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194
Integerprogramming software systems
, 2004
"... Recent developments in integer–programming software systems have tremendously improved our ability to solve large–scale instances. We review the major algorithmic components of state–of–the–art solvers and discuss the options available to users to adjust the behavior of these solvers when default s ..."
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Cited by 37 (0 self)
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Recent developments in integer–programming software systems have tremendously improved our ability to solve large–scale instances. We review the major algorithmic components of state–of–the–art solvers and discuss the options available to users to adjust the behavior of these solvers when default settings do not achieve the desired performance level. Furthermore, we highlight advances towards integrated modeling and solution environments. We conclude with a discussion of model characteristics and substructures that pose challenges for integer–programming software systems and a perspective on features we may expect to see in these systems in the near future.
Selected topics in robust convex optimization
 MATH. PROG. B, THIS ISSUE
, 2007
"... Robust Optimization is a rapidly developing methodology for handling optimization problems affected by nonstochastic “uncertainbutbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robu ..."
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Cited by 35 (2 self)
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Robust Optimization is a rapidly developing methodology for handling optimization problems affected by nonstochastic “uncertainbutbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links between RO and traditional chance constrained settings of problems with stochastic data, and (4) a novel generic application of the RO methodology in Robust Linear Control.
An Approximation Scheme for Stochastic Linear Programming and its Application to Stochastic Integer Programs
"... Stochastic optimization problems attempt to model uncertainty in the data by assuming that the input is specified by a probability distribution. We consider the wellstudied paradigm of 2stage models with recourse: first, given only distributional information about (some of) the data one commits on ..."
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Cited by 35 (6 self)
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Stochastic optimization problems attempt to model uncertainty in the data by assuming that the input is specified by a probability distribution. We consider the wellstudied paradigm of 2stage models with recourse: first, given only distributional information about (some of) the data one commits on initial actions, and then once the actual data is realized (according to the distribution), further (recourse) actions can be taken. We show that for a broad class of 2stage linear models with recourse, one can, for any ɛ> 0, in time polynomial in 1 ɛ and the size of the input, compute a solution of value within a factor (1 + ɛ) of the optimum, in spite of the fact that exponentially many secondstage scenarios may occur. In conjunction with a suitable rounding scheme, this yields the first approximation algorithms for 2stage stochastic integer optimization problems where the underlying random data is given by a “black box” and no restrictions are placed on the costs in the two stages. Our rounding approach for stochastic integer programs shows that an approximation algorithm for a deterministic analogue yields, with a small constantfactor loss, provably nearoptimal solutions for the stochastic generalization. Among the range of applications we consider are stochastic versions of the multicommodity flow, set cover, vertex cover, and facility location problems.
How to pay, come what may: Approximation algorithms for demandrobust covering problems
 In FOCS
, 2005
"... Robust optimization has traditionally focused on uncertainty in data and costs in optimization problems to formulate models whose solutions will be optimal in the worstcase among the various uncertain scenarios in the model. While these approaches may be thought of defining data or costrobust prob ..."
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Cited by 31 (9 self)
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Robust optimization has traditionally focused on uncertainty in data and costs in optimization problems to formulate models whose solutions will be optimal in the worstcase among the various uncertain scenarios in the model. While these approaches may be thought of defining data or costrobust problems, we formulate a new “demandrobust” model motivated by recent work on twostage stochastic optimization problems. We propose this in the framework of general covering problems and prove a general structural lemma about special types of firststage solutions for such problems: there exists a firststage solution that is a minimal feasible solution for the union of the demands for some subset of the scenarios and its objective function value is no more than twice the optimal. We then provide approximation algorithms for a variety of standard discrete covering problems in this setting, including minimum cut, minimum multicut, shortest paths, Steiner trees, vertex cover and uncapacitated facility location. While many of our results draw from rounding approaches recently developed for stochastic programming problems, we also show new applications of old metric rounding techniques for cut problems in this demandrobust setting.
A Linear DecisionBased Approximation Approach to Stochastic Programming
, 2008
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Robust unit commitment problem with demand response and wind energy
, 2010
"... To improve the efficiency in power generation and to reduce the greenhouse gas emission, both Demand Response (DR) strategy and intermittent renewable energy have been proposed or applied in electric power systems. However, the uncertainty and the generation pattern in wind farms and the complexity ..."
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Cited by 23 (3 self)
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To improve the efficiency in power generation and to reduce the greenhouse gas emission, both Demand Response (DR) strategy and intermittent renewable energy have been proposed or applied in electric power systems. However, the uncertainty and the generation pattern in wind farms and the complexity of demand side management pose huge challenges in power system operations. In this paper, we analytically investigate how to integrate DR and wind energy with fossil fuel generators to (i) minimize power generation cost; (2) fully take advantage wind energy with managed demand to reduce greenhouse emission. We first build a twostage robust unit commitment model to obtain dayahead generator schedules where wind uncertainty is captured by a polyhedron. Then, we extend our model to include DR strategy such that both price levels and generator schedule will be derived for the next day. For these two NPhard problems, we derive their mathematical properties and develop a novel and analytical solution method. Our computational study on a IEEE 118 system with 36 units shows that (i) the robust unit commitment model can significantly reduce total cost and fully make use of wind energy; (ii) the cutting plane method is computationally superior to known algorithms.
Minmax and minmax regret versions of combinatorial optimization problems: A survey
 European Journal of Operational Research
"... Minmax and minmax regret criteria are commonly used to define robust solutions. After motivating the use of these criteria, we present general results. Then, we survey complexity results for the minmax and minmax regret versions of some combinatorial optimization problems: shortest path, spannin ..."
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Cited by 21 (1 self)
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Minmax and minmax regret criteria are commonly used to define robust solutions. After motivating the use of these criteria, we present general results. Then, we survey complexity results for the minmax and minmax regret versions of some combinatorial optimization problems: shortest path, spanning tree, assignment, min cut, min st cut, knapsack. Since most of these problems are NPhard, we also investigate the approximability of these problems. Furthermore, we present algorithms to solve these problems to optimality.
Robust combinatorial optimization with exponential scenarios
 In IPCO
, 2007
"... Abstract. Following the wellstudied twostage optimization framework for stochastic optimization [15, 18], we study approximation algorithms for robust twostage optimization problems with an exponential number of scenarios. Prior to this work, Dhamdhere et al. [8] introduced approximation algorith ..."
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Cited by 21 (3 self)
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Abstract. Following the wellstudied twostage optimization framework for stochastic optimization [15, 18], we study approximation algorithms for robust twostage optimization problems with an exponential number of scenarios. Prior to this work, Dhamdhere et al. [8] introduced approximation algorithms for twostage robust optimization problems with explicitly given scenarios. In this paper, we assume the set of possible scenarios is given implicitly, for example by an upper bound on the number of active clients. In twostage robust optimization, we need to prepurchase some resources in the first stage before the adversary’s action. In the second stage, after the adversary chooses the clients that need to be covered, we need to complement our solution by purchasing additional resources at an inflated price. The goal is to minimize the cost in the worstcase scenario. We give a general approach for solving such problems using LP rounding. Our approach uncovers an interesting connection between robust optimization and online competitive algorithms. We use this approach, together with known online algorithms, to develop approximation algorithms for several robust covering problems, such as set cover, vertex cover, and edge cover. We also study a simple buyatonce algorithm that either covers all items in the first stage or does nothing in the first stage and waits to build the complete solution in the second stage. We show that this algorithm gives tight approximation factors for unweighted variants of these covering problems, but performs poorly for general weighted problems. 1
Provisioning virtual private networks under traffic uncertainty
 Networks
, 2004
"... We investigate a network design problem under traffic uncertainty which arises when provisioning Virtual Private Networks (VPNs): given a set of terminals that must communicate with one another, and a set of possible traffic matrices, sufficient capacity has to be reserved on the links of the large ..."
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Cited by 20 (3 self)
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We investigate a network design problem under traffic uncertainty which arises when provisioning Virtual Private Networks (VPNs): given a set of terminals that must communicate with one another, and a set of possible traffic matrices, sufficient capacity has to be reserved on the links of the large underlying public network so as to support all possible traffic matrices while minimizing the total reservation cost. The problem admits several variants depending on the desired topology of the reserved links, and the nature of the traffic data uncertainty. We present compact linear mixedinteger programming formulations for the problem with the classical hose traffic model and for a new, less conservative, robust variant relying on the traffic statistics that are often available. These flowbased formulations allow to solve optimally mediumtolargesize instances with commercial MIP solvers. We also propose a combined branchandprice and cutting plane algorithm to tackle larger instances. Computational results obtained for several classes of instances are reported and discussed. Key words: Virtual private networks, network design, traffic uncertainty, robust optimization, linear mixedinteger programs, branchandprice, cutting planes. 1