最优化理论与方法课件（英文版－南京大学）最优化-quiz2-ans

Optimization Theory and Method Fall 2009/2010 Quiz #2 Q1. Consider the problem (1) Determine the solution to this problem. (2) Formulate the dual, and determine whether its local solution gives the Lagrange multipliers at the optimal solution. Sol. Obviously, the optimal solution is . The Lagrangian function is The KKT conditions are as below: At the optimal solution, the Lagrange multiplier . In this case, the dual function is for . From Thus the dual function takes the form for . The dual problem is therefore It is easy to see that the solution is . Hence, the local solution of the dual problem does give the Lagrange multipliers at the optimal solution. Q2. Let be a solution to the problem where each is differentiable. Then there exists a number such that Sol. Let and be the Lagrangian multipliers associated with constraints and respectively. Then, the Lagrangian function of this problem is Since is the optimal solution of the concerned problem, it satisfies the KKT conditions. That is, The first equation means . Then, according to the last complementary conditions, there exists a number such that Q3. Consider the problem Suppose that the logarithmic barrier method is used to solve this problem. What is ? What is ? What is ? What is ? Sol. The logarithmic barrier function is The first-order condition for is It’s obvious that . Substituting this in the first equation above, we have that is when , . when , , . Q4. Glueco is planning to introduce a new product in three different regions. Present estimates are that the product will sell well in each region with respective probabilities 0.6, 0.5, and 0.3. The firm has available two top sales representatives that it can send to any of the three regions. The estimated probabilities that the product will sell well in each region when 0, 1, or 2 additional sales representatives are sent to a region are given in the following table. If Glueco wants to maximize the probability that its new product will sell well in all three regions, where should the company assign its sales representatives? You may assume that sales in the three regions are independent. No. of additional Sales Representatives Probability of Selling Well Region 1 Region 2 Region 3 0 0.6 0.5 0.3 1 0.8 0.7 0.55 2 0.85 0.85 0.7 Sol. Let Then, . where PAGE _1294168647.unknown _1321732176.unknown _1321732555.unknown _1324276335.unknown _1324278588.unknown _1324278780.unknown _1324278619.unknown _1324278429.unknown _1321732795.unknown _1321732815.unknown _1321775782.unknown _1321732673.unknown _1321732489.unknown _1321732493.unknown _1321732286.unknown _1321732478.unknown _1321438616.unknown _1321731094.unknown _1321732099.unknown _1321732126.unknown _1321731681.unknown _1321709551.unknown _1321709373.unknown _1294168681.unknown _1320220106.unknown _1320220129.unknown _1321438615.unknown _1320220118.unknown _1320220093.unknown _1294168663.unknown _1291536291.unknown _1294167594.unknown _1294168041.unknown _1294168644.unknown _1294168634.unknown _1294167603.unknown _1294167561.unknown _1291726092.unknown _1291747561.unknown _1291534949.unknown _1291535264.unknown _1291535709.unknown _1291535742.unknown _1291535114.unknown _1291533082.unknown _1291534751.unknown _1291532167.unknown