Quadratic-Equation-Constrained 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?
. 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
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
Yize Wang
2021-1-12
Hi Bruno, Thank you for your answer. I assume the toughness results from the complementary slackness, 
but the coneprog does not seem to deal with this since μ should also be part of the decision variable. Am I right about this?
but the coneprog does not seem to deal with this since μ should also be part of the decision variable. Am I right about this?
Bruno Luong
2021-1-12
编辑：Bruno Luong
2021-1-12
If you set the decision variables of coneprog() as a long vector of
X = [tau; x; mu; lambda]
Thus, the tau index is 1:n;
The x index is n+(1:n);
The mu index is 2*n+(1:n).
If the matrix Asc is defined with Asc(i,j) has value 1 for (i,j) == (n+k,2*n*k) for k=1,...n; and 0 elsewhere
Then
X'*Asc*X = dot(tau,x).
If you set bsc, dsc, gamma = 0, the cone constraint becomes
dot(mu,x) = 0
This is exactly what you want.
Yize Wang
2021-1-12
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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