# Homework 5


# Homework 5

Please answer the following questions in complete sentences in a typed manuscript and submit the solution to me in class on February 21th, 2012.

The previous homework was focused heavily on implementations and studying these methods in a practical setting. In this homework, we will return our focus back to theory.

## Problem 0: List your collaborators.

Please identify anyone, whether or not they are in the class, with whom you discussed your homework.

## Problem 1: Solving the trust region problem (Griva, Nash, and Sofer, 11.6.6)

In class, your professor mentioned that any solution to the trust region sub-problem,

could be characterized as follows:

• $(\mH(\vx_k) + \lambda \mI) \vp^* = -\vg$
• $\lambda (\normof{\vp^*} - \Delta) = 0$
• $\mH(\vx_k) + \lambda \mI \succeq 0$ (it’s positive semi-definite).

Use these conditions to show that, if $\lambda \not= 0$, then $\vp$ solves:

Hint First show that $m_k(\vp^*) \le m_k(\vp) + 1/2 \lambda (\vp^T \vp - {\vp^*}^T \vp^*)$ for any $\vp$.

Read the proof of Lemma 4.2. There is one missing step. Let $\mB$ be symmetric positive definite. Prove the inequality: