Homework 7

Deadline: Dec. 9, 22:00:00

Lagrange multiplier

1. Consider the following optimization problem:

   $\min \quad x_1^2+x_2^2$

   $\text{subject to} \quad x_1+2x_2=2.$

   • Sketch (draw a picture) the feasible set and the level set of the objective function.
   • Write down the Lagrange multiplier condition.

2. Consider the following optimization problem:

   $\min \quad e^{x_1}-x_2$

   $\text{subject to} \quad x_1+2x_2=2.$

   • Write down the Lagrange multiplier condition.
   • Does this optimization problem has finite optimal solution? Why?

3. Consider the following optimization problem:

   $\min \quad x_1^2+2x_1x_2+3x_2^2+4x_1+5x_2+6x_3$

   $\text{subject to} \quad x_1+2x_2=3, \, 4x_3+5x_1=6.$

   • Write down the Lagrange multiplier condition.
   • Find the point satisfying the condition and find the corresponding Lagrange multiplier.
   • For each point satisfying the condition, determine whether it is the optimal solution.