Directly solving the multi-objective optimization problem, min [(Wst)(epsilon)], in the energy-minimization multiscale (EMMS) model for gas-solid fluidization is an unsolved problem. Currently the solution is indirect based on the original stability condition, N-st = min, which is a single-objective optimization problem and deduced by combing the physical meaning of dominant mechanisms W-st = min and epsilon = min. In this work, a direct solution framework of min [(Wst)(epsilon)] in the EMMS model is performed, and the results are compared with those of N-st = min. The excellent agreement between these two schemes indicates that the proposed new framework might be promising for solving all multi-objective optimization problems that are frequently encountered in engineering study. In this framework, the impact indexes of dominant mechanisms (W-st = min and epsilon = min in the EMMS model) are first introduced under the premise that these dominant mechanisms are of equal importance. Consequently, an extremal formulation can be established, which is the mathematical equivalent to N-st = min for the EMMS model. Such an extremal expression can be adopted as a stability condition to close the mesoscale model, which seems to be an alternative of N-st = min for the EMMS model. Encouraged by its success in the gas-solid fluidization, it is being verified in more systems. (C) 2018 Elsevier Ltd. All rights reserved.