Deep lifted decision rules for two-stage adaptive optimization problems

This paper presents a novel method to generate flexible piecewise linear decision rules for two-stage adaptive optimization problems. Borrowing the idea of a neural network, the lifting network consists of multiple processing layers that enable the construction of more flexible piecewise linear functions used in decision rules whose quality and flexibility is superior to linear decision rules and axially-lifted ones. Two solution methods are proposed to optimize the weights and the decision rule approximation quality: a derivative-free method via an evolutionary algorithm and a derivative-based method using approximate derivative information. For the latter method, we suggest local-search heuristics that scale well and reduce the computational time by several folds while offering similar solution quality. We illustrate the flexibility of the proposed method in comparison to linear and axial piecewise linear decision rules via a transportation and an airlift operations scheduling problem.