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287
Conditional valueatrisk for general loss distributions
 Journal of Banking and Finance
, 2002
"... Abstract. Fundamental properties of conditional valueatrisk, as a measure of risk with significant advantages over valueatrisk, are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications because of the prevalence o ..."
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Cited by 322 (27 self)
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Abstract. Fundamental properties of conditional valueatrisk, as a measure of risk with significant advantages over valueatrisk, are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications because of the prevalence of models based on scenarios and finite sampling. Conditional valueatrisk is able to quantify dangers beyond valueatrisk, and moreover it is coherent. It provides optimization shortcuts which, through linear programming techniques, make practical many largescale calculations that could otherwise be out of reach. The numerical efficiency and stability of such calculations, shown in several case studies, are illustrated further with an example of index tracking. Key Words: Valueatrisk, conditional valueatrisk, mean shortfall, coherent risk measures, risk sampling, scenarios, hedging, index tracking, portfolio optimization, risk management
Fairness and optimal stochastic control for heterogeneous networks
 Proc. IEEE INFOCOM, March 2005. TRANSACTIONS ON NETWORKING, VOL
, 2008
"... Abstract — We consider optimal control for general networks with both wireless and wireline components and time varying channels. A dynamic strategy is developed to support all traffic whenever possible, and to make optimally fair decisions about which data to serve when inputs exceed network capaci ..."
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Cited by 257 (63 self)
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Abstract — We consider optimal control for general networks with both wireless and wireline components and time varying channels. A dynamic strategy is developed to support all traffic whenever possible, and to make optimally fair decisions about which data to serve when inputs exceed network capacity. The strategy is decoupled into separate algorithms for flow control, routing, and resource allocation, and allows each user to make decisions independent of the actions of others. The combined strategy is shown to yield data rates that are arbitrarily close to the optimal operating point achieved when all network controllers are coordinated and have perfect knowledge of future events. The cost of approaching this fair operating point is an endtoend delay increase for data that is served by the network.
The sample average approximation method for stochastic discrete optimization
 SIAM Journal on Optimization
, 2001
"... Abstract. In this paper we study a Monte Carlo simulation based approach to stochastic discrete optimization problems. The basic idea of such methods is that a random sample is generated and consequently the expected value function is approximated by the corresponding sample average function. The ob ..."
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Cited by 190 (19 self)
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Abstract. In this paper we study a Monte Carlo simulation based approach to stochastic discrete optimization problems. The basic idea of such methods is that a random sample is generated and consequently the expected value function is approximated by the corresponding sample average function. The obtained sample average optimization problem is solved, and the procedure is repeated several times until a stopping criterion is satisfied. We discuss convergence rates and stopping rules of this procedure and present a numerical example of the stochastic knapsack problem. Key words. Stochastic programming, discrete optimization, Monte Carlo sampling, Law of Large Numbers, Large Deviations theory, sample average approximation, stopping rules, stochastic knapsack problem AMS subject classifications. 90C10, 90C15
Theory and applications of Robust Optimization
, 2007
"... In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most pr ..."
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Cited by 90 (15 self)
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In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most prominent theoretical results of RO over the past decade, we will also present some recent results linking RO to adaptable models for multistage decisionmaking problems. Finally, we will highlight successful applications of RO across a wide spectrum of domains, including, but not limited to, finance, statistics, learning, and engineering.
Hedging uncertainty: Approximation algorithms for stochastic optimization problems
 In Proceedings of the 10th International Conference on Integer Programming and Combinatorial Optimization
, 2004
"... We initiate the design of approximation algorithms for stochastic combinatorial optimization problems; we formulate the problems in the framework of twostage stochastic optimization, and provide nearly tight approximation algorithms. Our problems range from the simple (shortest path, vertex cover, ..."
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Cited by 82 (12 self)
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We initiate the design of approximation algorithms for stochastic combinatorial optimization problems; we formulate the problems in the framework of twostage stochastic optimization, and provide nearly tight approximation algorithms. Our problems range from the simple (shortest path, vertex cover, bin packing) to complex (facility location, set cover), and contain representatives with different approximation ratios. The approximation ratio of the stochastic variant of a typical problem is of the same order of magnitude as its deterministic counterpart. Furthermore, common techniques for designing approximation algorithms such as LP rounding, the primaldual method, and the greedy algorithm, can be carefully adapted to obtain these results. 1
Optimal operation of multi reservoir systems: stateoftheart review
 J. Water Resour. Plann. Manag
, 2004
"... Abstract: With construction of new largescale water storage projects on the wane in the U.S. and other developed countries, attention must focus on improving the operational effectiveness and efficiency of existing reservoir systems for maximizing the beneficial uses of these projects. Optimal coor ..."
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Cited by 81 (0 self)
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Abstract: With construction of new largescale water storage projects on the wane in the U.S. and other developed countries, attention must focus on improving the operational effectiveness and efficiency of existing reservoir systems for maximizing the beneficial uses of these projects. Optimal coordination of the many facets of reservoir systems requires the assistance of computer modeling tools to provide information for rational management and operational decisions. The purpose of this review is to assess the stateoftheart in optimization of reservoir system management and operations and consider future directions for additional research and application. Optimization methods designed to prevail over the highdimensional, dynamic, nonlinear, and stochastic characteristics of reservoir systems are scrutinized, as well as extensions into multiobjective optimization. Application of heuristic programming methods using evolutionary and genetic algorithms are described, along with application of neural networks and fuzzy rulebased systems for inferring reservoir system operating rules.
Optimization under uncertainty: Stateoftheart and opportunities
 Computers and Chemical Engineering
, 2004
"... A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemi ..."
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Cited by 73 (0 self)
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A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemicals. A key difficulty in optimization under uncertainty is in dealing with an uncertainty space that is huge and frequently leads to very largescale optimization models. Decisionmaking under uncertainty is often further complicated by the presence of integer decision variables to model logical and other discrete decisions in a multiperiod or multistage setting. This paper reviews theory and methodology that have been developed to cope with the complexity of optimization problems under uncertainty. We discuss and contrast the classical recoursebased stochastic programming, robust stochastic programming, probabilistic (chanceconstraint) programming, fuzzy programming, and stochastic dynamic programming. The advantages and shortcomings of these models are reviewed and illustrated through examples. Applications and the stateoftheart in computations are also reviewed. Finally, we discuss several main areas for future development in this field. These include development of polynomialtime approximation schemes for multistage stochastic programs and the application of global optimization algorithms to twostage and chanceconstraint formulations.
Computational complexity of stochastic programming problems
, 2005
"... Stochastic programming is the subfield of mathematical programming that considers optimization in the presence of uncertainty. During the last four decades a vast quantity of literature on the subject has appeared. Developments in the theory of computational complexity allow us to establish the theo ..."
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Cited by 55 (1 self)
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Stochastic programming is the subfield of mathematical programming that considers optimization in the presence of uncertainty. During the last four decades a vast quantity of literature on the subject has appeared. Developments in the theory of computational complexity allow us to establish the theoretical complexity of a variety of stochastic programming problems studied in this literature. Under the assumption that the stochastic parameters are independently distributed, we show that twostage stochastic programming problems are ♯Phard. Under the same assumption we show that certain multistage stochastic programming problems are PSPACEhard. The problems we consider are nonstandard in that distributions of stochastic parameters in later stages depend on decisions made in earlier stages.
Adaptive leastexpected time paths in stochastic, timevarying transportation and data networks
 Networks
"... In congested transportation and data networks, travel (or transmission) times are timevarying quantities that are at best known a priori with uncertainty. In such stochastic, timevarying (or STV) networks, one can choose to use the a priori leastexpected time (LET) path or one can make improved r ..."
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Cited by 46 (0 self)
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In congested transportation and data networks, travel (or transmission) times are timevarying quantities that are at best known a priori with uncertainty. In such stochastic, timevarying (or STV) networks, one can choose to use the a priori leastexpected time (LET) path or one can make improved routing decisions en route as traversal times on traveled arcs are experienced and arrival times at intermediate locations are revealed. In this context, for a given origin–destination pair at a specific departure time, a single path may not provide an adequate solution, because the optimal path depends on intermediate information concerning experienced traversal times on traveled arcs. Thus, a set of strategies, referred to as hyperpaths, are generated to provide directions to the destination node conditioned upon arrival times at intermediate locations. In this paper, an efficient labelsettingbased algorithm is presented for determining the adaptive LET hyperpaths in STV networks. Such a procedure is useful in making critical routing decisions in Intelligent Transportation Systems (ITS) and data communication networks. A sidebyside comparison of this procedure with a labelcorrectingbased algorithm for solving the same problem is made. Results of extensive computational tests to assess and compare the performance of both algorithms, as well as to investigate the characteristics of the resulting hyperpaths, are presented. An illustrative example of both procedures is provided. © 2001 John Wiley & Sons, Inc.
A multistage stochastic integer programming approach for capacity expansion under uncertainty
 J. Global Opt
, 2003
"... This paper addresses a multiperiod investment model for capacity expansion in an uncertain environment. Using a scenario tree approach to model the evolution of uncertain demand and cost parameters, and fixedcharge cost functions to model the economies of scale in expansion costs, we develop a mul ..."
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Cited by 42 (3 self)
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This paper addresses a multiperiod investment model for capacity expansion in an uncertain environment. Using a scenario tree approach to model the evolution of uncertain demand and cost parameters, and fixedcharge cost functions to model the economies of scale in expansion costs, we develop a multistage stochastic integer programming formulation for the problem. A reformulation of the problem is proposed using variable disaggregation to exploit the lotsizing substructure of the problem. The reformulation significantly reduces the LP relaxation gap of this large scale integer program. A heuristic scheme is presented to perturb the LP relaxation solutions to produce good quality integer solutions. Finally, we outline a branch and bound algorithm that makes use of the reformulation strategy as a lower bounding scheme, and the heuristic as an upper bounding scheme, to solve the problem to global optimality. Our preliminary computational results indicate that the proposed strategy has significant advantages over straightforward use of commercial solvers. 1