High-utility Itemset Mining (HUIM) finds itemsets from a transaction database with utility no less than a user-defined threshold where the utility of an itemset is defined as the sum of the item-wise utilities. In this paper, we generalize this notion to utility functions that need not be a simple sum of individual utilities. In particular, we study generalized utility functions that are subadditive and monotone (SM). We also describe a novel function that allows us to include external information in the form of a relationship graph for computing utility. Next, we focus on algorithms for HUIM problems with SM utility functions. We note that the existing HUIM algorithms use upper-bounds like "Transaction Weighted Utility" and "Exact-Utility, Remaining-Utility" for efficient search-space exploration. We derive analogous and tighter upper-bounds for SM utility functions. We design a novel inverted-list data structure called SMI-list and a new algorithm called SM-Miner to mine HUIs for SM functions. We explain how existing tree-based and projection-based HUIM algorithms can be adapted using these bounds. We experimentally compare adaptations of some of the latest HUIM algorithms and point out some caveats that should be kept in mind while handling utility functions that allow integration of domain knowledge with a transaction database.