This article introduces the problem of searching locally optimal patterns within a set of patterns constrained by some anti-monotonic predicate: given any pattern scoring function, a locally optimal pattern has a maximal (or minimal) score locally among neighboring patterns. Some instances of this problem have produced patterns of interest in the framework of knowledge discovery since locally optimal patterns extracted from datasets are very few, informative and non-redundant compared to other pattern families derived from frequent patterns. This article then introduces the concept of variation consistency to characterize pattern functions and uses this notion to propose GALLOP, an algorithm that outperforms existing algorithms to extract locally optimal itemsets. Finally it shows how GALLOP can generically be applied to two classes of scoring functions useful in binary classification or clustering pattern mining problems.