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An approximate dynamic programming approach to network revenue management with customer choice. Transportation Science, 43:381–394, 2009. Use of Approximate Dynamic Programming for Production Optimization SPE 141677 (a) Comparison with baseline strategy (
"... We consider a network revenue management problem where customers choose among open fare products according to some prespecified choice model. Starting with a Markov decision process (MDP) formulation, we approximate the value function with an affine function of the state vector. We show that the re ..."
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We consider a network revenue management problem where customers choose among open fare products according to some prespecified choice model. Starting with a Markov decision process (MDP) formulation, we approximate the value function with an affine function of the state vector. We show that the resulting problem provides a tighter bound for the MDP value than the choicebased linear program proposed by Gallego et al. (2004) and Liu and van Ryzin (2007). We develop a column generation algorithm to solve the problem for a multinomial logit choice model with disjoint consideration sets. We also derive a bound as a byproduct of a decomposition heuristic. Our numerical study shows the policies from our solution approach can significantly outperform heuristics from the choicebased linear program. While a substantial amount of research has been done on methods for solving the network revenue management problem, much less work has been done in solving the version where customers choose among available network products. Usually, when airlines open up a menu of fares for a given set of flights, customers will make substitutions between those available, or purchase nothing. Although incorporating customer choice is important in practice, methodologically it is
REFORMULATIONS IN MATHEMATICAL PROGRAMMING: DEFINITIONS AND SYSTEMATICS
, 2008
"... A reformulation of a mathematical program is a formulation which shares some properties with, but is in some sense better than, the original program. Reformulations are important with respect to the choice and efficiency of the solution algorithms; furthermore, it is desirable that reformulations c ..."
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Cited by 17 (13 self)
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A reformulation of a mathematical program is a formulation which shares some properties with, but is in some sense better than, the original program. Reformulations are important with respect to the choice and efficiency of the solution algorithms; furthermore, it is desirable that reformulations can be carried out automatically. Reformulation techniques are very common in mathematical programming but interestingly they have never been studied under a common framework. This paper attempts to move some steps in this direction. We define a framework for storing and manipulating mathematical programming formulations, give several fundamental definitions categorizing reformulations in essentially four types (optreformulations, narrowings, relaxations and approximations). We establish some theoretical results and give reformulation examples for each type.
Reformulations in Mathematical Programming: A Computational Approach
"... Summary. Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathema ..."
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Cited by 16 (12 self)
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Summary. Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathematical expressions of the parameters and decision variables, and therefore excludes optimization of blackbox functions. A reformulation of a mathematical program P is a mathematical program Q obtained from P via symbolic transformations applied to the sets of variables, objectives and constraints. We present a survey of existing reformulations interpreted along these lines, some example applications, and describe the implementation of a software framework for reformulation and optimization. 1
Improved total variationtype regularization using higherorder edge detectors
 SIAM Journal on Imaging Sciences
"... Abstract. We present a novel deconvolution approach to accurately restore piecewise smooth signals from blurred data. The first stage uses Higher Order Total Variation restorations to obtain an estimate of the location of jump discontinuities from the blurred data. In the second stage the estimated ..."
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Cited by 7 (1 self)
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Abstract. We present a novel deconvolution approach to accurately restore piecewise smooth signals from blurred data. The first stage uses Higher Order Total Variation restorations to obtain an estimate of the location of jump discontinuities from the blurred data. In the second stage the estimated jump locations are used to determine the local orders of a Variable Order Total Variation restoration. The method replaces the first order derivative approximation used in standard Total Variation by a variable order derivative operator. Smooth segments as well as jump discontinuities are restored while the staircase effect typical for standard first order Total Variation regularization is avoided. As compared to first order Total Variation, signal restorations are more accurate representations of the true signal, as measured in a relative l 2 norm. The method can also be used to obtain an accurate estimation of the locations and sizes of the true jump discontinuities. The approach is independent of the algorithm used for the standard Total Variation problem and is, consequently, readily incorporated in existing Total Variation restoration codes.
Convexity and Concavity Detection in Computational Graphs Tree Walks for Convexity Assessment
, 2008
"... Abstract. In this paper, we examine sets of symbolic tools associated to modeling systems for mathematical programming which can be used to automatically detect the presence or lack of convexity and concavity in the objective and constraint functions. As a consequence, convexity of the feasible set ..."
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Cited by 2 (1 self)
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Abstract. In this paper, we examine sets of symbolic tools associated to modeling systems for mathematical programming which can be used to automatically detect the presence or lack of convexity and concavity in the objective and constraint functions. As a consequence, convexity of the feasible set may be assessed to some extent. The coconut solver system [Sch04b] focuses on nonlinear global continuous optimization and possesses its own modeling language and data structures. The Dr.ampl [FO07] metasolver aims to analyze nonlinear diffentiable optimization models and hooks into the ampl Solver Library [Gay02]. The symbolic analysis may ◭ be supplemented with a numerical disproving phase when the former returns inconclusive results. We report numerical results using these tools on sets of test problems for both global and local optimization. 1.
On Reoptimizing MultiClass Classifiers ∗
, 2006
"... Significant changes in the instance distribution or associated cost function of a learning problem require one to reoptimize a previously learned classifier to work under new conditions. We study the problem of reoptimizing a multiclass classifier based on its ROC hypersurface and a matrix describi ..."
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Significant changes in the instance distribution or associated cost function of a learning problem require one to reoptimize a previously learned classifier to work under new conditions. We study the problem of reoptimizing a multiclass classifier based on its ROC hypersurface and a matrix describing the costs of each type of prediction error. For a binary classifier, it is straightforward to find an optimal operating point based on its ROC curve and the relative cost of true positive to false positive error. However, the corresponding multiclass problem (finding an optimal operating point based on a ROC hypersurface and cost matrix) is more challenging and until now, it was unknown whether an efficient algorithm existed that found an optimal solution. We answer this question by first proving that the decision version of this problem is NPcomplete. As a complementary positive result, we give an algorithm that finds an optimal solution in polynomial time if the number of classes n is a constant. We also present several heuristics for this problem, including linear, nonlinear, and quadratic programming formulations, genetic algorithms, and a customized algorithm. Empirical results suggest that under uniform costs several methods exhibit significant improvements while genetic algorithms and margin maximization quadratic programs fare the best under nonuniform cost models.
2.2 Oblique Trees, MSMT.......................................................................10
, 2007
"... In this technical report a novel method is proposed that extends the decision tree framework, allowing standard decision tree classifiers to provide a unique certainty value for every input sample they classify. This value is calculated for every input sample individually and represents the classifi ..."
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In this technical report a novel method is proposed that extends the decision tree framework, allowing standard decision tree classifiers to provide a unique certainty value for every input sample they classify. This value is calculated for every input sample individually and represents the classifier's certainty in the classification. The algorithm proposed in this report is not limited to axisparallel trees, it can be applied to any kind of decision tree where the decisions are
cvx Users ’ Guide for cvx version 1.21 (build 790)
, 2010
"... 1.1 What is cvx?............................... 4 1.2 What is disciplined convex programming?............... 5 1.3 About this version............................ 5 ..."
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1.1 What is cvx?............................... 4 1.2 What is disciplined convex programming?............... 5 1.3 About this version............................ 5
cvx Users ’ Guide for cvx version 1.21 ∗
, 2010
"... 1.1 What is cvx?............................... 4 1.2 What is disciplined convex programming?............... 5 1.3 About this version............................ 5 ..."
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1.1 What is cvx?............................... 4 1.2 What is disciplined convex programming?............... 5 1.3 About this version............................ 5