A novel global optimization algorithm based on the smoothing function

. In this paper, a new global optimization algorithm based on the smoothing function is suggested. Firstly, one of the minimizer x ∗ 1 of the smooth function F is found by employing any local minimizer ﬁnder, then the function m ( x , x ∗ 1 ) : = min { F ( x ) , F ( x ∗ 1 ) } is considered instead of the objective function F and the minimizer x ∗ 2 of the function m is searched. Because of the non-smooth of the function m , the smoothing function for m is used in order to make the local minimizer ﬁnding algorithms for smooth optimization available. Finally, a global optimization algorithm based on the smoothing function is presented. The implementation of the algorithm on several test problems is reported with numerical results.


Introduction
We consider the box constrained global optimization problem as follows: min here F(x) : R n → R is a continuously differentiable function and D is a box.As F : R n → R is a coercive function, i.e., F(x) → +∞ when ||x|| → +∞, then the following problem: min x∈R n F(x) can be reformulated as the box constrained optimization problem in (1).
Global optimization has many applications in almost all fields of science and technology.Because a general non-convex objective function has many local minima, then it makes global optimization hard to solve ( [2,6,13]).Currently, Global optimization makes great progresses on theories and practical applications, and there are many methods have been presented, for example the integration-level set methods [1], the branch and bound methods [6], the outer approximation methods [7], the tunneling function methods [8], the interval methods [12], the filled function methods [3-5, 9-11, 14-22].Here, a novel global optimization algorithm based on the smoothing function is presented.
Most of global optimization algorithms must use local searches.The main difficulty for these types global optimization algorithms is to pass lower minimizers from the current one.Our minimization algorithm is also based on finding the lower local minimizer than the current local minimizer.Indeed, one of the minimizer x * 1 of the smooth function F is found by employing any local minimizer finder, then the function m(x) := min{F(x), F(x * 1 )} is considered instead of the objective function F and the minimizer x * 2 of the function m(x) is searched.This loop proceeds until the global minimizer is found.It has to be stated that, the function min{F(x), F(x * 1 )} has the same minimizers with the function F except x * 1 .Therefore, one of the main difficulties of global optimization process is eliminated.
On the other hand, the function m can be non-smooth because of the presence of "min" operator.Therefore, we propose using the smoothing functions for m in order to make the local minimizer finding algorithms for smooth optimization available.After smoothing process, the next local minimizer is found by considering any of local minimizer finder for smooth optimization.
The structure of the paper is as follows.After this introduction, a smoothing function for the function m(x, x * 1 ) = min{F(x), F(x * 1 )} is designed in section 2, and the properties of the smoothing function are analysed and discussed.Moreover, in section 3, a global optimization algorithm based on the smoothing function is proposed.Finally, the numerical experiment results are introduced in section 4. Compared with the numerical experiment results of [21], the results indicate that the global optimization algorithm in the paper is effective.Section 5 is the conclusion.

The smoothing function
Suppose that x * 1 is a current local minimizer of problem (1), then the smoothing function m(x, x * 1 ) := min{F(x), F(x * 1 )} is considered in stead of the objective function F and the minimizer x * 2 of the function m is searched.It can be stated that, the function m has the same minimizers with the function F except x * 1 .However, the function m can be non-smooth because of the presence of "min" operator.Therefore, we propose using the smooth-MATEC Web of Conferences 139, 00223 (2017) ing functions for m in order to make the local minimizer algorithms for smooth optimization available.
For the function m(x, x * 1 ) := min{F(x), F(x * 1 )}, we have the following: Thus, The following smoothing function is designed: where u > 0 and Proposition 2.1.For any (t, u) ∈ R 2 , we have the following results.

Global optimization algorithm
In this part, a novel global optimization algorithm for solving problem (1) is proposed based on the smoothing function S (x, x * 1 ), and the algorithm can find a global optimal solution or an approximate global optimal solution of problem (1).
The novel global optimization algorithm for problem (1) is described as follows.we refer the algorithm as SMGP(the global optimization algorithm based on the smoothing function for problem (1)).
Algorithm SMGP Step 0: Choose sufficiently small numbers u L > 0, λ L > 0. Choose a positive integer number I and directions e i , i = 1, . . ., I. Choose a starting point x 0 ∈ D. Set i := 0; Step 1: Seek for a local minimizer x * i of problem (1) by using local search methods starting from x i ; Step 2: where u > 0 and Set l = 1 and λ = 2; Step 3: Step 4: Seek for a local minimizer for the following optimization problem from the starting point y l i : Let y * i be a found local minimizer of problem (2).If Step 5: Let x s = x * i and stop.
Next, we give the explanation of the motivation and mechanism of the algorithm.
In Step 0, we choose the directions e i , i = 1, . . ., I as follows.For example, as n ≥ 2, set I = 2n, for i = 1, . . ., n, and the i-th component of e i is one, the other component of e i is zero; for i = n + 1, . . ., I, the i-th component of e i is −1, the other component of e i is zero.
In Step 0, we choose a starting point.Then search for a local minimizer x * i of problem (1) starting from the initial point by using a local minimization method.If x * i is not a global minimizer, then to find deeper local minimizers of problem ( 1) is our main task.
In step 3, we try to choose an appropriate initial point to find a local minimizer of problem (2) by using a local minimization method.If the set of directions {e i , i = 1, . . ., I} is large enough, no matter how to choose the initial points, local minimizers y * i of problem (2) will equal to x * i , and S (y * i , x * i , u) = f (x * i ), then we can say that the local minimizer x * i in Step 1 is an approximate global optimal solution of problem (1).When u is sufficiently small, the global optimal solution of problem (1) can be found.

Numerical experiments
In this part, several sets of numerical experiments are implemented and the results can show that Algorithm SMGP is effective.All the numerical experiments are implemented in Matlab2010b.We use The Matlab function 'fmincon' in the algorithm to search for local minimizers of problem (1) and problem (2).
The experiment results of each example are summarized in tables.The following symbols used in the tables are given: i The iteration number in finding the i-th local minimizer; x 0 i The initial point in the i-th iteration for searching for the i-th local minimizer of problem ( 1 and throughout our computational experiments, we set the parameters in Algorithm SMGP as follows:  I. Example 2. (The Goldstein and Price function in [3,22]). min where The Example has the global optimal solution x * = (0, −1) T and f (x * ) = 3.The computational results are summarized in Table II.

Conclusion
In this paper, A new global optimization algorithm based on the smoothing function is proposed.Firstly, one of the minimizer x * 1 of the smooth function F is found by employing any local minimizer finder, then the function m(x, x * 1 ) := min{F(x), F(x * 1 )} is considered instead of the objective function F and the minimizer x * 2 of the function m is searched.Because of the non-smooth of the function m, the smoothing function for m is used in order to make the local minimizer finding algorithms for smooth optimization available.Finally, a global optimization algorithm based on the smoothing function is proposed.The numerical experiments are carried out on several test problems.Compared with experiment results of [21], the results indicate Algorithm SMGP and Algorithm FFGP in [21] have the similar results, However the Cpu time of Algorithm SMGP is less than Algorithm FFGP in most case.Consequently Algorithm FFGP is effective.Nevertheless, the set of directions {e i , i = 1, . . ., I} in Algorithm SMGP is crucial.so without appropriate directions, The local minimizers of problem (2) can not be found.Next, we will pay attention to how to modify it.
(a)If l ≤ I, then go to 3(b); Otherwise, go to Step 5; (b)If λ ≥ λ L , then let y l i := x * i + λe l , go to 3(c); otherwise, let l := l + 1, λ = 2, go to 3(a); (c)If y l i ∈ D, then go to 3(d); otherwise, let λ := λ 2 , go to 3(b); (d)If F(y l i ) < F(x * i ), then let x i+1 := y l i , i := i + 1, go to Step 1; otherwise, go to Step 4; ); x * i The i-th local minimizer for problem (1); F(x * i ) The objective function value for x * i ; y 0 i The initial point in the i-th iteration for searching the i-th local minimizer of problem (2); y * i The i-th local minimizer for problem (2); F(y * i ) The objective function value for y * i ; CPU(s) The CPU time in seconds for the algorithms.

Table 1 .
Experiment results for Example 1

Table 2 .
Experiment results for Example 2

Table 3 .
Experiment results for Example 3