Updating Of Control Variable U



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 climbing method


       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


where is optimal step size length.


iv) Find
  and set    


v)  Compute the optimal step size length   in direction of Si
                      and find new point  
.
vi) Test for convergence and optimality of point Xi+1




or  





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


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.


15) Quasi-Newton method:

          Newton method since at the min solution X* for continuously differentiable function f(x) satisfied the necessary condition























2-Improved Newton Method

In some systems, the Newton method is diverged to:


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.

16) Variables Metric method (Davion-Fletcher powell method)

This is the best general purpose optimization technique:
i.) Start with initial point
 and n×n +ve definite symmetrical (Matrix H1(may be unit matrix)).

ii.) Set i=1 and compute 

iii.) Find the optimal stem length   and get:




iv.) Test   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).
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.

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Ahmad Mohammad

Hi. I’m Designer of Mawsoo3a 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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