Publications (111)97.79 Total impact
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ABSTRACT: The Stochastic User Equilibrium (SUE) model predicts traffic equilibrium flow assuming that users choose their perceived maximum utility paths (or perceived shortest paths) while accounting for the effects of congestion that arise due to users sharing links. Inspired by recent work on distributionally robust optimization, specifically a Cross Moment (CMM) choice model, we develop a new SUE model that uses the mean and covariance information on path utilities but does not assume the particular form of the distribution. Robustness to distributional assumptions is obtained in this model by minimizing the worstcase expected cost over all distributions with fixed two moments. We show that under mild conditions, the CMMSUE (Cross MomentStochastic User Equilibrium) exists and is unique. By combining a simple projected gradient ascent method to evaluate path choice probabilities with a gradient descent method to find flows, we show that the CMMSUE is efficiently computable. CMMSUE provides both modeling flexibility and computational advantages over approaches such as the wellknown MNPSUE (Multinomial ProbitStochastic User Equilibrium) model that require distributional (normality) assumptions to model correlation effects from overlapping paths. In particular, it avoids the use of simulation methods employed in computations for the distributionbased MNPSUE model. Preliminary computational results indicate that CMMSUE provides a practical distributionally robust alternative to MNPSUE.Transportation Research Part B Methodological 03/2015; DOI:10.1016/j.trb.2015.01.005 · 2.95 Impact Factor 

Conference Paper: From Boxes to bees: Active learning in freshmen calculus
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ABSTRACT: The vehicles of education have seen significant broadening with the proliferation of new technologies such as social media, microblogs, online references, multimedia, and interactive teaching tools. This paper summarizes research on the effect of using active learning methods to facilitate student learning and describes our experiences implementing group activities for a calculus course for first year university students at Singapore University of Technology and Design, a new designcentric university established in collaboration with Massachusetts Institute of Technology (MIT). We describe the educational impact of different pedagogical techniques, such as realtime response tools, hands on activities, mathematical modeling, visualization activities and motivational competitions on students with differing learning preferences in a unique cohort classroom setting. Based on faculty reflection and survey data, we provide guidelines on how to adopt the right set of active and group learning techniques to handle the changing learning preferences in the current and future generation of students.2013 IEEE Global Engineering Education Conference (EDUCON); 03/2013 
Article: Strongly Polynomial PrimalDual Algorithms for Concave Cost Combinatorial Optimization Problems
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ABSTRACT: We introduce an algorithm design technique for a class of combinatorial optimization problems with concave costs. This technique yields a strongly polynomial primaldual algorithm for a concave cost problem whenever such an algorithm exists for the fixedcharge counterpart of the problem. For many practical concave cost problems, the fixedcharge counterpart is a wellstudied combinatorial optimization problem. Our technique preserves constant factor approximation ratios, as well as ratios that depend only on certain problem parameters, and exact algorithms yield exact algorithms. Using our technique, we obtain a new 1.61approximation algorithm for the concave cost facility location problem. For inventory problems, we obtain a new exact algorithm for the economic lotsizing problem with general concave ordering costs, and a 4approximation algorithm for the joint replenishment problem with general concave individual ordering costs.  [Show abstract] [Hide abstract]
ABSTRACT: We study the problem of minimizing a nonnegative separable concave function over a compact feasible set. We approximate this problem to within a factor of 1+epsilon by a piecewiselinear minimization problem over the same feasible set. Our main result is that when the feasible set is a polyhedron, the number of resulting pieces is polynomial in the input size of the polyhedron and linear in 1/epsilon. For many practical concave cost problems, the resulting piecewiselinear cost problem can be formulated as a wellstudied discrete optimization problem. As a result, a variety of polynomialtime exact algorithms, approximation algorithms, and polynomialtime heuristics for discrete optimization problems immediately yield fully polynomialtime approximation schemes, approximation algorithms, and polynomialtime heuristics for the corresponding concave cost problems. We illustrate our approach on two problems. For the concave cost multicommodity flow problem, we devise a new heuristic and study its performance using computational experiments. We are able to approximately solve significantly larger test instances than previously possible, and obtain solutions on average within 4.27% of optimality. For the concave cost facility location problem, we obtain a new 1.4991+epsilon approximation algorithm.  [Show abstract] [Hide abstract]
ABSTRACT: This paper studies a core optimization modelthe survivable network design (SND) problemthat incorporates cost and survivability. Given an undirected network G: (N,E) with nodes N, nonnegative costs for each edge in E, and nonnegative integer connectivity requirements specifying the minimum number of edgedisjoint paths needed between pairs of nodes, the SND problem seeks the minimum cost network that satisfies all the connectivity requirements. Although simple to describe, this model captures, as special cases, many classic, but difficult, optimization problems like the Steiner tree problem and the traveling salesman problem. In this paper, we develop a family of new formulations for the SND problem, and show that these formulations are stronger than the traditional cutset formulation of the problem. Our proposed connectivitysplitting formulations have intuitive special cases that motivate the design and analysis of specialized heuristics for the survivable network design problem 
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ABSTRACT: We consider two types of hopindexed models for the unitdemand asymmetric Capacitated Vehicle Routing Problem (CVRP): (a) capacitated models guaranteeing that the number of commodities (paths) traversing any given arc does not exceed a specified capacity; and (b) hopconstrained models guaranteeing that any route length (number of nodes) does not exceed a given value. The latter might, in turn, be divided into two classes: (b1) those restricting the length of the path from the depot to any node k, and (b2) those restricting the length of the circuit passing through any node k. Our results indicate that formulations based upon circuit lengths (b2) lead to models with a linear programming relaxation that is tighter than the linear programming relaxation of models based upon path lengths (b1), and that combining features from capacitated models with those of circuit lengths can lead to formulations for the CVRP with a tight linear programming bound. Computational results on a small number of problem instances with up to 41 nodes and 440 edges show that the combined model with capacities and circuit lengths produce average gaps of less than one percent. We also briefly examine the asymmetric travelling salesman problem (ATSP), showing the potential use of the ideas developed for the vehicle routing problem to derive models for the ATSP with a linear programming relaxation bound that is tighter than the linear programming relaxation bound of the standard Dantzig, Fulkerson and Johnson [G. Dantzig, D. Fulkerson, D. Johnson, Solution of largescale travelling salesman problem, Operations Research 2 (1954) 393–410] formulation.Discrete Optimization 05/2008; 5(25):350372. DOI:10.1016/j.disopt.2007.05.001 · 0.70 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The function need not be continuous; it can have positive or negative jumps, though we do assume that the function is lower semicontinuous, that is, g a (x a ) lim inf x # a #xa g a (x # a ) for 1 any sequence x # a that approaches x a . Without loss of generality, we also assume, through a simple translation of the costs if necessary, that g a (0) = 0. Such a piecewise linear function can be fully characterized by its segments. On each arc a, each segment s of the function has a nonnegative variable cost, c a (the slope), a nonnegative fixed cost, f a (the intercept), and upper and lower bounds, b a and b a , on the flow of that segment. Since the total flow on each arc can always be bounded from above by either the arc capacity or the total demand flowing through the network, we assume that there is a finite number of segments on each arc a, which we represent by the set S a . We further introduce the following notation: K denotes the set of commodities, N is the VOperations Research 02/2007; 55(1):146157. DOI:10.1287/opre.1060.0314 · 1.74 Impact Factor 
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ABSTRACT: Heuristic programming algorithms frequently address large problems and require manipulation and operation on massive data sets. The algorithms can be improved by using efficient data structures. With this in mind, we consider heuristic algorithms for vehicle routing, comparing techniques of Clarke and Wright, Gillett and Miller, and Tyagi, and presenting modifications and extensions which permit problems involving hundreds of demand points to be solved in a matter of seconds. In addition, a multidepot routing algorithm is developed. The results are illustrated with a routing study for an urban newspaper with an evening circulation exceeding 100,000.Networks 10/2006; 7(2):113  148. DOI:10.1002/net.3230070203 · 0.83 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: "710277." Includes author index. Cover title. Supported in part by the U.S. Deaprtment of Transportation, Transportation Advanced Research Program (TARP) contract no. DOTTSC1058 by Bruce L. Golden and Thomas L. Magnanti.Networks 10/2006; 7(2):149  179. DOI:10.1002/net.3230070204 · 0.83 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In a previous paper, Gouveia and Magnanti (2003) found diameterconstrained minimal spanning and Steiner tree problems to be more difficult to solve when the tree diameter D is odd. In this paper, we provide an alternate modeling approach that views problems with odd diameters as the superposition of two problems with even diameters. We show how to tighten the resulting formulation to develop a model with a stronger linear programming relaxation. The linear programming gaps for the tightened model are very small, typically less than 0.5–, and are usually one third to one tenth of the gaps of the best previous model described in Gouveia and Magnanti (2003). Moreover, the new model permits us to solve large Euclidean problem instances that are not solvable by prior approaches.Annals of Operations Research 09/2006; 146(1):1939. DOI:10.1007/s1047900600490 · 1.22 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: By adding a set of redundant constraints, and by iteratively refining the approximation, we show that a commercial solver is able to routinely solve moderatesize strategic safety stock placement problems to optimality. The speedup arises because the solver automatically generates strong flow cover cuts using the redundant constraints.Operations Research Letters 03/2006; 34(234):228238. DOI:10.1016/j.orl.2005.04.004 · 0.62 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: This paper is an edited transcription of the author’s lecture given on February 25th, 2006, in honor of Saul Gass.Perspectives in Operations Research, 12/2005: pages 7798;  [Show abstract] [Hide abstract]
ABSTRACT: The network design problem with connectivity requirements (NDC) models a wide variety of celebrated combinatorial optimization problems including the minimum span ning tree, Steiner tree, and survivable network design problems. We develop strong for mulations for two versions of the edgeconnectivity NDC problem: unitary problems re quiring connected network designs, and nonunitary problems permitting nonconnected networks as solutions. We (i) present a new directed formulation for the unitary NDC problem that is stronger than a natural undirected formulation, (ii) project out several classes of valid inequalities—partition inequalities, oddhole inequalities, and combi natorial design inequalities—that generalize known classes of valid inequalities for the Steiner tree problem to the unitary NDC problem, and (iii) show how to strengthen and direct nonunitary problems. Our results provide a unifying framework for strengthening formulations for NDC problems, and demonstrate the strength and power of ßowbased formulations for net work design problems with connectivity requirements.Networks 03/2005; 45(2):6179. DOI:10.1002/net.20046 · 0.83 Impact Factor 
Article: Solving variational inequality and fixed point problems by line searches and potential optimization
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ABSTRACT: We introduce a general adaptive line search framework for solving fixed point and variational inequality problems. Our goals are to develop iterative schemes that (i) compute solutions when the underlying map satisfies properties weaker than contractiveness, for example, weaker forms of nonexpansiveness, (ii) are more efficient than the classical methods even when the underlying map is contractive, and (iii) unify and extend several convergence results from the fixed point and variational inequality literatures. To achieve these goals, we introduce and study joint compatibility conditions imposed upon the underlying map and the iterative step sizes at each iteration and consider line searches that optimize certain potential functions. As a special case, we introduce a modified steepest descent method for solving systems of equations that does not require a previous condition from the literature (the square of the Jacobian matrix is positive definite). Since the line searches we propose might be difficult to perform exactly, we also consider inexact line searches.Mathematical Programming 12/2004; 101(3):435461. DOI:10.1007/s1010700304765 · 1.80 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In a previous article, using underlying graph theoretical properties, Gouveia and Magnanti (2003) described several network flowbased formulations for diameterconstrained tree problems. Their computational results showed that, even with several enhancements, models for situations when the tree diameter D is odd proved to be more difficult to solve than those when D is even. In this article we provide an alternative modeling approach for the situation when D is odd. The approach views the diameterconstrained minimum spanning tree as being composed of a variant of a directed spanning tree (from an artificial root node) together with two constrained paths, a shortest and a longest path, from the root node to any node in the tree. We also show how to view the feasible set of the linear programming relaxation of the new formulation as the intersection of two integer polyhedra, a socalled triangletree polyhedron and a constrained path polyhedron. This characterization improves upon a model of Gouveia and Magnanti (2003) whose linear programming relaxation feasible set is the intersection of three rather than two integer polyhedra. The linear programming gaps for the tightened model are very small, typically less than 0.5%, and are usually one third to one tenth of the gaps of the best previous model described in Gouveia and Magnanti (2003). Moreover, using the new model, we have been able to solve large Euclidean problem instances that are not solvable by the previous approaches. © 2004 Wiley Periodicals, Inc.Networks 12/2004; 44(4):254265. DOI:10.1002/net.20034 · 0.83 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We develop an optimizationbased approach for a pointtopoint route planning problem that arises in many large scale delivery systems(for example, lessthantruckload freight, rail, mail and package delivery, communications). In these settings, a firm which must ship goods between many origin and destination pairs on a network needs to specify a route for each origindestination pair so as to minimize transportation costs and/or transit times. Typically, the cost structure is very complicated. The approach discussed in this paper exploits the structure of the problem to decompose it into two smaller subproblems, each amenable to solution by a combination of optimization and heuristic techniques. One subproblem is an 'assignment' problem with capacity constraints. The other subproblem is a mixedinteger multicommodity flow problem. We propose solution methods based on Lagrangian relaxation for each subproblem. Computational results with these methods and with a heuristic procedure for the multicommodity flow problem on a problem met in practice are encouraging and suggest that mathematical programming methods can be successfully applied to largescale problems in delivery systems planning and other problems in logistical system design.  [Show abstract] [Hide abstract]
ABSTRACT: We develop and study averaging schemes for solving fixed point and variational inequality problems. Typically, researchers have established convergence results for solution methods for these problems by establishing contractive estimates for their algorithmic maps. In this paper, we establish global convergence results using nonexpansive estimates. After first establishing convergence for a general iterative scheme for computing fixed points, we consider applications to projection and relaxation algorithms for solving variational inequality problems and to a generalized steepest descent method for solving systems of equations. As part of our development, we also establish a new interpretation of a norm condition typically used for establishing convergence of linearization schemes, by associating it with a strongfmonotonicity condition. We conclude by applying our results to transportation networks.
Publication Stats
8k  Citations  
97.79  Total Impact Points  
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Institutions

19742009

Massachusetts Institute of Technology
 • Department of Electrical Engineering and Computer Science
 • School of Engineering
 • MIT Sloan School of Management
Cambridge, Massachusetts, United States


1994

University of Pittsburgh
Pittsburgh, Pennsylvania, United States


1991

Indian Institute of Technology Kanpur
Cawnpore, Uttar Pradesh, India


1988

Texas A&M University
College Station, Texas, United States


1987

Cambridge Healthtech Institute
Needham, Massachusetts, United States
