Subject- Optimization

5. Using KKT. Consider the problem we saw in the lecture: min:s.t.:x126x1+x224x2x122x1+x22021x122x1x2+20x1x20 (a) Plot the constraints and shade the feasible region. 2 (b) Verify that the point x=(1,0) is an optimal solution, using the KKT equations. We did part of this in class. We did not show that complementary slackness holds. (c) Compute the other two corner points (the intersection of constraints 1 and 2, and then the intersection of constraints 1 and 3). Verify, using KKT, that neither of these is an optimal solution. (d) We have thus checked that of the three corner points, one is an optimal solution, and the other two are not. There are still points in the interior, that could potentially be optimal solutions. Use the KKT conditions to show that this cannot be the case, i.e, for any interior point x~, that is, any point for which the three constraints are strictly satisfied, x is not an optimal solution