb*(T) is χ-bounded for every tree T, and he conjectured that Forb*(H) is χ-bounded for every graph H. Recently, a group of authors constructed a counterexample to Scott's conjecture [51]. This raises the following question: for which graphs H is Scott's conjecture true? In this thesis, we present the proof of Scott's conjecture for the cases when H is the paw (i.e. a four-vertex graph consisting of a triangle and a pendant edge), the bull, and a necklace (i.e. a graph obtained from a path by choosing a matching such that no edge of the matching is incident with an endpoint of the path, and for each edge of the matching, adding a vertex adjacent to the ends of this edge). This is joint work with Chudnovsky, Scott, and Trotignon, and it originally appeared in [13]. Finally, we consider several operations (namely, "substitution," "gluing along a clique," and "gluing along a bounded number of vertices"), and we show that the closure of a χ-bounded class under any one of them, as well as under certain combinations of these three operations (in particular, the combination of substitution and gluing along a clique, as well as the combination of gluing along a clique and gluing along a bounded number of vertices) is again χ-bounded. This is joint work with Chudnovsky, Scott, and Trotignon, and it originally appeared in [14].Mathematicsip2158Industrial Engineering and Operations Research, MathematicsDissertationsEssays 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 ResearchDissertationsMultiproduct Pricing Management and Design of New Service Products
http://academiccommons.columbia.edu/catalog/ac:144706
Wang, Ruxianhttp://hdl.handle.net/10022/AC:P:12603Fri, 17 Feb 2012 00:00:00 +0000In this thesis, we study price optimization and competition of multiple differentiated substitutable products under the general Nested Logit model and also consider the designing and pricing of new service products, e.g., flexible warranty and refundable warranty, under customers' strategic claim behavior. Chapter 2 considers firms that sell multiple differentiated substitutable products and customers whose purchase behavior follows the Nested Logit model, of which the Multinomial Logit model is a special case. In the Nested Logit model, customers make product selection decision sequentially: they first select a class or a nest of products and subsequently choose a product within the selected class. We consider the general Nested Logit model with product-differentiated price coefficients and general nest-heterogenous degrees. We show that the adjusted markup, which is defined as price minus cost minus the reciprocal of the price coefficient, is constant across all the products in each nest. When optimizing multiple nests of products, the adjusted nested markup is also constant within a nest. By using this result, the multi-product optimization problem can be reduced to a single-dimensional problem in a bounded interval, which is easy to solve. We also use this result to simplify the oligopolistic price competition and characterize the Nash equilibrium. Furthermore, we investigate its application to dynamic pricing and revenue management. In Chapter 3, we investigate the flexible monthly warranty, which offers flexibility to customers and allow them to cancel it at anytime without any penalty. Frequent technological innovations and price declines severely affect sales of extended warranties as product replacement upon failure becomes an increasingly attractive alternative. To increase sales and profitability, we propose offering flexible-duration extended warranties. These warranties can appeal to customers who are uncertain about how long they will keep the product as well as to customers who are uncertain about the product's reliability. Flexibility may be added to existing services in the form of monthly-billing with month-by-month commitments, or by making existing warranties easier to cancel, with pro-rated refunds. This thesis studies flexible warranties from the perspectives of both the customer and the provider. We present a model of the customer's optimal coverage decisions under the objective of minimizing expected support costs over a random planning horizon. We show that under some mild conditions the customer's optimal coverage policy has a threshold structure. We also show through an analytical study and through numerical examples how flexible warranties can result in higher profits and higher attach rates. Chapter 4 examines the designing and pricing of residual value warranty that refunds customers at the end of warranty period based on customers' claim history. Traditional extended warranties for IT products do not differentiate customers according to their usage rates or operating environment. These warranties are priced to cover the costs of high-usage customers who tend to experience more failures and are therefore more costly to support. This makes traditional warranties economically unattractive to low-usage customers. In this chapter, we introduce, design and price residual value warranties. These warranties refund a part of the upfront price to customers who have zero or few claims according to a pre-determined refund schedule. By design, the net cost of these warranties is lower for light users than for heavy users. As a result, a residual value warranty can enable the provider to price-discriminate based on usage rates or operating conditions without the need to monitor individual customers' usage. Theoretic results and numerical experiments demonstrate how residual value warranties can appeal to a broader range of customers and significantly increase the provider's profits.Operations research, Industrial engineeringrw2267Industrial Engineering and Operations ResearchDissertationsA Simulation Model to Analyze the Impact of Golf Skills and a Scenario-based Approach to Options Portfolio Optimization
http://academiccommons.columbia.edu/catalog/ac:143076
Ko, Soonminhttp://hdl.handle.net/10022/AC:P:12166Tue, 10 Jan 2012 00:00:00 +0000A simulation model of the game of golf is developed to analyze the impact of various skills (e.g., driving distance, directional accuracy, putting skill, and others) on golf scores. The golf course model includes realistic features of a golf course including rough, sand, water, and trees. Golfer shot patterns are modeled with t distributions and mixtures of t and normal distributions since normal distributions do not provide good fits to the data. The model is calibrated to extensive data for amateur and professional golfers. The golf simulation is used to assess the impact on scores of distance and direction, determine what factors separate pros from amateurs, and to determine the impact of course length on scores. In the second part of the thesis, we use a scenario-based approach to solve a portfolio optimization problem with options. The solution provides the optimal payoff profile given an investor's view of the future, his utility function or risk appetite, and the market prices of options. The scenario-based approach has several advantages over the traditional covariance matrix method, including additional flexibility in the choice of constraints and objective function.Engineering, Operations researchsk2822Industrial Engineering and Operations Research, BusinessDissertations