How write these equation in Matlab for optimization

2020 7 5
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Dear Friends,
I want to write this optimization problem in matlab.
Anybody here to help me write thi in Matlab please.
the only one decision variable is

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First you don't need the variable w, your problem is to maximize corr(Q,S), which is a unconstrained problem in x. Second, by your problem definition, Q is a single value, and correlation between single values doesn't make much sense, so I believe what you mean by corr is that the values should be equal or Q is a vector and you forgot a summation index. Considering the second hypothesis an example of how to do it could be seen here:
% Random data
S = 1:100;
c = randn(100,4);1
% Function to minimize
f = @(x,S,c)-min(min((corrcoef(S', (x*c')/sum(x) )) )); % Negative so min = max. Two min's since corrcoef gives a matrix as output
% Minimizer
[x,fval] = fminsearch(@(x)f(x,S,c),[1,1,1,1])
You can use this code as basis to implement your problem.

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Muhammad Usama
Muhammad Usama 2020-7-5
why we are not using corr?
as corr(Vector A, Vector B) would be a single value
You can use corr, it will give the same answer, I just forgot about it. You would just need to change the line to
f = @(x,S,c)-corr(S', ((x*c')/sum(x))' );
Muhammad Usama
Muhammad Usama 2020-7-5
编辑：Muhammad Usama 2020-7-5
Actully in f
there was
sum(x*c')/sum(x)
%instead of
(x*c')/sum(x)
however when I write sum with numerator it not woks
Also, c would be rectangular matrix, so
sum(x.*c(:,i))/sum(x)
I replaced sum with a matrix operation. If you want to have the sum it would be:
sum(x.*c,2)'/sum(x)
Which is the same
Muhammad Usama
Muhammad Usama 2020-7-5
how can we restrict 'x' value to be non-negtive in this ?
Muhammad Usama
Muhammad Usama 2020-7-5
But it is giving -ve values of x as well
I want to constraint x >= 0
Suppose the optimizer finds a value of -2 for a x entry. When it goes to the function the -2 becomes +2 because of the absolute. Which means that if you do
[x,fval] = fminsearch(@(x)f(x,S,c),[1,1,1,1])
x = abs(x);
You will still get the same cost function, i.e., the abs(x) is exact the same minimum.
Muhammad Usama
Muhammad Usama 2020-7-5
Thank you Sir.
one more thing, I want to add sum(x) ==1
is it possible?
You have this already. Note that you divide by sum(x). This is a linear operation, so it would be the same as:
(x/(sum(x)))*c'
which would give: sum(x) ==1. In other words, if you normalize x by it's sum you'll get the same result as the not normalized one.

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