الفهرس 
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Abstract
We have presented a new trust-region algorithm for solving the nonlinear systems
of equalities and inequalities without any restriction on the number of the equalities or the
inequalities. This algorithm can be viewed as an extension to Waleed’s approach (Ref.
[23]).
Our algorithm uses Dennis, El-Alem and Williamson’s approach (Ref. [5]) for
treating the inequalities.
We have used three methods in our algorithm to solve the nonlinear system of
equalities and inequalities. The first method (Newton’s method) is used if we have a
square system of equations. The second method (Craig’s method) is used if we have an
under-determined system. The third method (Gauss Newton’s method) is used if we have
an over-determined system. To ensure global convergence, we have applied trust-region
method as a globalization strategy. This strategy will ensure convergence of the algorithm
from any starting point. We have used the dogleg Algorithm for computing the trial step.
We have implemented our algorithm using MATLAB software package version
7.0 and run under the Windows XP operating system. Our algorithm has been applied to
several nonlinear problems of deferent dimensions. The performance of the algorithm was
compared against the minimization approach algorithm (Algorithm 2.2).
We have found that the algorithm behaves predictably and reliably and its speed of
convergence was quite satisfactory. The algorithm merit further investigation.
For future work, there are many questions that should be answered.
Although we have implemented the algorithm and tested it, we believe that the
implementation of the algorithm should be refined with efficiency in mined. In particular a
better way in computing the trial steps should be suggested which ensures that the
algorithm can handle large-scale problems.
A related important question that has to be looked at is how can we prove that the
algorithm is globally convergent?
Appendix A
59.
APPENDIX
In this appendix, we list the problems which were used to test our algorithm. The
numerical results for these test problems were shown in Chapter 4. By HS (i) we mean the
constraint set of the test problem number i of Hock and Schittkowski (Ref. [12]). By S (i),
we mean the constraint set of the test problem number i of Schittkowski (Ref. [19]). In
some problems the symbol * is added to its name, to indicate that the corresponding
problems have been modified. Finally in some problems the symbol A is used to mean
external examples.
Test Problems
1- HS (40):
C1(x) = x3
1 + x2
2 - 1 = 0
C2(x) = x2
1 x4 - x3 = 0
C3(x) = x2
4 - x2 = 0.
2- HS (262):
C1(x) = x1 + x2 + x3 - 2 x4 - 6 = 0
C2(x) = -10 + x1 + x2 + x3 + x4 0
C3(x) = -10 + 0.2 x1 + 0.5 x2 + x3 + 2 x4 0
C4(x) = -10 + 2 x1 + x2 + 0.5 x3 + 0.2 x4 0
C5(x) = -x1 0
C6(x) = -x2 0
C7(x) = -x3 0
C8(x) = -x4 0.
3- HS (340):
C1(x) = -1.8 + x1 + 2 x2 + 2 x3 0
Appendix A
60