Comparative study between Reduced Space Searching RSS

Comparative study between Reduced Space Searching method RSS & Simplex Search method S^2:

SUPERVISOUR: ASST.PROF. MOHAMMED ALHANJOURI By RAED T. ALDAHDOoH MAHMOUD E. ALZAALAN Comparative study between Reduced Space Searching method RSS & Simplex Search method S^2

Introduction:

Introduction Simplex search S 2. Simplex search earliest Direct-search methods attempts to solve unconstrained problems by direct search produced methods modeled essentially on single variable methods. Reduce Space Searching RSS algorithms. (RSS) is a new search and optimization algorithm.

Simplex Search S^2 Method:

Simplex Search S^2 Method The method begins by setting up a regular simplex in the space of the independent variables and evaluating the function at each vertex. This ‘‘worst’’ vertex is then redirected through the centroid to generate a new point. iterations move along crabwise until either the minimum is straddled or the iterations begin to cycle between two or more simplexes.

Cont..:

Cont.. X 2 X 1 X 3 Old simplex X 1 , X 2 , and X3 New simplex X 2 , X 3 , and X4 X 4 X 2 X 1 X 3 M=1.65N + 0.05N 2

S^2 implementation :

S^2 implementation Generation of a regular simplex given a base point and appropriate scale factor. For i and j 1, 2, 3, . . . , N. The increments &1 and &2, which depend only on N

cont:

cont The second calculation, refection through the centroid, is equally straight forward. The new reflected pint calculation

Reduced Space Searching (RSS):

Reduced Space Searching (RSS) This new algorithm originates from an idea which relates to a simple experience when humans search for an optimal solution to a 'real-life' problem Develop an algorithm like RSS method we must first define how to portions the search space into small parts and how to rank each parts. How to divide search space into parts? How to rank the parts?

Cont (RSS):

Cont (RSS) The rectangular region 1 is the search space of an optimization problem. A and B tow point are selected randomly. If C solution is better than A, we set it as the solution and divide the original space around C with a smaller scale than that in (a) After several repetitions, one should obtain a 'good' solution.

( 1- 2 ) RSS method summarized as:

( 1- 2 ) RSS method summarized as Randomly select one point Xa (Xl, X2, •••, xN ) in the original search space as the solution Xbest. Randomly select another pointXb . If j( Xb ) <j(Xbest), then Xbest = Xb .

(3) RSS method summarized as cont..:

(3) RSS method summarized as cont.. Reduce the search space with the ratio of K (0 < K < 1, in this paper K = 1/2) and Xbest is located in the centre of the new space. Ymin is the lower bound of the ith variable in the new search space and Ymax is the upper bound. To avoid the new bounds stepping outside the original constraints as shown in Figure l(a), the following variables are defined: Y min = max(X min , Xbest( i ) - KL( i )); Y max = min(X maxi, Xbest( i ) + KL( i )); i = 1, 2, ... N; n is the counter of repetition of step 3; 1 < n <m; L( i ) = Xmax i – Xmin i Randomly select one point Xb in the new space. If j( Xb ) <j(Xbest), then Xbest= Xb .

(4) RSS method summarized as cont..:

(4) RSS method summarized as cont.. Repeat (3) m times, m depends on the precision needed and relates to the value of K. If K = 1/2, a value of m = 15~25 should prove adequate.

(5) RSS method summarized as cont..:

(5) RSS method summarized as cont.. Change the search space back to the original one and repeat (2 ~ 4) until the solution is found. In our applications, the process is stopped if the algorithm cannot locate a new Xbest after several loops.

Demo:

Demo

Results:

Results Lampa 2 Epsilon .01 α 2 Equation Initial Point [x,..] S 2 RSS X 2 3 1 0.006 14 0 0.00009 33 1 0.00007 (X 1 2 + X 2 - 11) 2 + (X 1 + X 2 2 - 7) 2 [5,5] 0.19 0.14 [7,-3] 0.74 0.69 [10.-5] 0.33 0.26 X 1 2 +2 X 2 2 - 3X 3 2 - 6X 1 X 2 + 8X 1 X 3 - 4X 2 X 3 [0,0,0] -148.4 -168752.09 100(X 2 + X 1 2 ) 2 + (1-X 1 ) 2 [-1.2,1] 27.2 28.2 [-1.3,1.07] 45.4 46.7 (X 1 + 10X 2 ) 2 +5(X 3 –X 4 ) 2 +(X 2 –2X 3 ) 4 +10(X 1 –X 4 ) 4 [3,-1,0,1] 24.84 24.82 [2,3,5,-6] 2.48 2.30 [0,0,0,0] 0 0.12

Conclusion:

Conclusion Reduced Space Searching is better than Simplex method in some cases: when the initial point is so far from the optimal. When the number of variables equal one. Simplex better than RSS as need number of iteration less than RSS