A cross-decomposition scheme with integrated primal–dual multi-cuts for two-stage stochastic programming investment planning problems

Abstract

We describe a decomposition algorithm that combines Benders and scenariobased Lagrangean decomposition for two-stage stochastic programming investment planning problems with complete recourse, where the first-stage variables are mixedinteger and the second-stage variables are continuous. The algorithm is based on the cross-decomposition scheme and fully integrates primal and dual information in terms of primal-dual multi-cuts added to the Benders and the Lagrangean master problems for each scenario. The potential benefits of the cross-decomposition scheme are demonstrated with an illustrative case study for a facility location problem under disruptions, where the underlying LP relaxation is weak, and hence, multicut Benders decomposition converges only slowly. If the LP relaxation is improved by adding a tightening constraint, the performance of multi-cut Benders decomposition improves but the cross-decomposition scheme stays competitive and outperforms Benders for the largest problem instance.Keywords Cross-decomposition · Two-stage stochastic programming · Investment planning MotivationTwo-stage stochastic programming investment planning problems (Birge and Louveaux, 2011) can be hard to solve since the resulting deterministic equivalent programs can lead to very large-scale problems. There are two main approaches, which

The Benders decomposition algorithm: A literature review

“…Algorithms based on the single-search-tree strategy often use this approach to tighten the root node and reduce the size of the tree (see e.g., Botton et al, 2013). Cross-decomposition has received much less attention, although a recent study by Mitra et al (2016) has demonstrated that it can be superior to the BD method when the underlying LP relaxation is weak. Stabilization techniques significantly reduce the number of iterations, but the cost of each iteration may increase because of the additional complexities that they introduce in the MP.…”

Optimization of CCUS Supply Chains for Some European Countries under the Uncertainty

“…In addition to the above constraints, the non-anticipativity constraints are assumed for the first stage variables: their value is the same for all scenarios [51,52]. With these constraints, it is assumed that the decision should depend only on information available at the time of the decision and not on future observations [53,54].…”

“…When considering a deterministic equivalent problem for the two-stage stochastic model of the supply chains, the objective function is defined by the following relation [53,54,64] (see Equation ( 23)):…”

Improved Benders Decomposition and Feasibility Validation for Two-Stage Chance-Constrained Programs in Process Optimization

“…Directly using an optimization solver to handle all scenarios simultaneously is not efficient because its computational time grows exponentially as the number of scenarios increases. Several decomposition strategies thus have been developed to solve each scenario separately, for example, Lagrangian decomposition, 23−25 Benders decomposition, 26,27 and cross decomposition; 29 their computational time grows linearly as the number of scenarios increases. A more complicated case, which combines the chance constraints and two-stage stochastic program, is proposed by Liu et al 30 for the two-stage decision process with dual mode operations.…”