  This paper presents a piecewise convexification method to approximate the
whole approximate optimal solution set of non-convex optimization problems with
box constraints. In the process of box division, we first classify the
sub-boxes and only continue to divide only some sub-boxes in the subsequent
division. At the same time, applying the $\alpha$-based Branch-and-Bound
({\rm$\alpha$BB}) method, we construct a series of piecewise convex relax
sub-problems, which are collectively called the piecewise convexification
problem of the original problem. Then, we define the (approximate) solution set
of the piecewise convexification problem based on the classification result of
sub-boxes. Subsequently, we derive that these sets can be used to approximate
the global solution set with a predefined quality. Finally, a piecewise
convexification algorithm with a new selection rule of sub-box for the division
and two new termination tests is proposed. Several instances verify that these
techniques are beneficial to improve the performance of the algorithm.
