A hybrid linearized approximate proximal point algorithm

In this article, a new approximate proximal point method for minimizing the sum of two convex functions is introduce. It replaces the optimization problem in the proximal point algorithm by a series of subproblems whose objective functions are approximal functions to get the optimal solution of the original optimization problem. In one subproblem, the function with less nonlinearity is replaced by its linearly approximating function and in the next subproblem, the function with more nonlinearity is replaced by the maximum of two linear functions, and so on. Three numerical examples are given to illustrate the effectiveness of the present algorithm.