The optimized-inner-projection (OIP) method is implemented and applied to determine the lower bounds to the ground-state correlation energy for the Pariser-Parr-Pople and Hubbard models of cyclic polyenes ${\mathrm{C}}_{\mathit{N}}$${\mathrm{H}}_{\mathit{N}}$, N=4\ensuremath{\nu}+2, \ensuremath{\nu}=3--5, in the whole range of the coupling constant. In actual calculations, the orthogonally spin-adapted form of the OIP equations is used, and full advantage is taken of the high symmetry of the cyclic polyene model, so that systems with N\ensuremath{\ge}14 can be explored. Comparison is made with the results of both full and limited configuration-interaction calculations and with the correlation energies obtained with certain approximate coupled-cluster schemes. Contrary to the six- and ten-membered rings, the OIP lower bounds to the ground-state correlation energy of cyclic polyenes with larger number of sites are very poor and cannot serve as a meaningful source of information. Since the eigenvalues of the intermediate Hamiltonian, in terms of which the OIP technique is defined, are highly degenerate even in the weakly correlated region, and in view of the random behavior of the OIP bracketing function, the OIP method becomes computationally very demanding for N\ensuremath{\ge}14 rings. Moreover, simple iterative schemes usually diverge and sophisticated root searching procedures have to be applied. The usefulness of the OIP technique for the determination of lower bounds to energy eigenvalues in larger systems is thus questioned.