Question

2. Consider the problem of minimizing $\qquad f(x_1, x_2) = (x_1 - x_2)^3 + (x_1 - x_2)^2 + (x_2 - 1)^2$ over $\mathbb{R}^2$. (a) Perform one complete iteration with Newton's method, starting from the point $x_0 = (0, 0)^T$. (b) Find all the local minimum points of $f$ over $\mathbb{R}^2$. (c) Does $f$ have any global minimum point over $\mathbb{R}^2$? Justify your answer. (d) Find two real numbers $a_1$ and $a_2$ such that $f$ is a convex function over the convex set $\qquad \Omega = \{(x_1, x_2)^T \in \mathbb{R}^2 : a_1 x_1 + a_2 x_2 \le 1\}$.

          2. Consider the problem of minimizing
$\qquad f(x_1, x_2) = (x_1 - x_2)^3 + (x_1 - x_2)^2 + (x_2 - 1)^2$
over $\mathbb{R}^2$.
(a) Perform one complete iteration with Newton's method, starting from the point $x_0 = (0, 0)^T$.
(b) Find all the local minimum points of $f$ over $\mathbb{R}^2$.
(c) Does $f$ have any global minimum point over $\mathbb{R}^2$? Justify your answer.
(d) Find two real numbers $a_1$ and $a_2$ such that $f$ is a convex function over the convex set
$\qquad \Omega = \{(x_1, x_2)^T \in \mathbb{R}^2 : a_1 x_1 + a_2 x_2 \le 1\}$.

$2. Consider the problem of minimizing f(x1, x2) = (x1 - x2)^3 + (x1 - x2)^2 + (x2 - 1)^2 over ℝ^2. (a) Perform one complete iteration with Newton's method, starting from the point x0 = (0, 0)^T. (b) Find all the local minimum points of f over ℝ^2. (c) Does f have any global minimum point over ℝ^2? Justify your answer. (d) Find two real numbers a1 and a2 such that f is a convex function over the convex set Ω = {(x1, x2)^T ∈ℝ^2 : a1 x1 + a2 x2 ≤ 1}.$

Added by Vicenta K.

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