QP with quadratic constraints using fmincon

I have to compute an objective function asfollows :

min q(y) = 0.5*y'*Q*y + f'y
  subject to norm(y,2)=1

I am using the following code in matlab

 options=optimoptions('fmincon','Algorithm','interior-point');

 [x,fval] = fmincon(@(x)(0.5*x'*Q*x + F'*x),x0,[],[],[],[],[], [],@nonlncon,options);

function [c,ceq ] = nonlncon(x )

c = [];

ceq =  norm(x,2)-1;

end

Is this formulation correct ?

Answer 1

[Matt J]

It might be a bit safer to formulate ceq differentiably

 ceq=norm(x)^2-1

You could also consider providing gradient calculations for the objective function and constraints, since the formulae for those are fairly simple.

Lastly, I would point out that the problem has a closed form solution, which you might consider if your problem dimension is not too large,

https://www.mathworks.com/matlabcentral/fileexchange/53191-quadratic-minimization-with-norm-constraint?s_tid=srchtitle

[Saurabh]

Thanks for your code.

I forgot to mention that Q is p.s.d and there will be no inequality constraints. My Q is large, of the order of 4096-by-4096. If I use my code, will I get incorrect answers ? I tried running my code and the solution vector x does not satisfy norm(x,2)-1 = 0

[Matt J]

I forgot to mention that Q is p.s.d and there will be no inequality constraints.

No harm that you didn't. It doesn't affect my response above.

fmincon might work better with the following options

 options=optimoptions(@fmincon,'GradObj','on','GradConstr','on','Algorithm','interior-point','Display','off');

and by providing analytical gradients as I mentioned earlier. You should, of course, always run with at least 4 arguments

 [x,fval, exitflag, output]=fmincon(...)

and use exitflag/output to check that the optimization was successful.