  We introduce the framework of qualitative optimization problems (or, simply,
optimization problems) to represent preference theories. The formalism uses
separate modules to describe the space of outcomes to be compared (the
generator) and the preferences on outcomes (the selector). We consider two
types of optimization problems. They differ in the way the generator, which we
model by a propositional theory, is interpreted: by the standard propositional
logic semantics, and by the equilibrium-model (answer-set) semantics. Under the
latter interpretation of generators, optimization problems directly generalize
answer-set optimization programs proposed previously. We study strong
equivalence of optimization problems, which guarantees their interchangeability
within any larger context. We characterize several versions of strong
equivalence obtained by restricting the class of optimization problems that can
be used as extensions and establish the complexity of associated reasoning
tasks. Understanding strong equivalence is essential for modular representation
of optimization problems and rewriting techniques to simplify them without
changing their inherent properties.
