Mean-Risk Stochastic Linear Programming Methods

Abstract

In this chapter, we study decomposition methods for mean-risk two-stage stochastic linear programming (MR-SLP). We use structural properties of MR-SLP derived in Chap. 2 and decomposition techniques from Chap. 6 to derive solution algorithms for MR-SLP for quantile and deviation risk measures. Definitions of risk measures and deterministic equivalent problem (DEP) formulations are derived in Chap. 2. The risk measures quantile deviation (QDEV), conditional value-at-risk (CVaR), and expected excess EE have DEPs with dual block angular structure amenable to Benders decomposition. Therefore, we derive an L-shaped algorithm termed, $\mathbb {D}$-AGG algorithm, for $\mathbb {D} \in \{\text{QDEV, CVaR, EE}\}$ involving a single (aggregated) optimality cut at each iteration of the algorithm in Sect. 7.2. This is followed by the derivation of the $\mathbb {D}$-SEP algorithm in Sect. 7.3 involving two separate optimality cuts, one for the expectation term and the other for the quantile (deviation) term of the DEP objective function. For the remainder of the chapter, we turn to the derivation of two subgradient-based algorithms for the deviation risk measure absolute semideviation (ASD), termed ASD-AGG and ASD-SEP algorithms. Unlike MR-SLP with QDEV, CVaR, and EE, the DEP for ASD has a block angular structure due to a set of linking constraints. Therefore, the L-shaped method is not applicable in this case, and that is why we consider a subgradient-based approach to tackle the problem. We derive the single optimality cut ASD-AGG algorithm in Sect. 7.4 and the separate optimality cut ASD-SEP algorithm in Sect. 7.5. Implementing (coding) algorithms for MR-SLP on a computer is not a trivial matter. Therefore, we include detailed numerical examples to illustrate the algorithms and provide some insights and guidelines for computer implementation.