Exact Penalty Algorithm of Strong Convertible Nonconvex Optimization

TL;DR

This paper introduces a novel exact penalty algorithm for strong convertible nonconvex functions, enabling efficient solutions to unconstrained nonconvex and nonsmooth optimization problems without subdifferentiation.

Contribution

The paper defines SCN functions, explores their properties, and develops an exact penalty algorithm with proven convergence for solving complex nonconvex optimization problems.

Findings

SCN functions include many nonconvex and nonsmooth examples.

The proposed algorithm converges to solutions of minmax reformulations.

Avoids the need for subdifferentiation or smoothing techniques.

Abstract

This paper defines a strong convertible nonconvex(SCN) function for solving the unconstrained optimization problems with the nonconvex or nonsmooth(nondifferentiable) function. First, many examples of SCN function are given, where the SCN functions are nonconvex or nonsmooth. Second, the operational properties of the SCN functions are proved, including addition, multiplication, compound operations and so on. Third, the SCN forms of some special functions common in machine learning and engineering applications are presented respectively where these SCN function optimization problems can be transformed into minmax problems with a convex and concave objective function. Fourth,a minmax optimization problem of SCN function and its penalty function are defined. The optimization condition,exactness and stability of the minmax optimization problem are proved. Finally, an algorithm of penalty function to solve the minmax optimization problem and its convergence are given. This paper provides an efficient technique for solving unconstrained nonconvex or nonsmooth(nondifferentiable) optimization problems to avoid using subdifferentiation or smoothing techniques.