Optimization conditions and decomposable algorithms for convertible nonconvex optimization

Abstract

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 decomposable CN functions. 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 CN functions are proved, including addition, subtraction, multiplication, division, and compound operations. Second, optimization conditions of the global optimal solution to the unconstrained optimization with weak uniform CN function are proved. Based on the unconstrained optimization problem with decomposable CN functions, a decomposable algorithm is proposed by its augmented Lagrangian penalty function and its convergence is proved. Numerical results demonstrate that an approximate global optimal solution to unconstrained optimization with CN function may be obtained by the decomposable algorithm. The decomposable algorithm can effectively reduce the scale in solving the unconstrained optimization problem with decomposable CN function. This paper provides a new idea for solving unconstrained nonconvex optimization problems.