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A novel filled function is given in this paper to find a global minima for a nonsmooth constrained optimization problem. First, a modified concept of the filled function for nonsmooth constrained global optimization is introduced, and a filled function, which makes use of the idea of the filled function for unconstrained optimization and penalty function for constrained optimization, is proposed. Then, a solution algorithm based on the proposed filled function is developed. At last, some preliminary numerical results are reported. The results show that the proposed approach is promising.

Recently, since more accurate precisions demanded by real-world problems, studies on global optimization have become a hot topic. Many theories and algorithms for global optimization have been proposed. Among these methods, filled function method is a particularly popular one. The filled function method was originally introduced in [

In general, there are two difficulties in global optimization: the first is how to leave the current local minimizer of

The rest of this paper is organized as follows. In Section

Consider the following problem (

In this section, we first list some definitions and lemmas from [

Letting

Let

for all

Considering problem (

the number of the different value of local minimizer of

Now, we give the definition of filled function for problem (

A function

If

Consider the problem (

Define

Next, we will prove that

Since

In this case, note that

In this case,

For any

For any

Suppose that Assumptions (1)–(

Since

Denotes

Therefore, one has

In the previous section, several properties of the proposed filled function are discussed. Now a solution algorithm based on these properties is described as follows.

Choose a disturbance constant

Choose an upper bound of

Choose a constant

Choose direction

Set

Start from an initial point

Let

Construct the filled function:

Increase

The motivation and mechanism behind the algorithm are explained as below.

A set of

In Step

Recall from Theorem

The proposed filled function method can also apply to smooth constrained global optimization.

In this section, we perform a numerical test to give an initial feeling of the potential application of the proposed function approach in real-world problems. In our programs, the filled function is of the form

The main iterative results of Algorithm NFFA applying on four test examples are listed in Tables

the

the

the function value of the

the function value of the

Numerical results for Problem 1.

1 | — | ( | 6.1184 | ( | 5.7164 |

2 | 1 | ( | 2.1433 | (0.0001, | |

3 | 10 | (0.0007, | (0.0000,0.0000) |

Numerical results for Problem 2.

1 | — | ( | 0.8125 | ( | |

2 | 1 | (1.1931,0.6332, | (1.9889, |

Numerical results for Problem 3.

1 | — | (2,2,2,2) | 42.0000 | (0.0000,1.0000,0.0000,2.0000) | |

2 | 1 | (0.6078,2.0003,0.0003,0.0319) | (0.9289,0.8620,0.2453,0.0803) | ||

3 | 100 | (0.4012,0.2524,0.2288,0.0000) | (0.0000,1.0000,1.0000,1.0000) |

Numerical results for Problem 4.

1 | — | (2,2,2,2,2,2) | (5,1,5,6,5,4) | ||

2 | 1 | (5,1,5,1.7581,5,4) | (5,1,5,0,5,4) | ||

3 | 1000 | (5,1,5,1.7579,5,10) | (5,1,5,0,5,10) |

We have

Algorithm NFFA succeeds in finding a global minimizer

We have

Algorithm NFFA successfully finds an approximate global solution

We have

Algorithm NFFA successfully finds a global solution

We have

The proposed algorithm successfully finds a global solution

In this paper, we extend the concept of the filled function for unconstrained global optimization to nonsmooth constrained global optimization. Firstly, we give the definition of the filled function for constrained optimization and construct a new filled function with one parameter. Then, we design a solution algorithm based on this filled function. Finally, we perform some numerical experiments. The preliminary numerical results show that the new algorithm is promising.

The paper was supported by the National Natural Science Foundation of China under Grant nos. 10971053, 11001248.