In this and two accompanying papers, we report the development of our recently proposed method for optimizing potential-energy functions for protein-structure prediction and folding simulations (Liwo, A.; Arlukowicz, P.; Czaplewski, C.; Oldziej, S.; Pillardy, J.; Scheraga, H. A. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 1937-1942) aimed at obtaining hierarchical energy landscapes of the protein(s) chosen to calibrate a force field (in which the free energy decreases with increasing degree of nativelikeness). The structure of each benchmark protein is described in terms of structural levels, the number of native structural elements increasing in a given order with level number. Optimization is aimed at lowering the free energy with increasing level of the structural hierarchy. In this paper, we implemented cubic-lattice models of 12-bead polypeptide chains to study the properties of the method and to assess its advantage over a single energy gap and Z-score optimization. The interaction potential consisted of the hydrophobic-polar HP-type potential, a contact potential for peptide-group interaction (simulating hydrogen bonding), and a local potential consisting of bending and torsional terms. Canonical Monte Carlo simulations were carried out at various temperatures for each of several systems and with each of several sets of energy-function parameters to determine the stability (tau(f)(min)) and the folding time (tau(s)(max)). We found that, with moderate energy gaps, nonhierarchical optimization results in both poor (resulting in large tau(f)(min) and small tau(s)(max)) and good (resulting in small tau(f)(min) and large tau(s)(max)) folders, whereas increasing the energy gap improves the folding properties. Good folders are characterized by low-energy structures with some nativelike elements and high-energy states with no native elements, whereas the reverse is true for poor folders. Subsequently, hierarchical optimizations were carried out with various hierarchies, starting by assembling more and more nativelike elements with increasing level number and then carefully considering the folding pathway. We found that explicit introduction of an appropriate structural hierarchy into the optimization procedure substantially improves foldability; however, a wrongly designed hierarchy can impair it. We also found that the best ordering should follow the most efficient path to the native structure, without creating low-energy intermediates separated by large energy barriers. For an appropriate hierarchy, tau(f)(min) is low, and tau(s)(max) increases with the energy gap, which reflects the increased thermodynamic stability of the native structure. It was also found that an analysis of the variation of the heat capacity and the fluctuations of the overlap of the average structure with the native structure with temperature can provide some information about the correspondence between nativelikeness and energy.