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New formulations for optimization under stochastic dominance constraints
 SIAM Journal on Optimization
"... Stochastic dominance constraints allow a decisionmaker to manage risk in an optimization setting by requiring their decision to yield a random outcome which stochastically dominates a reference random outcome. We present new integer and linear programming formulations for optimization under first ..."
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Cited by 10 (1 self)
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Stochastic dominance constraints allow a decisionmaker to manage risk in an optimization setting by requiring their decision to yield a random outcome which stochastically dominates a reference random outcome. We present new integer and linear programming formulations for optimization under first and secondorder stochastic dominance constraints, respectively. These formulations are more compact than existing formulations, and relaxing integrality in the firstorder formulation yields a secondorder formulation, demonstrating the tightness of this formulation. We also present a specialized branching strategy and heuristics which can be used with the new firstorder formulation. Computational tests illustrate the potential benefits of the new formulations.
Risk modeling via stochastic dominance in power systems with dispersed generation
, 2007
"... We propose a new approach to risk modeling in power optimization employing the concept of stochastic dominance. This leads to new classes of largescale blockstructured mixedinteger linear programs for which we present decomposition algorithms. The new methodology is applied to stochastic optimiz ..."
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Cited by 9 (0 self)
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We propose a new approach to risk modeling in power optimization employing the concept of stochastic dominance. This leads to new classes of largescale blockstructured mixedinteger linear programs for which we present decomposition algorithms. The new methodology is applied to stochastic optimization problems related to operation and investment planning in a power system with dispersed generation.
Secondorder stochastic dominance constraints induced by mixedinteger linear recourse
, 2007
"... We introduce stochastic integer programs with dominance constraints induced by mixedinteger linear recourse. Closedness of the constraint set mapping with respect to perturbations of the underlying probability measure is derived. For discrete probability measures, largescale, blockstructured, m ..."
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Cited by 8 (2 self)
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We introduce stochastic integer programs with dominance constraints induced by mixedinteger linear recourse. Closedness of the constraint set mapping with respect to perturbations of the underlying probability measure is derived. For discrete probability measures, largescale, blockstructured, mixedinteger linear programming equivalents to the dominance constrained stochastic programs are identified. For these models, a decomposition algorithm is proposed. Computational tests with instances from power optimization and Sudoku puzzling conclude the paper. Key Words. Stochastic integer programming, stochastic dominance, mixedinteger optimization. AMS subject classifications. 90C15, 90C11, 60E15. 1
INTEGER PROGRAMMING APPROACHES FOR SOME NONCONVEX AND STOCHASTIC OPTIMIZATION PROBLEMS
, 2007
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STABILITY AND SENSITIVITY OF STOCHASTIC DOMINANCE CONSTRAINED OPTIMIZATION MODELS
"... Abstract. We consider convex optimization problems with kth order stochastic dominance constraints for k ≥ 2. We discuss distances of random variables that are relevant for the dominance relation and establish quantitative stability results for optimal values and solution sets of the optimization pr ..."
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Abstract. We consider convex optimization problems with kth order stochastic dominance constraints for k ≥ 2. We discuss distances of random variables that are relevant for the dominance relation and establish quantitative stability results for optimal values and solution sets of the optimization problems in terms of a suitably selected probability metric. Moreover, we provide conditions ensuring Hadamard directional differentiability of the optimal value function. We introduce the notion of a shadow utility, which determines the changes of the optimal value when the underlying random variables are perturbed. Finally, we derive a limit theorem for the optimal values of empirical (Monte Carlo, sample average) approximations of dominance constrained optimization models.
User’s guide to ddsip.vSD – A C Package for the Dual Decomposition of Stochastic Programs with Dominance Constraints Induced by MixedInteger Linear Recourse
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AssetLiability Management Modelling with . . .
, 2009
"... An AssetLiability Management model with a novel strategy for controlling risk of underfunding is presented in this paper. The basic model involves multiperiod decisions (portfolio rebalancing) and deals with the usual uncertainty of investment returns and future liabilities. Therefore it is wellsu ..."
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An AssetLiability Management model with a novel strategy for controlling risk of underfunding is presented in this paper. The basic model involves multiperiod decisions (portfolio rebalancing) and deals with the usual uncertainty of investment returns and future liabilities. Therefore it is wellsuited to a stochastic programming approach. A stochastic dominance concept is applied to measure (and control) risk of underfunding. A small numerical example is provided to demonstrate advantages of this new model which includes stochastic dominance constraints over the basic model. Adding stochastic dominance constraints comes with a price. It complicates the structure of the underlying stochastic program. Indeed, new constraints create a link between variables associated with different scenarios of the same time stage. This destroys the usual treestructure of the constraint matrix in the stochastic program and prevents the application of standard stochastic programming approaches such as (nested) Benders decomposition. A structureexploiting interior point method is applied to this problem. A specialized interior point solver OOPS can deal efficiently with such problems and outperforms the industrial strength commercial solver CPLEX. Computational results on medium scale problems with sizes reaching about one million of variables demonstrate the efficiency of the specialized solution technique. The solution time for these nontrivial asset liability models seems to grow sublinearly with the key parameters of the model such as the number of assets and the number of realizations of the benchmark portfolio, and this makes the method applicable to truly large scale problems.