Question 6 (duality and optimality conditions) Consider the following investment problem. Given $ 100 to invest, you need to allocate this money into six possible investments. The returns of each investment is given in Table 3. Table 3: Returns (per dollar invested) Investment 1 2 3 4 5 6 Return 1.5 2 1.2 1.3 2 1.1 In addition to the budget constraint, you have the following diversification constraints: You cannot allocate more than 30 dollars in investments 1 and 2. Investment 5 is very risky, while investment 6 is risk-free. Thus, for every dollar in excess of $10 that you allocate to investment 5, you need to allocate one dollar to investment 6. For example, if you invest $13 dollars in investment 5, then you need to invest $3 in investment six. You can allocate at most $50 in investments 3, or 4. Letting ; be the money allocated to investment i, the investment problem can be formulated as max 1.5x1 + 2x2 + 1.2x3 + 1.3x4 + 2x5 +1.1x6 s.t. x1+x2+ 3+ 4+ 5+ x6 = 100 130 x230 x3 + x450 x6-25-10 1, 2, 3, 4, 5, 6 0. (y1) (y2) (Y3) (34) (Y5) Let y1,...,ys be the dual variables associated with each constraint. Answer the following questions. 1. Formulate the dual problem. 2. Write the optimality conditions. 3. An optimal dual solution vector is y = 1.55, y2 = 0, y3 y = -0.45. Compute an optimal solution for the primal. = 0.45, 94 == 0 and