  This paper defines a convertible nonconvex function(CN function for short)
and a weak (strong) uniform (decomposable, exact) CN function, proves the
optimization conditions for their global solutions and proposes algorithms for
solving the unconstrained optimization problems with the decomposable CN
function. First, to illustrate the fact that some nonconvex functions,
nonsmooth or discontinuous, are actually weak uniform CN functions, examples
are given. The operational properties of the CN functions are proved, including
addition, subtraction, multiplication, division and compound operations.
Second, optimization conditions of the global optimal solution to unconstrained
optimization with a weak uniform CN function are proved. Based on the
unconstrained optimization problem with the decomposable CN function, a
decomposable algorithm is proposed by its augmented Lagrangian penalty function
and its convergence is proved. Numerical results show that an approximate
global optimal solution to unconstrained optimization with a CN function may be
obtained by the decomposable algorithms. The decomposable algorithm can
effectively reduce the scale in solving the unconstrained optimization problem
with the decomposable CN function. This paper provides a new idea for solving
unconstrained nonconvex optimization problems.
