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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. [ABSTRACT FROM AUTHOR]