Operational Properties of SCN Function, Optimization Condition, and Exactness of Penalty Function for SCN Optimization

This paper defines a strong convertible nonconvex (SCN) function for solving the unconstrained optimization problems with the nonconvex or nonsmooth (nondifferentiable) function. First, the concept of SCN function is defined, where the SCN functions are nonconvex or nonsmooth. Second, the operational properties of SCN functions are proved, including addition, multiplication, and compound operations. Third, an SCN optimization with the SCN function and its penalty function is defined. The optimization condition, exactness, and stability of the penalty function for SCN optimization are proved. This paper provides a way for solving unconstrained nonconvex or nonsmooth (nondifferentiable) optimization problems to avoid using subdifferentiation or smoothing techniques.