Genetically controlled random search: a global optimization method for continuous multidimensional functions

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Abstract

A new stochastic method for locating the global minimum of a multidimensional function inside a rectangular hyperbox is presented. A sampling technique is employed that makes use of the procedure known as grammatical evolution. The method can be considered as a “genetic” modification of the Controlled Random Search procedure due to Price. The user may code the objective function either in C++ or in Fortran 77. We offer a comparison of the new method with others of similar structure, by presenting results of computational experiments on a set of test functions.

Program summary

Title of program: GenPrice

Catalogue identifier:ADWP

Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADWP

Program available from: CPC Program Library, Queen's University of Belfast, N. Ireland

Computer for which the program is designed and others on which it has been tested: the tool is designed to be portable in all systems running the GNU C++ compiler

Installation: University of Ioannina, Greece

Programming language used: GNU-C++, GNU-C, GNU Fortran-77

Memory required to execute with typical data: 200 KB

No. of bits in a word: 32

No. of processors used: 1

Has the code been vectorized or parallelized?: no

No. of lines in distributed program, including test data, etc.:13 135

No. of bytes in distributed program, including test data, etc.: 78 512

Distribution format: tar. gz

Nature of physical problem: A multitude of problems in science and engineering are often reduced to minimizing a function of many variables. There are instances that a local optimum does not correspond to the desired physical solution and hence the search for a better solution is required. Local optimization techniques are frequently trapped in local minima. Global optimization is hence the appropriate tool. For example, solving a nonlinear system of equations via optimization, employing a “least squares” type of objective, one may encounter many local minima that do not correspond to solutions, i.e. minima with values far from zero.

Method of solution: Grammatical Evolution is used to accelerate the process of finding the global minimum of a multidimensional, multimodal function, in the framework of the original “Controlled Random Search” algorithm.

Typical running time: Depending on the objective function.

Introduction

A recurring problem in many applications is that of finding the global minimum of a function. This problem may be stated as: Determinex=argminxSf(x). The nonempty set SRn considered here, is a hyper box defined as:S=[a1,b1][a2,b2][an,bn]

Recently several methods have been proposed for the solution of the global optimization problem. These methods can be divided in two main categories, deterministic and stochastic. Random search methods are widely used in the field of global optimization, because they are easy to implement and also since they do not depend critically on a priori information about the objective function. Various random search methods have been proposed, such as the Random Line Search [1], Adaptive Random Search [2], Competitive Evolution [3], Controlled Random Search [4], Simulated Annealing [5], [6], [7], [8], Genetic Algorithms [9], [10], Differential Evolution [11], [12], methods based on Tabu Search [23], etc. This article introduces a new sampling technique for use with conjunction with Controlled Random Search. The method is based on the genetic programming procedure known as Grammatical Evolution. Performance comparison to other methods is quite favorable as might be verified by inspecting the reported results of our computational experiments in Table 1, Section 3.2. The suggested approach uses a population of randomly created moves, that guide the underlying stochastic search towards the global minimum. These random moves are produced by applying the method of grammatical evolution. Grammatical evolution is an evolutionary process that can produce code in an arbitrary language. The production is performed using a mapping process governed by a grammar expressed in Backus Naur Form. Grammatical evolution has been applied successfully to problems such as symbolic regression [14], discovery of trigonometric identities [15], robot control [16], caching algorithms [17], financial prediction [18], etc. The rest of this article is organized as follows: in Section 2 we give a brief presentation of the grammatical evolution and of the suggested algorithms. In Section 3 we list some experimental results from the application of the proposed method and a comparison is made against traditional global optimization methods and in Section 4 we present the installation and the execution procedures of the GenPrice.

Section snippets

Grammatical evolution

Grammatical evolution is an evolutionary algorithm that can produce code in any programming language. The algorithm requires the grammar of the target language in BNF syntax and the proper fitness function. Chromosomes in grammatical evolution, in contrast to classical genetic programming [20], are not expressed as parse trees, but as vectors of integers. Each integer denotes a production rule from the BNF grammar. The algorithm starts from the start symbol of the grammar and gradually creates

Experimental results

The Genetically Controlled Random Search (GCRS) was tested against

  • 1.

    The original Controlled Random Search (CRS).

  • 2.

    The modified Controlled Random Search (PCRS) as described in [26].

We list also results from the Simulated Annealing (SA) as modified by Goffe et al. [8] not for immediate comparison since the methods are quite different, but only as a reference point (their code simann.f is available from the URL: http://www.netlib.org).

The comparison is made using a suite of well-known test problems.

Distribution

The package is distributed in a tar.gz file named GenPrice.tar.gz and under UNIX systems the user must issue the following commands to extract the associated files:

  • 1.

    gunzip GenPrice.tar.gz.

  • 2.

    tar xfv GenPrice.tar.

These steps create a directory named GenPrice with the following contents:

  • 1.

    bin: A directory which is initially empty. After compilation of the package, it will contain the executable make_genprice.

  • 2.

    examples: A directory that contains the test functions used in this article, written in ANSI

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    This paper and its associated computer program are available via the Computer Physics Communications homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).

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