A revised cut-peak function method for box constrained continuous global optimization

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Abstract

In this paper, we modify the concept of cut-peak function given in Wang et al. [Y. Wang, W. Fang, T. Wu, A deterministic algorithm of global optimization using cut-peak functions, Technique Report, in: The Conference of Mathematical Programming of China, 2006], and then propose a revised cut-peak function algorithm for solving box constrained continuous optimization problems. The smoothing technique is used to overcome the difficulty arising from the non-smoothness of the constructed function. By using the exterior penalty function method we find iteratively a better minimizer from the current local minimizer till a global minimizer of the concerned problem is found. Some preliminary numerical results are reported.

Introduction

Global optimization is concerned with the theory and algorithms on seeking global minimizers of multi-modal functions. It has wide applications in almost all fields of engineering, finance, management as well as social science. Many methods have been proposed in the literature to solve the global optimization problems, however, how to solve such problems efficiently is still a problem with a great challenge. The existing literature in the field of global optimization can be usually divided into two classes: stochastic methods and deterministic methods. Usually, deterministic methods are more efficient than stochastic methods.

In the last decades, some deterministic method (such as the filled function method [1], [2], [3], [7], [8], [9], [10], [11], [13], [14]) has attracted much interest in the field of global optimization. The basic ideas of these methods is that they find a sequence of local minimizers with monotonically decreasing objective function, where the minimization sequence could leave from a local minimizer to a better minimizer of objective function through minimizing an auxiliary function constructed at the local minimizer. In the numerical experiments, one of the main difficult points is how to give a proper termination rule.

Recently, Wang et al. [15] proposed a cut-peak function method for solving unconstrained continuous global optimization (see Section 2 for the details). The implementation of this method is relatively easier than those existing in deterministic methods. However, when all other minimizers of the concerned problem is far away from the current local minimizer, it is possible that the global minimizers are cut down so that the global minimizer cannot be found.

In this paper, we give a modification of the concept of cut-peak function introduced in [15], and then propose a revised cut-peak function method for solving box constrained continuous global optimization. The revised cut-peak function method may overcome the disadvantage mentioned above. To show the efficiency of the method, we will report some numerical results of the method for solving fifteen testing problems which were tested in some recent literature. For every problem tested in this paper, the method presented here can find a global minimizer of the concerned problem.

This paper is organized as follows: In Section 2, we review some concepts and ideas of cut-peak function method in [15]. In Section 3, we describe the idea of the revised cut-peak function method and present a revised cut-peak function algorithm. In Section 4, we give some numerical implementation of the algorithm. Some conclusions are drawn in Section 5.

Section snippets

Cut-peak function method

In this section, we review some concepts and the idea of the cut-peak function method introduced in [15].

In [15] the authors considered the following unconstrained continuous optimization:minxΩf(x),where f is Lipschitz continuous and differentiable on a compact region ΩRn. The following important concepts were introduced.

Definition 2.1

w(r,x(k),x) is said a cut-peak function of f at the point x(k) with a positive parameter r if the following two conditions are satisfied:

  • (i)

    x(k) is the unique maximum point of w(

Basic ideas and algorithm

In this paper, we consider the following box constrained continuous optimization:minf(x)s.t.lxu,where l,uRn with l<u, and f is differentiable on [l,u].

Let x(k) be a local minimizer of (3.1), which can be found by some usual local optimization methods.

Definition 3.1

w(x(k),x)f(x(k)) is said a revised cut-peak function of f at the point x(k).

It should be noted that the function w(x(k),·) in Definition 3.1 is not satisfied the conditions of Definition 2.1, and hence it is not cut-peak function according to

Testing problems

We will test the following problems, which were tested in the literature of various global optimization methods.

Problem 4.1

The three-hump camelback problem [1], [14]:f(x)=2x12-1.05x14+x16/6-x1x2+x22,l=(-3,-3)T,u=(3,3)T.The known global minimizer is x=(0,0)T with the optimal value is f(x)=0. The initial point in our numerical computation is (2,2)T.

Problem 4.2

The six-hump camelback problem [6], [1], [8], [14], [15]:f(x)=4x12-2.1x14+x16/3+x1x2-4x22+4x24,l=(-3,-1.5)T,u=(3,1.5)T.The two known global minimizers are

Some remarks

In this paper, we propose a revised cut-peak function algorithm for solving box constrained continuous global optimization problems. The new algorithm has a simple termination rule. Some preliminary numerical results are reported.

From our numerical results, it is not difficult to see that

  • the algorithm may find a global minimizer for every tested problem given in this paper; and

  • the numbers of evaluations of the objective function and its gradient is less than those in the cut-peak function

Acknowledgements

This work was partially supported by the National Nature Science Foundation of China (No. 10571134) and the Science and Technology Development Plan of Tianjin (No. 06YFGZGX05600). This work was also partially supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry and the Scientific Research Foundation of Tianjin University for the Returned Overseas Chinese Scholars.

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