Linear Programming, optimal location for "main" Station on a Line

Hello there,

I'am having trouble solving an Exercise regarding a linear programming optimization.

fmincon(@dist, x0, [], [], [], [], lb, ub, @d_constr)

First I want to generate a Random set of numbers of "little" stations and their distribution on a Line:

j = randi([1 1000]);
n = randi([2 20]);
m = randi([j],n,1);

Then I need to find the location for the "Main" Station,

which needs to bet at the point on the Line where it has the minimum distance to all other Stations.

the function for the sum is this:

function d = dist(x)

L = x(1);
d = @(m) sum(m - L);
end

Now a function for the constrains is needed:

As the only one is L>0 && L<j ; where j is the total lenght of the line.

function [g,h] = d_constr(x)

L = x(1);

g(1) = L < j;
g(2) = L > 0;
h = [];

end

Well it doesn't work and Iam blank.

This is the Error I am getting all the Time:

Error using fmincon (line 700)
FMINCON requires all values returned by functions to be of data type double.

Error in distance (line 69)
    fmincon(@dist, x0, [], [], [], [], x1, x2, @d_constr)

Thank you for your time <3

[Matt J]

the function for the sum is ... d=@(m) sum(m-L)

That doesn't really make sense for multiple reasons. First of all, you have written this as a function of m, whereas the unknown variable to be optimized is L. Second, the values m(i)-L can be negative and therefore cannot represent distances. Presumably you mean either the sum of absolute values or a sum of squares.

Answer 1

[Matt J]

which needs to bet at the point on the Line where it has the minimum distance to all other Stations.

If you mean the minimum least squares distance, then this can be found with fmincon as follows

Linitial = mean(m);

Loptimal = fmincon(@(L) sum(m-L).^2, Linitial, [],[],[],[],0,j)

However, you should expect this to converge in zero iterations because the initial guess here mean(m) is the known analytical solution to this problem.

If you mean the minimum absolute distance sum(abs(m-L)), then you shouldn't be using fmincon at all. You should be using linprog to solve the linear program,

This makes a little more sense, since you said this was supposed to be a linear programming problem.