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.

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Bruno Luong
Bruno Luong 2021-1-12
编辑:Bruno Luong 2021-1-12
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 个评论
Johan Löfberg
Johan Löfberg 2021-3-31
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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