Lagrangian Duality for Robust Problems with Decomposable Functions: The Case of a Robust Inventory Problem

We consider a class of min-max robust problems in which the functions that need to be “robustified” can be decomposed as the sum of arbitrary functions. This class of problems includes many practical problems, such as the lot-sizing problem under demand uncertainty. By considering a Lagrangian relaxation of the uncertainty set, we derive a tractable approximation, called the dual Lagrangian approach, that we relate with both the classical dualization approximation approach and an exact approach. Moreover, we show that the dual Lagrangian approach coincides with the affine decision rule approximation approach. The dual Lagrangian approach is applied to a lot-sizing problem, in which demands are assumed to be uncertain and to belong to the uncertainty set with a budget constraint for each time period. Using the insights provided by the interpretation of the Lagrangian multipliers as penalties in the proposed approach, two heuristic strategies, a new guided iterated local search heuristic, and a subgradient optimization method are designed to solve more complex lot-sizing problems in which additional practical aspects, such as setup costs, are considered. Computational results show the efficiency of the proposed heuristics that provide a good compromise between the quality of the robust solutions and the running time required in their computation. Summary of Contribution: The paper includes both theoretical and algorithmic contributions for a class of min-max robust optimization problems where the objective function includes the maximum of a sum of affine functions. From the theoretical point of view, a tractable Lagrangian dual model resulting from a relaxation of the well-known adversarial problem is proposed, providing a new perspective of well-known models, such as the affinely adjustable robust counterpart (AARC) and the dualization technique introduced by Bertsimas and Sim. These results are particularized to lot-sizing problems. From the algorithm point of view, efficient heuristic schemes—which exploit the information based on the interpretation of the Lagrangian multipliers to solve large size robust problems—are proposed, and their performance is evaluated through extensive computational results based on the lot-sizing problem. In particular, a guided iterated local search and a subgradient optimization method are proposed and compared against the dualization approach proposed by Bertsimas and Sim and with several heuristics based on the AARC approach, which include an iterated local search heuristic and a Benders decomposition approach. Computational results show the efficiency of the proposed heuristics, which provide a good compromise between the quality of the robust solutions and the running time.