Hi,
(I posted this in stack exchange 2 days back but didn't get a response. Hope it works here).
I'm trying to generate start points for my optimization problem in Matlab. At this point Im not worried about feasibility but only a fast and yet systematic way to generate the points from which I could test the performance of different algorithms available within fmincon. Though am not worried about feasibility I don't want to use random number generator.
One approach that I have been using successfully (which I guess is a very common approach) is, if I have 2 decision variables with given min/max range for each of the 2 decision variables and let's suppose I want 400 start points. I divide each of the individual decision variables min/max range to Square root of 400 thereby giving 20 points in one decision variable and 20 in another decision variable and for all the possible combinations I will have 400 points. I'm considering alternate approaches to generating start points and one of them is described below.
As a simple example, assume I have 2 decision variables x1 and x2 and I can provide the min/max ranges for each of these 2 variables.
Like,
One start point that I can use is
Random_Point1=(x1ow+xhigh)/2;
After running the optimization algorithm with above start point, let's assume I want to generate more start points. Now I can generate 4 additional start points based on the mid-point Random_Point1 and the 4 original vertices (another way to think about this is by dividing the original rectangle in to 4 equal size smaller rectangles and finding the centre of each of the rectangle)
Random_Point21 = (Random_Point1 + [xlow(1),xlow(2)])/2;
Random_Point22 = (Random_Point1 + [xlow(1),xhigh(2)])/2;
Random_Point23 = (Random_Point1 + [xhigh(1),xlow(2))/2;
Random_Point24 = (Random_Point1 + [xhigh(1),xhigh(2)])/2;
Now I again run the fmincon with above 4 start points.
After looking at the results, I decide to generate more start points. This time I can generate 16 start points by using the previously generated above start points. The way to think about this is dividing each of the 4 smaller rectangles in previous iteration in to 4 pieces each so that we have 16 equal pieces along the respective axis and finding the centre of each of the 16 rectangles
I would continue the process of generating start points and running fmincon till my solution is deemed satisfactory (I have some criteria on when I want to stop optimization process)
My question is, what approach (coding) should I use to generate these start points in the most efficient manner (quick). Whatever I wrote (similar to above hard-coding) is definitely not the right approach. I guess I need to use recursion
Stating the problem in general terms:-
a) Assume I have k decision variables and for each decision variable I have the min/max ranges
b) I want to start with generating the mid-point (easy)
c) In second iteration, first take the mid-point from b) and divide the original space in to 2^k sub-spaces (parallel to each axis) and find the centre of these thereby generating 2^k points
d) In 3rd iteration, repeat c) thereby generating 2^(2k) points
e) In 4th iteration, repeat d) generate 2^(3k) points and so on.
Note, in my application the number of decision variable is not expected to be large (may be 3 or maximum 4)..the complexity I have is more in functional form of objective rather than # of decision variables. I'm stating this so that there are no worries about generating unpractical number of start points (like I will apply a condition that if in the pth iteration if 2^((p-1)k) was greater than 10000 abort the generation process.
