1. In this problem, we will see that our intuition about optimization does not hold. (a) Consider the optimization problem below min[z11z24][1001][z11z24]s.t.z1,z22 Our intuition suggests that we can solve this optimization problem by finding the unconstrained optimal zunc=(1,4) and then projecting onto the constraints i.e. za=min{min{zunc,2},0} where the min and max are element-wise on the vector z. Sketch or plot the cost level-sets and feasible regions. Is za the optimal? HINT: MATL B command contour can be used to plot the level-sets of a cost. (b) The intuition in part (a) only held because the cost and constraints were simple. Next, consider the optimization problem below with a more complex cost min[z11z24][2111][z11z24]s.t.z1,z21 This problem has the same unconstrained optimal zunc=(1,4). And since the constraints are the same, the projection zp=min{min{zunc,2},0} is the same. Sketch the cost and feasible regions. Is za still the optimal? If no, what is the optimal zb ? (c) Next, consider the optimization problem below with a simple cost, but more complex constraints min[z11z25][1001][z11z25]s.t.0z12,0z2z12 Sketch the cost and feasible regions. What is the optimal zc ? (d) How does Tzb compare with zc where T=[1101]? How are the optimization problems in parts (b) and (c) related? What does this tell us about transforming an optimization problem to make it simple like in part (a)