In this paper we study the d-dimensional assignment problem in which entries of the cost array satisfy the Monge property, except for ∞-entries, which may violate it. We assume that the ∞-entries are incurred by incompatible partner indices and their number is bounded by an upper bound λ for each index. We show that the problem can be solved in linear time for fixed d and λ, and it becomes strongly NP-hard if d or λ is part of the input.

This article presents a review and a classification of interactive optimization methods. These interactive methods are used for solving optimization problems. The interaction with an end user or decision maker aims at improving the efficiency of the optimization procedure, enriching the optimization model, or informing the user regarding the solutions proposed by the optimization system. First, we present the challenges of using optimization methods as a tool for supporting decision making, and we justify the integration of the user in the optimization process. This integration is generally achieved via a dynamic interaction between the user and the system. Next, the different classes of interactive optimization approaches are presented. This detailed review includes trial and error, interactive reoptimization, interactive multiobjective optimization, interactive evolutionary algorithms, human-guided search, and other approaches that are less well covered in the research literature. On the basis of this review, we propose a classification that aims to better describe and compare interaction mechanisms. This classification offers two complementary views on interactive optimization methods. The first perspective focuses on the user’s contribution to the optimization process, and the second concerns the components of interactive optimization systems. Finally, on the basis of this review and classification, we identify some open issues and potential perspectives for interactive optimization methods.