We present a generalization of the matrix effective potential (MEP) formalism presented previously [D. G. Truhlar and K. Onda, Phys. Lett. A 76, 119 (1980); K. Onda and D. G. Truhlar, Phys. Rev. A 22, 86 (1980)]. In the original MEP we used perturbation theory to define pseudochannels that gave the correct linear response of a system to a perturbing particle in the adiabatic limit at all geometries of approach. The pseudochannel was used in conjunction with the true ground-state channel for nonadiabatic calculations of the elastic scattering and the total inelastic cross section. In the present work we use a variational approach to define pseudochannels that give the correct infinite-order (full) response of the adiabatically perturbed system. The pseudochannels are used with more than one true channel of the system; this allows the study of selected inelastic processes as well as the total inelastic cross section. The new approach is compared to the original one for vibrationally inelastic collinear collisions of He with H2 and Cl2. It is also compared to conventional close coupling calculations; this comparison illustrates how the method can serve as the basis of a general convergence-acceleration technique for coupled-channels calculations of inelastic cross sections.