This paper presents an image space branch-reduction-bound algorithm for solving a class of multiplicative problems (MP). First of all, by introducing auxiliary variables and taking the logarithm of the objective function, an equivalent problem (EP) of the problem (MP) is obtained. Next, by using a new linear relaxation technique, the parametric linear relaxation programming (PLRP) of the equivalence problem (EP) can be established for acquiring the lower bound of the optimal value to the problem (EP). Based on the characteristics of the objective function of the equivalent problem and the structure of the branch-and-bound algorithm, some region reduction techniques are constructed for improving the convergence speed of the algorithm. Finally, the global convergence of the algorithm is proved and its computational complexity is estimated, and numerical experiments are reported to indicate the higher computational performance of the algorithm.

This article presents and validates a potential practical algorithm for minimizing the sum of affine fractional functions over a polyhedron. During the branch and bound search, this algorithm computes the lower bounds by solving the affine relaxation problems of the equivalent problem, which are derived by utilizing a two-level affine relaxation technique. By successive refinement and successively solving a series of affine relaxation problems, the algorithm is convergent to the global minimum of the primal problem. Moreover, the gap between the objective function and its affine relaxation function is derived for the first time, and when the partition interval tends to be infinitesimal, the gap is infinitely close to zero. Furthermore, the maximum iterations of the proposed algorithm are derived by estimating its computational complexity. Some test problems are solved to verify the potential practical and computational advantages of the algorithm.

This article presents and validates a new branch-and-bound algorithm for effectively solving the generalized polynomial problem (GPP). In this algorithm, a new affine relaxed technique is derived for establishing the relaxed linear programs problem of the GPP. In addition, some box reducing manipulations are employed to improve the speed of branch-and-bound search of the algorithm. Combining the relaxed linear programs problem with the box reducing manipulations, a new branch-and-bound algorithm is constructed. Some numerical examples are solved to verify the potential practical and computing advantages of the algorithm. At last, several engineering design problems are solved to validate the usefulness of the algorithm.