A novel filled function is constructed to locate a global optimizer or an approximate global optimizer of smooth or nonsmooth constrained global minimization problems. The constructed filled function contains only one parameter which can be easily adjusted during the minimization. The theoretical properties of the filled function are discussed and a corresponding solution algorithm is proposed. The solution algorithm comprises two phases: local minimization and filling. The first phase minimizes the original problem and obtains one of its local optimizers, while the second phase minimizes the constructed filled function and identifies a better initial point for the first phase. Some preliminary numerical results are also reported.
Science and economics rely on the increasing demand for locating the global optimization optimizer, and therefore global optimization has become one of the most attractive research areas in optimization. However, the existence of multiple local minimizers that differ from the global solution confronts us with two difficult issues, that is, how to escape from a local minimizer to a smaller one and how to verify that the current minimizer is a global one. These two issues make most of global optimization problems unsolvable directly by classical local optimization algorithms. Up to now, various kinds of new theories and algorithms on global optimization have been presented [
The filled function approach, initially proposed for smooth optimization by Ge and Qin [
The paper is organized as follows. In Section
In the rest of this paper, the generalized gradient of a nonsmooth function
Consider the following nonsmooth constrained global optimization problem
To proceed, we assume that the number of minimizers of problem
A function
for any
if
Definition
We propose in this section a oneparameter filled function as follows:
Theorems
Assume that
Since
The above discussion indicates that
Assuming that
For any
Assume that
By the conditions, there exists an
Then, for any
The proof is by contradiction. Suppose that
Based on the properties of the proposed filled function, we now give a corresponding filled function algorithm as follows.
Set a disturbance constant
Select an upper bound of
Select directions
Set
Start from an initial point
Let
Construct a filled function at
If
If
If
Increase
If
At the end of this section, we make a few remarks below.
(1) Algorithm FFAM is comprised of two stages: local minimization and filling. In stage 1, a local minimizer
(2) The motivation and mechanism behind the algorithm FFAM are given below.
In Step (3) of the Initialization Step, we can choose directions
In Step (1) and Step (6) of the Main Step, we can obtain a local optimizer of the problem
(3) The proposed filled function algorithm can also be applied to smooth constrained optimization. Any smooth local minimization procedure in the minimization phase can be used, such as conjugate gradient method and quasiNewton method.
We perform the numerical tests for three examples. All the numerical tests are programmed in Fortran 95. In nonsmooth case, we search for the local minimizers by using Hybrid Hooke and JeevesDirect Method for Nonsmooth Optimization [
The following are the three examples and their numerical results. And the symbols used in the tables are explained below:
Consider
Computational results with initial point (−1, −1, −1).





1  (−1, −1, −1)  ( 

2  (1.1931, 0.6332, −1.1932)  (1.9889, −0.0001, −0.0111) 

Consider
Computational results with initial point (−1, −1).





1  (−1, −1)  (−15.0000, 0.0000)  5.7164 
2  (−1.0585, 0.5165)  (0.0001, −0.2094)  −0.3690 
3  (0.0007, −0.0435)  (0.0000, 0.0000)  −2.7183 
Consider
Computational results with initial point (0, 0).





1  (0, 0)  (0.6116, 3.4423)  −4.0541 
2  (2.1653, 2.2546)  (2.3295, 3.1780)  −5.5081 
In this paper, we present a new filled function for both nonsmooth and smooth constrained global optimization and investigate its properties. The filled function contains only one parameter which can be readily adjusted in the process of minimization. We also design a corresponding filled function algorithm. Moreover, in order to demonstrate the performance of the proposed filled function method, we make three numerical tests. The preliminary computational results show that the proposed filled function approach is promising.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the NNSF of China (nos. 11471102 and 11001248), the SEDF under Grant no. 12YZ178, the Key Discipline “Applied Mathematics” of SSPU under no. A30XK1322100, and NNSF of Zhejiang (no. LY13A010006).