Given a set of n vectors a_i, I am trying to find a vector x to minimize the expression |a_1-x|^p+|a_2-x|^p+|a_3-x|^p+…+|a_n-x|^p, where p is greater than 1.

I am having trouble with writing this sum expression in cvx syntax. Could someone explain what is wrong with the code below? In the code I try to initialize 5 15x1 vectors in sample, and try to find the correct value of x.

sample - x is not dimensionally consistent; that is a (15 by 5) matrix minus a (5 by 1) vector. Also, |a_1-x|^p doesn’t even make sense when x is a vector, not a scalar. Perhaps you want p norm to the pth power of a_1 - x ?

Thanks for the reply! Yes, I was hoping to minimize the pnorm between the a_is and x, so ||a_1-x||+||a_2-x||+||a_3-x||+…+||a_n-x||. I have trouble seeing how to do this using cvx.

If p = 1 or 2, you can use a very compact and vectorized implementation:

m = 15; n= 5;
samplee = rand(m,n);
p = 2;
cvx_begin
variable x(m)
minimize(sum(norms(samplee-repmat(x,1,n),p)))
cvx_end

Here is a more brute force way which will work for any p >= 1.

p = 2; % p = 1.5 also works, even p = inf
cvx_begin
variable x(m)
Obj = 0;
for i = 1:n
Obj = Obj + norm(samplee(:,i)-x,p);;
end
minimize(Obj)
cvx_end

For p =1 or 2, both programs work, and for the same instantiation of samplee, produce the same results, within numerical tolerance

For any p >= 1 (but not inf), you could instead make your objective be the sum of norm^p by changing Obj = Obj + norm(samplee(:,i)-x,p);
in the 2nd version to Obj = Obj + pow_pos(norm(samplee(:,i)-x,p),p);