Optimization Method 2016

HW 7 Due November 3

1. Solve

min_{x∈R^{2}} (x_{1}^{2} + x_{2}^{2} + 2ax_{1}x_{2}) subject to: |x_{1}| + |x_{2}| ≤ 1  (1)

where a is a parameter, −∞ < a < ∞

2. Find

min_{x∈R^{n}} (x^{T} A x) subject to: x^{T} x = 1  (2)

where A is a symmetric positively defined n × n matrix.

3. a. Solve

min_{x∈R^{3}} ∑_{i=1}^{3} (x_{i}^{2} + x_{i})  (3)

subject to

x_{1} − x_{2} + 2x_{3} = 1,  (4)
2x_{1} + x_{2} − 3x_{3} ≤ a,  (5)

where a is a parameter, −∞ < a < ∞ b. Formulate and solve a dual problem.