In this paper, we consider a class of mixed integer weakly concave
programming problems (MIWCPP) consisting of minimizing a difference of a
quadratic function and a convex function. A new necessary global optimality
conditions for MIWCPP is presented in this paper. A new local optimization method
for MIWCPP is designed based on the necessary global optimality conditions,
which is different from the traditional local optimization method. A global optimization
method is proposed by combining some auxiliary functions and the new
local optimization method. Furthermore, numerical examples are also presented to
show that the proposed global optimization method for MIWCPP is efficient.