Global Optimization Portal - Resource
[ GOP ]

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    GOP | Resource

The software program developed at GOP is easy to use. The user needs to prepare two input files to run the codes. Three examples are listed below. They are for one dimension, two dimensions and four dimensions. The so-called delta width (Δ) values employed in the examples are calculated by the sample number, domain range values and dimensionality. For every example, default delta width value is calculated and used for each dimension involved.

(1) One dimension (one variable) example: 12 input sample locations in design space and location range is between -10 and +10. They are listed below;

10 , 8.18 , 6.36 , 4.54 , 2.72 , 0.9 , -0.92 , -2.74 , -4.56 , -6.38 , -8.2 , -10

They need to be formatted as,

[[10],[8.18],[6.36],[4.54], [2.72],[0.9],[-0.92],[-2.74],[-4.56],[-6.38],[-8.2],[-10] ]

in accordance with Javascript format. That is, there are 12 brackets within one large bracket. Each bracket inside gives one location. The target function (which is listed at here) values at these12 input locations are formatted as,

[246.9,191.4,143.9,104.6,73.3,50.1,35,27.9,29,38.1,55.3,80.3]

There are 12 values,separated by commas, in one bracket. The first value "246.9" is for first location "10" and the last value "80.3" is for the last location "-10". All other values are assigned to locations in accordance with the same order in both lists.

(2) Two dimensions (two variables) example: 12 input sample locations in design space and location ranges of both variables are between 0 and +11. They are formatted and listed below;

[[0,4],[1,9],[2,1],[3,6], [4,11],[5,3],[6,8],[7,0],[8,5],[9,10],[10,2],[11,7] ]

There are 12 brackets within one large bracket. Each bracket inside gives one location consisting of two values. The target function (which is listed at here values at these12 input locations are formatted as,

[-31,-37,-20.54,-39.9,-12.48,-27.1,13.1,1.47,2.04,49.44,32.9,66.9]

The first value "-31" is for first location "0,4" and the last value "66.9" is for the last location "11,7". All other values are assigned according to their order in the location list.

(3) Four dimensions (four variables) example: 72 input sample locations in design space are listed in File A and 72 target function values are listed in File B.

Input data format for higher dimensions and sample numbers ought to be generalized in accordance with the format stated above.


Using GOP software with the input data listed above, the optimum locations found, together with the initial starting location and the optimization iteration number used, are reported below.

For the one-dimension example, two optimization runs are made. the first one with delta=0.53 (default value), the second one with delta=1.5. (The true optimum locaiton is at -3.41) 

Initial location       Optimum location       Iteration number
        0		-1.22			5
       -7		-4.49			8
       -8		-4.49			12
        8		-2.79			17
        5		-2.79			16
       -5		-4.49			5
       -1		-1.22			9
		
Initial location       Optimum location       Iteration number
        0		-3.78			28
       -7		-3.81			28
       -8		-3.81			29
        8		-3.78			34
        5		-3.78			32
       -5		-3.8			23
       -1		-3.78			27
It appears that with the larger delta width, the local optimum locations have been removed and the global (single) optimum can be found (near -3.8). To enlarge delta width will reduce the interpolation accuracy but increase the opportunity to track down the global optimum with more iterations.

For the two dimension example stated above, the global minimum is located at (1.74, 6.37). That is, x1=+1.74, x2=+6.37. The sample data locations and the global minimum are presented in the picture below.



Two optimization runs are made. The first one with delta width=1.01 (default value) and the second one with delta width=1.5. Results are given below.

Initial location	Optimum location	Iteration number
       5, 5		2.8, 5.96			8
       1, 6		2.79, 5.96			15
       2, 6		2.79, 5.96			9
       3, 6		2.8, 5.96			5
       4, 6		2.8, 5.96			7
       0, 5		2.79, 5.96			26
       7, 8		2.79, 5.96			19
       7, 2		2.8, 5.96			24
       8, 8		2.79, 5.96			18
       9, 4		2.8, 5.96			27
      10, 5		2.8, 5.96			28
      11, 5		2.8, 5.96			30
       1, 8		0.8, 8.9			11
       0, 6		2.79, 5.96			21
       1, 10		0.79, 8.9			9
      10, 1		2.8, 5.96			38


Initial location	Optimum location	Iteration number
       5, 5		2.35, 5.91			19
       1, 6		2.32, 5.91			30
       2, 6		2.32, 5.91			16
       3, 6		2.35, 5.91			14
       4, 6		2.35, 5.91			17
       0, 5		2.32, 5.91			51
       7, 8		2.35, 5.91			18
       7, 2		2.35, 5.9			27 
       8, 8		2.35, 5.91			20 
       9, 4		2.35, 5.9			29
      10, 5		2.35, 5.9			29
      11, 5		2.3, 5.9			32
       1, 8		0.57, 8.75			17
       0, 6		2.32, 5.9			42
       1, 10		0.56, 8.76			14
       10, 1		2.33, 5.9			41

It can be seen that both runs produced two optimum locations. It is straightfoward to use the target function to select the best optimum at (2.3, 5.9).

The picture below indicates how the optimization process begins at (10, 1) location and proceeds in design space to the final location at (2.3, 5.9) with 41 iterations/steps.



4-dimension test case presented here by using the target function listed in this page. All 4 variables are bounded between -10 and +10. 72 samples are randomly selected by use of Sobol low-discrepancy sequence, together with 72 target function values. The global minimum location is located at (2.5, 2.5, 5.25, -3.5).

Two optimization runs are made. The first one with delta width=2.18 (default value) and the second one with delta width=3. Results are shown below. 

Initial location 		Optimum location	Iteration #
2.52, 6.38,-6.76, 1.6		1.8,4.8,5.6,-0.4	14
0, 0, 0, 0			1.8,4.8,5.6,-0.4	22
4.001,4.001,4.001,4.001		1.8,4.8,5.6,-0.4	13
-4,-4,-4,-4			-1.9,-2.5,3.5,-7	23
-4,-4,4,4			1.8,4.8,5.6,-0.4	17
4,4,-4,-4			4,6.2,6.5 -6.1		21
4,-4,4,-4			3.9,6.1,6.5,-6		81
4,-4,-4,4			1.8,4.8,5.6,-0.4	33
-4,4,4,-4			1.8,4.8,5.6,-0.4	26
1, 3, 5, 7			1.8,4.8,5.6,-0.4	13
7, 5, 3, 1			1.8,4.8,5.6,-0.4	21
-7,5,-3, 1			1.8,4.8,5.6,-0.4	32
7, -5,-3,1			3.9,6.1,6.5,-6.1	88
0,7,7,-7			3.9,6.2,6.5,-6.1	17

Initial location 		Optimum location	Iteration #
2.52, 6.38,-6.76, 1.6		2.7,5.4,5.9,-1.3	22	
0, 0, 0, 0			2.7,5.4,5.9,-1.3	33	
4.001,4.001,4.001,4.001		2.7,5.4,5.9,-1.3	23	
-4,-4,-4,-4			4.1,6.3,6.8,-6.2	58
-4,-4,4,4			2.7,5.4,5.9,-1.3	29
4,4,-4,-4			4.1,6.3,6.8,-6.2	28
4,-4,4,-4			4.1,6.3,6.8,-6.2	53
4,-4,-4,4			2.7,5.4,5.9,-1.3	37
-4,4,4,-4			4.1,6.4,6.8,-6.2	35
1, 3, 5, 7			2.7,5.4,5.9,-1.3	31
7, 5, 3, 1			2.7,5.4,5.9,-1.3	46
-7,5,-3, 1			2.7,5.4,5.9,-1.3	43
7, -5,-3,1			4.1,6.3,6.8,-6.2	58
0,7,7,-7			4.1,6.4,6.8,-6.2	22

It can be seen that the second run removed one local optimum in the first run. There are still two optima found. Again, use target function to select the best optimum at (2.7,5.4,5.9,-1.3).

Two projections from the used Sobol's 4-dimensional 72 locations onto 2-dimensional domain space are presented in the following pictures. As can be seen, sample locations are distributed fairly uniformly across the entire domain (no cluster).



According to the methodology presented in RDIC, the following table presents the default delta (Δ) width values with respect to the sample number and the target function dimensionality. For the hypercube with side length L in m-Dimension, the default delta width can be calculated as follows: N is the number of sample data points, m is the dimension number and the formula is,



		m	N	L	delta
		1	12	20	0.53
		2	12	11	1.01
		4	72	20	2.18
		10	100	20	4.0
		10	500	20	3.4
		14	1,024	20	3.69
		18	2,048	20	4.01
		20	100,000	20	3.56
		100	3x1010	20	5.0




GOP is a by-product of RDIC. The optimum locations found are "local optima". The user needs to examine the results. If two or more local optima are close to one another, then it is likely to have a global optimum among their locations. The following diagrams (close-up views) explain and show what is happening in one dimension using the same target function presented above. Note that the target function is a parabola.





Suggestions & Comments:
  • User needs to read/review the methodology given in RDIC before using GOP software.
  • Reduce the dimensionality (independent variables) of the problem.
  • Input locations must be scaled with the same domain range. That is the same length "L" in hypercube Lm, m is the dimension number.
  • Choose the sampling methods/pattern (random, periodic, latin hypercube, sparse grid, Sobol sequence, etc.)
  • Cut down the input sample number according to the tolerance and the accuracy requirements.
  • With limited samples, GOP can be used as a baseline surrogate model. With ample samples, GOP delivers results according to the Central Limit Theorem in statistics.
  • Default delta width value helps prevent overfitting issue.
  • A couple of more optimizaiton runs by using a slightly larger (than default) delta width values can help choose the best optimum location.
  • Unlike machine learning algorithm, GOP software needs no training. It can produce results once the input sample data are given .
  • GOP software does not need fine tuning and no hyper-parameter optimization employed.
  • The delta width square (Δ2) used in GOP corresponds to the so-called "R2" term employed in the Radial-Basis-Function (RBF) method, For example, in the two dimension example presented above gives the default delta value1.01. In comparison with the RBF method with the same size of domain space and sample number, it yields the R value about 0.89 which is compatible to 1.01.
  • GOP software delivers the global optimum in accordance with the sample number and the dimensionality with minimum computing resource consumption.

 

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(All information including software provided at GOP should be used by the discretion of the user.
FANG, INC. is not responsible for the outcome of the usage of GOP.)

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