Computational methods in optimization

David Gleich

Purdue University

Spring 2012

Course number CS 52000

Tuesday and Thursday, 12:00-1:15pm

Lawson 1106

Homework 4

$\newcommand{\eps}{\varepsilon} \newcommand{\kron}{\otimes} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\trace}{trace} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator*{\minimize}{minimize} \DeclareMathOperator*{\maximize}{maximize} \DeclareMathOperator{\subjectto}{subject to} \newcommand{\mat}[1]{\boldsymbol{#1}} \renewcommand{\vec}[1]{\boldsymbol{\mathrm{#1}}} \newcommand{\vecalt}[1]{\boldsymbol{#1}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\normof}[1]{\|#1\|} \newcommand{\onormof}[2]{\|#1\|_{#2}} \newcommand{\MIN}[2]{\begin{array}{ll} \minimize_{#1} & {#2} \end{array}} \newcommand{\MINone}[3]{\begin{array}{ll} \minimize_{#1} & {#2} \\ \subjectto & {#3} \end{array}} \newcommand{\MINthree}[5]{\begin{array}{ll} \minimize_{#1} & {#2} \\ \subjectto & {#3} \\ & {#4} \\ & {#5} \end{array}} \newcommand{\MAX}[2]{\begin{array}{ll} \maximize_{#1} & {#2} \end{array}} \newcommand{\MAXone}[3]{\begin{array}{ll} \maximize_{#1} & {#2} \\ \subjectto & {#3} \end{array}} \newcommand{\itr}[2]{#1^{(#2)}} \newcommand{\itn}[1]{^{(#1)}} \newcommand{\prob}{\mathbb{P}} \newcommand{\probof}[1]{\prob\left\{ #1 \right\}} \newcommand{\pmat}[1]{\begin{pmatrix} #1 \end{pmatrix}} \newcommand{\bmat}[1]{\begin{bmatrix} #1 \end{bmatrix}} \newcommand{\spmat}[1]{\left(\begin{smallmatrix} #1 \end{smallmatrix}\right)} \newcommand{\sbmat}[1]{\left[\begin{smallmatrix} #1 \end{smallmatrix}\right]} \newcommand{\RR}{\mathbb{R}} \newcommand{\CC}{\mathbb{C}} \newcommand{\eye}{\mat{I}} \newcommand{\mA}{\mat{A}} \newcommand{\mB}{\mat{B}} \newcommand{\mC}{\mat{C}} \newcommand{\mD}{\mat{D}} \newcommand{\mE}{\mat{E}} \newcommand{\mF}{\mat{F}} \newcommand{\mG}{\mat{G}} \newcommand{\mH}{\mat{H}} \newcommand{\mI}{\mat{I}} \newcommand{\mJ}{\mat{J}} \newcommand{\mK}{\mat{K}} \newcommand{\mL}{\mat{L}} \newcommand{\mM}{\mat{M}} \newcommand{\mN}{\mat{N}} \newcommand{\mO}{\mat{O}} \newcommand{\mP}{\mat{P}} \newcommand{\mQ}{\mat{Q}} \newcommand{\mR}{\mat{R}} \newcommand{\mS}{\mat{S}} \newcommand{\mT}{\mat{T}} \newcommand{\mU}{\mat{U}} \newcommand{\mV}{\mat{V}} \newcommand{\mW}{\mat{W}} \newcommand{\mX}{\mat{X}} \newcommand{\mY}{\mat{Y}} \newcommand{\mZ}{\mat{Z}} \newcommand{\mLambda}{\mat{\Lambda}} \newcommand{\mPbar}{\bar{\mP}} \newcommand{\ones}{\vec{e}} \newcommand{\va}{\vec{a}} \newcommand{\vb}{\vec{b}} \newcommand{\vc}{\vec{c}} \newcommand{\vd}{\vec{d}} \newcommand{\ve}{\vec{e}} \newcommand{\vf}{\vec{f}} \newcommand{\vg}{\vec{g}} \newcommand{\vh}{\vec{h}} \newcommand{\vi}{\vec{i}} \newcommand{\vj}{\vec{j}} \newcommand{\vk}{\vec{k}} \newcommand{\vl}{\vec{l}} \newcommand{\vm}{\vec{l}} \newcommand{\vn}{\vec{n}} \newcommand{\vo}{\vec{o}} \newcommand{\vp}{\vec{p}} \newcommand{\vq}{\vec{q}} \newcommand{\vr}{\vec{r}} \newcommand{\vs}{\vec{s}} \newcommand{\vt}{\vec{t}} \newcommand{\vu}{\vec{u}} \newcommand{\vv}{\vec{v}} \newcommand{\vw}{\vec{w}} \newcommand{\vx}{\vec{x}} \newcommand{\vy}{\vec{y}} \newcommand{\vz}{\vec{z}} \newcommand{\vpi}{\vecalt{\pi}} \newcommand{\vlambda}{\vecalt{\lambda}}$

Please answer the following questions in complete sentences in a typed manuscript and submit the solution to me on blackboard on February 8th, 2012, by 5pm.

Homework 4:

Problem 0: List your collaborators.

Please identify anyone, whether or not they are in the class, with whom you discussed your homework. This problem is worth 1 point, but on a multiplicative scale. (Note that collaboration is not allowed on the bonus question below.)

Problem 1: Steepest descent

(Nocedal and Wright, Exercise 3.6) Let’s conclude with a quick problem to show that steepest descent can converge very rapidly! Consider the steepest descent method with exact line search for the function $f(\vx) = (1/2) \vx^T \mQ \vx - \vx^T \vb$. Suppose that we know $\vx_0 - \vx^*$ is parallel to an eigenvector of $\mQ$. Show that the method will converge in a single iteration.

Problem 2: Inequality constraints

Draw a picture of the feasible region for the constraints:

$$\bmat{1 - x_1 - x_2 \\ 1 - x_1 + x_2 \\ 1 + x_1 - x_2 \\ 1 + x_1 + x_2} \ge 0.$$

Problem 3: Necessary and sufficient conditions

Let $f(\vx) = \frac{1}{2} \vx^T \mQ \vx - \vx^T \vc$.

1. Write down the necessary conditions for the problem:

   $$\MINone{}{f(\vx)}{\vx \ge 0}.$$

2. Write down the sufficient conditions for the same problem.

3. Consider the two-dimensional case with

   $$\mQ = \bmat{1 & 2 \\ 2 & 1} \qquad \vc = \bmat{0 \\ -1.5}.$$

   Determine the solution to this problem by any means you can, and justify your work.

4. Produce a Matlab or hand illustration of the solution showing the function contours, gradient, the constraint normal. What are the active constraints at the solution? What is the value of $\lambda$ in $\mA^T lambda = \vg$?