Real-world optimization problems have been studied in the past, but the work resulted in approaches tailored to individual problems that could not be easily generalized. The reason for this limitation was the lack of appropriate models for the systematic study of salient aspects of real-world problems. The aim of this article is to study one of such aspects: multi-hardness. We propose a variety of decomposition-based algorithms for an abstract multi-hard problem and compare them against the most promising heuristics.