Quadratic-​Equation-C​onstrained Optimization

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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.

Accepted Answer

Bruno Luong
Bruno Luong on 12 Jan 2021
Edited: Bruno Luong on 12 Jan 2021
There is
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?
  4 Comments
Johan Löfberg
Johan Löfberg on 31 Mar 2021
Late to the game here, but the discussion above is not correct. A constraint of the form mu'*x=0 is nonconvex and cannot be represented using second-order cones. If that was the case, P=NP as it would allow us to solve linear bilevel programming problems in polynomial time, as these can be used to encode integer programs...
It appears the discussion confuses x'*Asc*x == 0 (a nonconvex quadratic constraint) and the generation of a SOCP constraints ||Asc*x + 0|| <= 0 + 0*x (note the linear operator inside the norm)

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