We consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures. Thereby we place emphasis on dynamic models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favorable properties. Polyhedral risk measures are defined as optimal values of certain linear stochastic programs where the arguments of the risk measure appear on the right-hand side of the dynamic constraints. Dual representations for polyhedral risk measures are derived and used to deduce criteria for convexity and coherence. As examples of polyhedral risk measures we propose multiperiod extensions of the Conditional-Value-at-Risk.

The standard approach to formulating stochastic programs is based on the assumption that the stochastic process is independent of the optimization decisions. We address a class of problems where the optimization decisions influence the time of information discovery for a subset of the uncertain parameters. We extend the standard modeling approach by presenting a disjunctive programming formulation that accommodates stochastic programs for this class of problems. A set of theoretical properties that lead to reduction in the size of the model is identified. A Lagrangean duality based branch and bound algorithm is also presented.

Polyhedral discrepancies are relevant for the quantitative stability of mixed-integer two-stage and chance constrained stochastic programs. We study the problem of optimal scenario reduction for a discrete probability distribution with respect to certain polyhedral discrepancies and develop algorithms for determining the optimally reduced distribution approximately. Encouraging numerical experience for optimal scenario reduction is provided.

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This paper presents a decomposition approach for linear multistage stochastic programs, that is based on the concept of recombining scenario trees. The latter, widely applied in Mathematical Finance, may prevent the node number of the scenario tree to grow exponentially with the number of time stages. It is shown how this property may be exploited within a non-Markovian framework and under time-coupling constraints. Being close to the well-established Nested Benders Decomposition, our approach uses the special structure of recombining trees for simultaneous cutting plane approximations. Convergence is proved and stopping criteria are deduced. Techniques for the generation of suitable scenario trees and some numerical examples are presented.

We propose a new approach to risk modeling in power optimization employing the concept of stochastic dominance. This leads to new classes of large-scale block-structured mixed-integer 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.

Models and algorithms for risk neutral and risk averse power optimization under uncertainty are presented. The approach differs from previous ones by incorporating the transmission network explicitly.

informs ® doi 10.1287/opre.1060.0303

Traditional stochastic programming is risk neutral in the sense that it is concerned with the optimization of an expectation criterion. A common approach to addressing risk in decision making problems is to consider a weighted mean-risk objective, where some dispersion statistic is used as a measure of risk. We investigate the computational suitability of various mean-risk objective functions in addressing risk in stochastic programming models. We prove that the classical mean-variance criterion leads to computational intractability even in the simplest stochastic programs. On the other hand, a number of alternative mean-risk functions are shown to be computationally tractable using slight variants of existing stochastic programming decomposition algorithms. We propose a parametric cutting plane algorithm to generate the entire mean-risk e#cient frontier for a particular mean-risk objective.

For a real world problem – transporting pallets between warehouses in order to guarantee sufficient supply for known and additional stochastic demand – we propose a solution approach via convex relaxation of an integer programming formulation, suitable for online optimization. The essential new element linking routing and inventory management is a convex piecewise linear cost function that is based on minimizing the expected number of pallets that still need transportation. For speed, the convex relaxation is solved approximately by a bundle approach yielding an online schedule in 5 to 12 minutes for up to 3 warehouses and 40000 articles; in contrast, computation times of state of the art LP-solvers are prohibitive for online application. In extensive numerical experiments on a real world data stream, the approximate solutions exhibit negligible loss in quality; in long term simulations the proposed method reduces the average number of pallets needing transportation due to short term demand to less than half the number observed in the data stream.