There are many methods
by which we change the vector U from one iteration to next as follow :-
1) 1st
gradient method ,and their modification
2) Optimal gradient method
Where:
is optimal step size length which can be obtained
from next items
3) Reduced optimal gradient
4) Optimal search direction
5)
Newton
Which also called 2nd order
gradient method
6) The reduced Hessian method
7) Conjugate direction method
8) P-Q decomposition method
negative reduced
gradient vector in the direction of steepest descent.
9) Improve of 2nd order
method
given the best results.
10) Using the Han-Powell method
and quadratic programming problem.
where S is obtained
from
11)
Decomposition and
system reduction
Where DV obtained from reduced
quadratic programming.
12) Modified O. G. M.
Where:
is gradient vector and
used the diagonal direction for energy six or seven iteration and
instead of
hence the speed of convergence is faster.
1-The Algorithm Of conjugate
i) Start with initial arbitrary point x1.
ii) Set the 1st search direction
iii) Find the
point x2 according to
iv) Find
v) Compute the
optimal step size length in direction of Si
in direction of Si
and find new point
vi) Test for
convergence and optimality of point Xi+1
If Xi+1 is optimal, stop the process, otherwise set the value of i =
i+1 and repeat steps (iv), (v), (vi) until the convergence is achieved .This
procedure is indicated in flow chart shown in the following figure:
13) Parallel tangent: for
minimization of f(x):
- Start with initial point X1.
- Find 1st search
direction
= Optimal step size length.
- Take new search direction as:
The new value of point is
14) Conjugate gradient method (Fletcher-Reeves method).
In order to minimize the round off error, we used
Newton
instead of the used form where
m = n+1 is no. of variables in spite of this, the
Fletcher-Reeves is vastly superior to the steepest descent method and pattern-search method (Parallel tangent) but it turn out be rather than the Quasi-Newton and variables metric methods but in Newton and Metric method, we need to storage a matrix of order n×n, hence if the storage is one of main consideration hence Fletcher-Reeves is not to be ignored.
Fletcher-Reeves is vastly superior to the steepest descent method and pattern-search method (Parallel tangent) but it turn out be rather than the Quasi-Newton and variables metric methods but in Newton and Metric method, we need to storage a matrix of order n×n, hence if the storage is one of main consideration hence Fletcher-Reeves is not to be ignored.
15) Quasi-Newton method:
2-Improved Newton Method
In some systems, the
Which has the following advantages:
-It reduces to no. of iterations.
-It find, the optimal point for all systems.
But the disadvantages are:
-It's required to storage matrix with
order n×n ---- [H]
-It's required to compute the elements
of H and it's inverse to
each iteration.
-It's required the multiplication
of 
at each iteration.
at each iteration.
16) Variables Metric method
(Davion-Fletcher powell method)
This is the best general
purpose optimization technique:
i.) Start with initial
point
ii.) Set i=1 and compute
iii.) Find the optimal stem length 
and get:
and get:
iv.) Test 
for optimality if it is optimal hence terminate the process if no we go
to
for optimality if it is optimal hence terminate the process if no we go
to
step
v.
v.) Set the new iteration number 
and go to step (ii).
and go to step (ii).
This method is powerful and
convergence quadratically (as conjugate
gradient method),and no need for 2nd
order derivatives or matrix
inversion.
Lagrange multiplier 
must be (+ve) for
minimization problem and (-ve) for maximization.
must be (+ve) for
minimization problem and (-ve) for maximization.Hi. I’m Designer of Engineering Topics Blog. I’m Electrical Engineer And Blogger Specializing In Electrical Engineering Topics. I’m Creative.I’m Working Now As Maintenance Head Section In An Industrial Company.
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