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Format: PDF
Pages: 50
Content: Lagos State University Project Topic (Mathematics)
Date: JUNE 2022
Abstract:
ABSTRACT
Although it is a very old theme, unconstrained optimization is an
area which is always actual for many scientists.
Today, the results of unconstrained optimization are applied in
different branches of science, as well as generally in practice. Here,
we consider some important classes of unconstrained optimization
methods which are, Newton’s Method, Trust Region Method and
Conjugate Gradient Method.
These classes of methods are very interesting; it seems that they
are never out of date. This thesis clearly gives review of the meth-
ods mentioned above for solving unconstrained minimization of real-
valued functions.
An analysis of the pertinent mathematical theories and mini-
mization methods are presented and tested using a well known set
of benchmark problem.
Traditionally researchers use one of two computational tools when
seeking approximations to their numerical analysis and optimiza-
tion problems. They either use readily available software packages
or write their own tailor made programs using some high-level pro-
gramming languages.
Both of these are capable of handling fairly complicated and large
problems effectively.
In this thesis, we then illustrate the practical behavior of each
methods mentioned, and their respective algorithms are well stated
in Chapter 3 , we try to present, analyze, and state an illustrative
example with the use of MATHEMATICA.
Furthemore, the comparisons between the three unconstrained
optimisation methods are highlighted in the Result and Discussion
Chapter.
Table of Content:
Contents
Certification . . . . . . . . . . . . . . . . . . . . . . . . . . i
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1 INTRODUCTION 1
1.1 BACKGROUND OF STUDY . . . . . . . . . . . . . 2
1.2 STATEMENT OF PROBLEM . . . . . . . . . . . . 5
1.3 AIM AND OBJECTIVES OF STUDY . . . . . . . . 6
1.4 SIGNIFICANCE OF STUDY . . . . . . . . . . . . . 7
1.5 SCOPE OF STUDY . . . . . . . . . . . . . . . . . . 8
2 LITERATURE REVIEW 9
2.1 REVIEW OF RELATED LITERATURE . . . . . . . 9
3 METHODS OF STUDY 14
3.1 NEWTON’S METHOD . . . . . . . . . . . . . . . . 15
3.2 TRUST-REGION METHODS . . . . . . . . . . . . . 20
3.3 CONJUGATE GRADIENT METHOD . . . . . . . 22
vii
4 ANALYSIS, RESULTS AND DISCUSSIONS 28
4.1 ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 RESULTS . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2.1 COMPARISONS . . . . . . . . . . . . . . . . 39
4.3 DISCUSSIONS . . . . . . . . . . . . . . . . . . . . . 42
4.3.1 BENCH MARK PROBLEM . . . . . . . . . . 42
5 CONCLUSION 45
5.1 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 CONCLUSION REACHED . . . . . . . . . . . . . . 46
5.3 RECOMMENDATION . . . . . . . . . . . . . . . . . 46
5.4 SUGGESTIONS FOR FURTHER STUDIES. . . . . 47
References . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
viii
Introduction:
Chapter 1
INTRODUCTION
In our daily life, we always choose the best possible solutions
for several problems. We encounter the problems of maximum and
minimum. In mathematics, the study of maximum and minimum
problems began a very long time ago. There has been increasing in-
terest in the problem of minimizing functions of n variables numer-
ically. There were no uniform ways to find the maxima or minima
of problems.
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