Hi,

What do you mean by distance? To compute a distance from a point (vector) to a hyperplane you must have the coeficients of the hyperplane to find the normal vector from the hyperplane to the point. If you want to compute the distance from a specific point (vector) to another point, they have to be in the same feature space, i.e. the vectors must be of the same size. You z formula probably won't work due to dimension mismatch.

Supposing your z works, you can apply GA decoding individuals as binary 1x8 vectors, where ones represent columns being used to form C and zeros otherwise. For instance, an individual i= [1 1 0 0 0 0 0 0] means that your C is composed of columns 1 and 2 of A. Then you can compute your optimisation function using:

C = A(:, logical (i)). You generate n random individuals to form your initial population and start the search process (mutation, crossover, selection...).

In you specific case, I would perform an exhautive search though. The number combinations (8C2 + 8C3 + ... + 8C8) and the size of your matrix allow to do that in a reasonable time and you guarantee you find the global optima, which is not the case when using GA.

Regards.

Hi Ricardo,

Thank you for your replay.

The equation z=b-C*(posodo invers of (C))*b, is derived from the notion that you have the equation Cx=b and there is no solution, but you want to find a solution C times x that takes you as close to b as possible

( Least squares approximation). So if we say that we have 8!/(8-2)!=56 combinations of A or ([C1].....[C56]) the z value is the shortest distance between the endividual plane Cn and the vector b ,(b is constant), I want to finde that specific combination af of A i.e. that specific combination of columns in A that gives the smallest z-value, out of the 56 possible ones.

Some ekstra info: in my real system the A-matrix is 54*54 and the combinations are therefore 54!/(54-8)!=4.2e^13 and that is why i thought genetic algorithm was the solution to this problem.

I am using MATLAB as programming tool

I will impliment your proposal in MATLAB(as soon as i find out how to code it) and write back when I've fixed my problem :)

It makes more sense to use a convex quadratic optimization formulation - it is guaranteed to find the solution with a very high precision. This is an instance where a metaheuristic simply is not a wise choice.

Hi Ricardo,

I have tried to solve my problem using GA according to the method you described, but it seems like I am making a mistake somewhere. I have written my code for this optimization problem in tree separate files.

File 1:

A=[1 5 8 8 9 6 2 5

8 7 5 5 5 1 1 7

7 7 7 2 5 4 4 4

6 5 2 2 6 6 2 1

5 6 7 9 8 2 2 1

4 1 4 7 4 8 4 6

5 5 5 5 4 8 4 7

1 4 7 8 9 6 7 4];

b=[ 7

5

2

4

5

6

6

4];

i = 8; %creates a random 1 by 8 vector whith zeros and ones

Pop = randi([0, 1], [1, i])

File 2:

function z=myFitness(A,b,Pop)

z=b-A(:, logical ( Pop))*pinv(A(:, logical ( Pop)))*b

end

File 3:

FitFcn = @myFitness;

nvars=8

[x,fval]=ga(FitFcn,nvars,[],[],[],[],[],[],[])

I get the following error :

Error in testfunction (line 5): [x,fval]=ga(FitFcn,nvars,[],[],[],[],[],[],[])

Caused by: Failure in initial user-supplied fitness function evaluation. GA cannot continue.

Hi Michael Patriksson,

Thank you for your suggestion, as I understand it when reading the theory, it only works if the plane/function is concave, and I can only get a vector ‘’x_min’’ as a result:

x_min=1/2*x’*A+x’*b

I have tried to implement the convex quadratic optimization formulation in MATLAB, using the following code:

[x,fval]=quadprog(A,b,[],[],[],[],[],[]);

I get the fallowing error:

Warning: Your Hessian is not symmetric. Resetting H=(H+H')/2.

> In quadprog (line 324)

In K_b (line 55) %line where the code is written and file name%

The problem is non-convex.

This error leads me to think that this method is not suitable for my application.

I need a matrix as the result, which is composed of some optimal combination of the columns in A i.e. a minimum of to columns. I need these columns because they represent degrees of freedom in a structural system, and it is these degrees of freedom I need to obtain.

Hi Michael,

A and b in your example are treated as constants (you are changing the index of the columns), so you should load them inside your fitness function. Also, since GA is a minimization algorithm and your output is a vector, the GA can't say which vector is "lower" than the other. So you have to convert it to a measure you want to minimize. For me, what makes more sense is the mean squared error (or mean squared distances), so I incluced z=sum(z.^2) (you can change as you wish but must be a scalar to be minimized). Your fitness function should look like this:

function z=myFitness(x)

A=[1 5 8 8 9 6 2 5

8 7 5 5 5 1 1 7

7 7 7 2 5 4 4 4

6 5 2 2 6 6 2 1

5 6 7 9 8 2 2 1

4 1 4 7 4 8 4 6

5 5 5 5 4 8 4 7

1 4 7 8 9 6 7 4];

b=[ 7

5

2

4

5

6

6

4];

z=b-A(:, logical ( x))*pinv(A(:, logical ( x)))*b;

z = sum(z.^2);

end

In your file 3, since matlab's GA doesn't work with binaries, you should include upper and lower bounds and also tell matlab that you want all you variables (x1, x2, ... x8) as integers, so it will only return 0 ou 1 for each element of x.

FitFcn = @myFitness;

nvars=8;

lb = zeros(8,1);

ub = ones(8,1);

integers = 1:8;

[x,fval]=ga(FitFcn,nvars,[],[],[],[],lb,ub,[],integers);

This should work, but since you mentioned Least squares approximation, I really don't understand what are you trying to do. If your Cx=b have no solution, you should seek for an x that mininizes Cx-b, not a combination C. If one row of the Cx=b looks like: c1.x1 + c2.x2 + c3.x3 = b, when you select a combination of C, you are simply throwing some variables away and changing the dimension of the space under analysis, and that doesn't make sense.

To minimize the error (z or distance) you should compute:

x = (Ct*C)-1*Ct*b

alternatively: x = pseudoInverse(C)*b

where x must have the same number of elements as columns in C...

Good luck!