NPO-PB is the class of NP optimization problems with polynomially bounded values. In this paper we provide a new characterization for the class: That is, NPO-PB = MIN Σ0. This result shows that quantifiers are not relevant in characterizing approximability for minimization problems, unlike maximization problems. In proving the result, we develop a generic reduction, which combines maximization and minimization problems. Based on the new characterization, several problems are shown to be NPO-PB-complete. All of these problems are shown to be hard to approximate and tighter bounds are given.