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23
Exploiting knowledge about future demands for realtime vehicle dispatching
 Transportation Science
"... doi 10.1287/trsc.1050.0114 ..."
A CostShaping Linear Program for AverageCost Approximate Dynamic Programming with Performance Guarantees
, 2006
"... ..."
The Smoothed Approximate Linear Program
, 2009
"... We present a novel linear program for the approximation of the dynamic programming costtogo function in highdimensional stochastic control problems. LP approaches to approximate DP have typically relied on a natural ‘projection ’ of a well studied linear program for exact dynamic programming. Such ..."
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Cited by 7 (0 self)
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We present a novel linear program for the approximation of the dynamic programming costtogo function in highdimensional stochastic control problems. 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 costtogo 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. Doing so appears to have several advantages: First, we demonstrate substantially superior bounds on the quality of approximation to the optimal costtogo function afforded by our approach. Second, experiments with our approach on a challenging problem (the game of Tetris) show that the approach outperforms the existing LP approach (which has previously been shown to be competitive with several ADP algorithms) by an order of magnitude. 1.
Pathwise Optimization for Optimal Stopping Problems ∗
, 2010
"... 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 a highdimensional optimal stopping problem. The PO method builds on a dual characterization of optimal stopping problems as optimization pr ..."
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Cited by 4 (1 self)
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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 a highdimensional optimal stopping problem. 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 of a quality comparable with stateoftheart approaches, but in a fraction of the time required for those approaches. As a byproduct, it yields lower bounds (and suboptimal exercise policies) that are substantially superior to those produced by stateoftheart methods. The PO method thus constitutes a practical and desirable approach to highdimensional pricing problems. Further, we develop an approximation theory relevant to martingale duality approaches in general and the PO method in particular. Our analysis provides a guarantee on the quality of upper bounds resulting from these approaches, and identifies three key determinants of their performance: the quality of an input value function approximation, the square root of the effective time horizon of the problem, and a certain spectral measure of ‘predictability ’ of the underlying Markov chain. As a corollary to this analysis we develop approximation guarantees specific to the PO method. Finally, we view the PO method and several approximate dynamic programming (ADP) methods for highdimensional pricing problems through a common lens and in doing so show that the PO method dominates those alternatives. 1.
INTEGRATED SUPPLY CHAIN DESIGN MODELS: A SURVEY AND FUTURE RESEARCH DIRECTIONS
"... (Communicated by Joseph Geunes) Abstract. Optimization models, especially nonlinear optimization models, have been widely used to solve integrated supply chain design problems. In integrated supply chain design, the decision maker needs to take into consideration inventory costs and distribution cos ..."
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Cited by 4 (2 self)
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(Communicated by Joseph Geunes) Abstract. Optimization models, especially nonlinear optimization models, have been widely used to solve integrated supply chain design problems. In integrated supply chain design, the decision maker needs to take into consideration inventory costs and distribution costs when the number and locations of the facilities are determined. The objective is to minimize the total cost that includes location costs and inventory costs at the facilities, and distribution costs in the supply chain. We provide a survey of recent developments in this research area. 1. Introduction. A
Approximate dynamic programming via iterated Bellman inequalities,” http://www.stanford.edu/ ∼boyd/papers/ adp iter bellman.html
, 2010
"... In this paper we introduce new methods for finding functions that lower bound the value function of a stochastic control problem, using an iterated form of the Bellman inequality. Our method is based on solving linear or semidefinite programs, and produces both a bound on the optimal objective, as w ..."
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Cited by 4 (4 self)
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In this paper we introduce new methods for finding functions that lower bound the value function of a stochastic control problem, using an iterated form of the Bellman inequality. Our method is based on solving linear or semidefinite programs, and produces both a bound on the optimal objective, as well as a suboptimal policy that appears to works very well. These results extend and improve bounds obtained by authors in a previous paper using a single Bellman inequality condition. We describe the methods in a general setting, and show how they can be applied in specific cases including the finite state case, constrained linear quadratic control, switched affine control, and multiperiod portfolio investment. 1
Approximate Dynamic Programming via a Smoothed Linear Program
"... We present a novel linear program for the approximation of the dynamic programming costtogo function in highdimensional stochastic control problems. LP approaches to approximate DP have typically relied on a natural ‘projection ’ of a well studied linear program for exact dynamic programming. Such ..."
Abstract

Cited by 4 (2 self)
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We present a novel linear program for the approximation of the dynamic programming costtogo function in highdimensional stochastic control problems. 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 costtogo 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. Doing so appears to have several advantages: First, we demonstrate substantially superior bounds on the quality of approximation to the optimal costtogo function afforded by our approach. Second, experiments with our approach on a challenging problem (the game of Tetris) show that the approach outperforms the existing LP approach (which has previously been shown to be competitive with several ADP algorithms) by an order of magnitude. 1.
Relaxations of Approximate Linear Programs for the Real Option Management of Commodity Storage, Working paper, Carnegie Mellon Univ., 2012
 Mellon Univ., 2012
, 2009
"... The real option management of commodity conversion assets gives rise to intractable Markov decision processes (MDPs). This is due primarily to the high dimensionality of a commodity forward curve, which is part of the MDP state when using high dimensional models of the evolution of this curve, as co ..."
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Cited by 3 (3 self)
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The real option management of commodity conversion assets gives rise to intractable Markov decision processes (MDPs). This is due primarily to the high dimensionality of a commodity forward curve, which is part of the MDP state when using high dimensional models of the evolution of this curve, as commonly done in practice. Focusing on commodity storage, we develop a novel approximate dynamic programming methodology that hinges on the relaxation of approximate linear programs (ALPs) obtained using value function approximations based on reducing the number of futures prices that are part of the MDP state. We derive equivalent approximate dynamic programs (ADPs) for a class of these ALPs, also subsuming a known ADP. We obtain two new ADPs, the value functions of which induce feasible policies for the original MDP, and lower and upper bounds, estimated via Monte Carlo simulation, on the value of an optimal policy of this MDP. We investigate the performance of our ADPs on existing natural gas instances and new crude oil instances. Our approach has potential relevance for the approximate solution of MDPs that arise in the real option management of other commodity conversion assets, as well as the valuation and management of real and financial options that depend on forward curve dynamics. 1
Inventory Routing with Continuous Moves
"... The typical inventory routing problem deals with the repeated distribution of a single product from a single facility with an unlimited supply to a set of customers that can all be reached with outandback trips. Unfortunately, this is not always the reality. We introduce the inventory routing prob ..."
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Cited by 3 (0 self)
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The typical inventory routing problem deals with the repeated distribution of a single product from a single facility with an unlimited supply to a set of customers that can all be reached with outandback trips. Unfortunately, this is not always the reality. We introduce the inventory routing problem with continuous moves to study two important reallife complexities: limited product availabilities at facilities and customers that cannot be served using outandback tours. We need to design delivery tours spanning several days, covering huge geographic areas, and involving product pickups at different facilities. We develop an innovative randomized greedy algorithm, which includes linear programming based postprocessing technology, and we demonstrate its effectiveness in an extensive computational study. 1
Approximate Linear Programming for AverageCost Dynamic Programming
, 2003
"... This paper extends our earlier analysis on approximate linear programming as an approach to approximating the costtogo function in a discountedcost dynamic program [6]. In this paper, we consider the averagecost criterion and a version of approximate linear programming that generates approximati ..."
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Cited by 2 (2 self)
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This paper extends our earlier analysis on approximate linear programming as an approach to approximating the costtogo function in a discountedcost dynamic program [6]. In this paper, we consider the averagecost criterion and a version of approximate linear programming that generates approximations to the optimal average cost and differential cost function. We demonstrate that a naive version of approximate linear programming prioritizes approximation of the optimal average cost and that this may not be wellaligned with the objective of deriving a policy with low average cost. For that, the algorithm should aim at producing a good approximation of the differential cost function. We propose a twophase variant of approximate linear programming that allows for external control of the relative accuracy of the approximation of the differential cost function over different portions of the state space via staterelevance weights. Performance bounds suggest that the new algorithm is compatible with the objective of optimizing performance and provide guidance on appropriate choices for staterelevance weights.