Optimization Conditions and Decomposable Algorithms for Convertible Nonconvex Optimization

TL;DR

This paper introduces a new class of nonconvex functions called convertible nonconvex functions, establishes their optimization conditions, and proposes decomposable algorithms with proven convergence for solving related unconstrained problems.

Contribution

It defines CN functions and their properties, proves optimization conditions, and develops a decomposable algorithm with convergence analysis for nonconvex optimization.

Findings

Numerical results demonstrate the effectiveness of the decomposable algorithms.

The algorithms can approximate global solutions efficiently.

The approach reduces problem scale and complexity.

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 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.