The duality between conflicts and diagnoses in the field of diagnosis, or between plans and landmarks in the field of planning, or between unsatisfiable cores and minimal co-satisfiable sets in SAT or CSP solving, has been known for many years. Recent work in these communities (Davies and Bacchus, CP 2011, Bonet and Helmert, ECAI 2010, Haslum et al., ICAPS 2012, Stern et al., AAAI 2012) has brought it to the fore as a topic of current interest. The present paper lays out the set-theoretic basis of the concept, and introduces a generic implementation of an algorithm based on it. This algorithm provides a method for converting decision procedures into optimisation ones across a wide range of applications without the need to rewrite the decision procedure implementations. Initial experimental validation shows good performance on a number of benchmark problems from AI planning.