I am trying to solve the problem of the following form:
$\min \frac{1}{2} x'Hx + f'x$
subject to,
$Ax = b$, and $lb \leq x \leq ub$
I have utilized 'linprog' to check whether the set $Ax = b$, and $lb \leq x \leq ub$ is empty, by solving these two constraints with the empty objective function. It returns me with a solution. However, when I am using 'quadprog' to obtain the optima, it returns with no solution. I am using interior point method in both cases.
This is a pathological case, and this is why I am not posting A, B, H, and f matrices here.
Actually, I am trying to solve another problem of the form:
$\min \frac{1}{2} x'Px + c'x$
subject to,
$Qx = b$, and $lb \leq x \leq ub$
I have represented this problem with its KKT condition. While the equalities and bounds as a part of the KKT conditions can be easily represented by linear functions, the complementary slackness condition makes one of the constraints quadratic. Therefore, in the modified problem, I am trying to minimize the complementary slackness condition itself. It is trivial to note that, in the modified problem the optimal solution should be zero.
Please note that, I am doing this just to check whether I have correctly represented the KKT condition itself. I am not able to figure out why I am observing the same. I have taken the solution to the original problem and plugged it into modified problem, and I can see no constraint violation.