UNCONSTRAINED NONSMOOTH NONCONVEX OPTIMIZATION BASED ON CN FUNCTION

Finding a global optimal solution to an unconstrained nonsmooth nonconvex optimization is not an easy job. To tackle the problem, we introduce in this paper the concept of CN function. Examples are given to show that some nonsmooth or nonconvex functions are actually CN functions. Furthermore, operations such as addition, subtraction, multiplication or division on CN functions still lead to CN functions. Sufficient conditions of optimal solution to unconstrained optimization based on CN function are presented, which are equivalent to Karush-Kuhn-Tucker (KKT) condition. Lagrange function and proper Lagrange function based on CN function are defined respectively, whose dual problems and strong duality properties show that the global optimal objective function value of the dual problem is equal to the global optimal objective function value of the CN optimization. Augmented Lagrangian penalty function and proper augmented Lagrangian pena1ty function of a CN function are a1so introduced respectively. With these augmented functions, we devise an augmented Lagrangian penalty function method to find the optimal solution to the CN optimization. Overall, this paper provides a new approach to solving unconstrained optimization problems. We obtain some new results on nonconvex nonsmooth unconstrained optimization problems without using subdifferential.

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