A new procedure for optimizing the potential function for proteins is presented, and the folding of lattice chain protein models was studied with optimized potentials. The pairwise contact energies between residues in the lattice protein models were optimized by an iterative algorithm that maximizes the free-energy difference between the (given) native structure and the non-native stales of the protein. Optimization of the energy parameters for a variety of sequences of the model protein was carried out. The statistical-mechanical properties of the model proteins were analyzed with different sets of energy parameters to investigate the effects of optimization of the energy parameters on protein folding. For many sequences, optimization of the potential leads to strongly foldable protein models that can be folded to the target native structures in relatively short Monte Carlo simulations. It is found that consistency among the different components of the interactions in the native structure of a protein is a necessary condition for the existence of an exact and efficient folding potential. The results of this work reveal some crucial correlations between the sequence and the native structure of a protein, which determine the unique folding of the protein.