  We give a unified treatment to optimization problems that can be expressed in
the form of nonnegative-real-weighted Boolean constraint satisfaction problems.
Creignou, Khanna, Sudan, Trevisan, and Williamson studied the complexity of
approximating their optimal solutions whose optimality is measured by the sums
of outcomes of constraints. To explore a wider range of optimization constraint
satisfaction problems, following an early work of Marchetti-Spaccamela and
Romano, we study the case where the optimality is measured by products of
constraints' outcomes. We completely classify those problems into three
categories: PO problems, NPO-hard problems, and intermediate problems that lie
between the former two categories. To prove this trichotomy theorem, we analyze
characteristics of nonnegative-real-weighted constraints using a variant of the
notion of T-constructibility developed earlier for complex-weighted counting
constraint satisfaction problems.
