A New Unified Path to Smoothing Nonsmooth Exact Penalty Function for the Constrained Optimization

We propose a new unified path to approximately smoothing the nonsmooth exact penalty function in this paper. Based on the new smooth penalty function, we give a penalty algorithm to solve the constrained optimization problem, and discuss the convergence of the algorithm under mild conditions.


Introduction
We consider the following constrained optimization problem In the many penalty functions that have been proposed, the exact penalty function is often discussed, such as the 1 l penalty function and the p l penalty function. The classical 1 l penalty function (Zangwill [1]) is given as where 0 β > is a penalty parameter. The p l penalty function is given as where 0 β > is a penalty parameter and 0 1 p < < . But these exact penalty functions are often nonsmooth, which hampers the use of fast convergent algorithms such as the conjugate gradient method, the Newton method, and the quasi-Newton method. Many scholars have proposed smooth approximations to the classical exact penalty functions, which can be found in the references ([2]- [14]), and different penalty algorithms have been given to solve different optimization problems. In [10] [11] and [14], smooth approximations to 1 l penalty function were proposed for nonlinear inequality constrained optimization problems. Different smoothing penalty functions were also proposed in [13] to solve the global optimization problems. To solve the problem (P), [7] proposed two smooth approximations to the exact penalty function In [6] and [12], some smoothing techniques for the above exact penalty function were also given.
Smoothed penalty methods can also be applied to solve the optimization problems with large scale such as the network-structured problems and the minimax problems in [3], and the traffic flow network models in [8].
[5] gave a family of smoothing penalty functions to the 1 l penalty function and established a simple penalty algorithm. In this paper, a new unified smooth approximation path to the p l penalty function is proposed for the problem (P). On the basis of the proposed smoothing penalty functions, a new approximate algorithm is established, and the convergence of the algorithm is discussed under appropriate conditions. Remark 1 we assume in this paper that The above assumption is common since if it is not satisfied, then we can take the place of ( )

Approximately Smoothing Exact Penalty Functions
For the p l penalty function (3), we give a new family of smooth approximation in this section as follows, is a parameter and the function : Here we assume the function ψ satisfies the following properties: It is easy to show that the following functions are all examples of the function   Since ( ) 0 ψ ⋅ ≥ and ( ) ψ ⋅ is increasing, we can easily get the properties (b3) and (b4). 

Smooth Penalty Algorithm and Its Convergence
We propose an algorithm based on the penalty function Step 2. Let Step 3. Set  Thus 0 Ω can denote the feasible set of (P). In this paper we always suppose that 0 Ω ≠ ∅ . We denote the optimal solution set of (P) by * 0 Ω .
The perturbation function of (P) is defined as Open Journal of Optimization Now we give the following lemma. Let 0 x ∈ Ω , then ( ) 0 g x ≤ . By Proposition 2.1, we know that where 1 σ is given by Proposition 2.1.
For sufficiently large 0 k k ≥ , by Proposition 2.1, we have that which contradicts with (10). 