• 2.12: Only a few students managed to get close to a solution here. A common mistake was letting f(x) = Bx-b, i.e. f is a vector-valued function instead of a real-valued function.
• 2.13: Some students were able to show using Farkas's lemma that if there was a primal optimal solution then there is a dual optimal solution, but no one was able to show strict complementarity.
• 2.15: Many students tried to apply previous theorems and corollaries to show (xbar,ybar) saddle point iff (xbar,ybar) KKT point. However, this often failed since these results conclude the existence of a ybar, but you are not guaranteed to get the same ybar that you started with. The direct approach using the definition of saddle point and KKT point was more successful.
• 3.1: Most students tried to use Corollary 2.31, but did not use Weak Duality (Theorem 3.4, Corollary 3.5) to conclude ybar was optimal for (LD) or that xbar was optimal for (CO), and then had difficultly concluding optimality.
• 3.2: This problem was quite difficult for the students. One major factor was that they were unsure about what it meant that the Lagrange and Wolfe duals are equivalent. A common mistake was not using the Slater regular assumption, which is clearly needed (see Example 3.12 in text which is not Slater regular and (LD) is not the same as (WD)).
• 3.3: This problem was well done. However, one common mistake was not checking all four KKT conditions, especially complemenatary slackness.
• 3.4: The first two parts were well done. In the third part, only a couple of students realized that a Slater point must be in the relative interior of R^3 x C.