最优化理论与方法（英文版）quiz2-ans.doc

Optimization Theory and Method Fall 2009/2010
Quiz #2
Consider the problem

Determine the solution to this problem.
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.
Let be a solution to the problem
Optimization Theory and Method Fall 2009/2010

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
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 logari