MATLAB: Quadratic-​Equation-C​onstrained Optimization

Question

Dear all,

I am trying to solve a bilevel optimization as follows,

I then transformed the lower-level optimization with KKT conditions and obtained a new optimization problem:

The toughness is the constraint . I am wondering whether there exists a solver that can efficient deal with this constraint?

Thank you all in advance.

Answer 1

There is

https://www.mathworks.com/help/optim/ug/coneprog.html

but only for linear objective function.

You migh iterate on by relaxing succesively the cone constraint and second order objective like this

while not converge
   x1 = quadprog(...) % ignoring tau change (remove it as opt variable)
   x2 = coneprog(...) % replace quadratic objective (x2'*H*x2 + f'*x2) by linear (2*x1'*H + f')*x2
until converge

Otherwise you can always call FMINCON but I guess you already know that?