Worksheet 8-1: Convex Optimization

Download: CMSE382-WS8_1.pdf

Warning

This is an AI-generated transcript of the worksheet and may contain errors or inaccuracies. Please refer to the original course materials for authoritative content.

---

Worksheet 8-1: Q1

For each problem:

• Sketch the feasible set. Is it convex?

• Is the objective function convex? You may use the atoms below.

Atom	Property
\(t^m\)	convex for \(m=2,4,6\)
\(|\mathbf{x}|\)	convex
\(e^t\)	convex, nondecreasing

• Based on this, state whether the problem is a convex optimization problem or not.

(a)

\[\begin{split} \begin{aligned} & \text{min} && x^2-y\\ & \text{s.t.} && x^2+y^2=3 \end{aligned} \end{split}\]

(b)

\[\begin{split} \begin{aligned} & \text{min} && x+y^4\\ & \text{s.t.} && x+5y\le 10 \end{aligned} \end{split}\]

(c)

\[\begin{split} \begin{aligned} & \text{min} && -e^{x^2}+y^2\\ & \text{s.t.} && \|\mathbf{x}\|\le 5 \end{aligned} \end{split}\]

(d)

\[\begin{split} \begin{aligned} & \text{min} && (1+x^2+x^{10})^2\\ & \text{s.t.} && 5x+1\le 4 \end{aligned} \end{split}\]

(e)

\[\begin{split} \begin{aligned} & \text{min} && -x^4+3y\\ & \text{s.t.} && x+3y\le 3 \end{aligned} \end{split}\]

---

Worksheet 8-1: Q2

For a position in a company, we need to schedule job interviews for 3 candidates, Alice, Bob, and Charlie, who are coming in that order.

The available interview time windows are:

Candidate	Times	Interval as hours since 9:00am
Alice	9:00 - 10:30	\([0,1.5]\)
Bob	10:00 - 11:30	\([1,2.5]\)
Charlie	9:30 - 12:30	\([0.5,3.5]\)

The goal is to schedule starting times \(t_A,t_B,t_C\) to maximize the minimal starting-time difference between consecutive interviews.

(a) The objective can be written as

\[ f(t_A,t_B,t_C)=\min\{t_B-t_A,\,t_C-t_B\}. \]

Write the full optimization problem.

(b) We can pull a mathematical trick by introducing a new variable \(s\) to rewrite the optimization problem as

\[\begin{split} \begin{aligned} \max_{t_A,t_B,t_C,s}\quad & s\\ \text{s.t.}\quad & \min\{t_B-t_A,\,t_C-t_B\}\ge s,\\ & \text{(constraints from part (a))} \end{aligned} \end{split}\]

Then break up the min function and fill in the blanks:

\[\begin{split} \begin{aligned} \max_{t_A,t_B,t_C,s}\quad & s\\ \text{s.t.}\quad & t_B-t_A\;\boxed{\phantom{\prod}}\;s,\\ & t_C-t_B\;\boxed{\phantom{\prod}}\;s,\\ & t_A\ge \boxed{\phantom{\prod}}\\ & t_A\le \boxed{\phantom{\prod}}\\ & t_B\ge \boxed{\phantom{\prod}}\\ & t_B\le \boxed{\phantom{\prod}}\\ & t_C\ge \boxed{\phantom{\prod}}\\ & t_C\le \boxed{\phantom{\prod}} \end{aligned} \end{split}\]

(c) Use the above to write the problem in standard form

\[\begin{split} \begin{aligned} \max_{\mathbf{x}}\quad & \mathbf{c}^\top\mathbf{x}\\ \text{s.t.}\quad & A\mathbf{x}\le \mathbf{b}\\ & \mathbf{x}\ge 0 \end{aligned} \end{split}\]

by defining \(\mathbf{x}\), \(\mathbf{c}\), \(A\), and \(\mathbf{b}\).

---

Worksheet 8-1: Q3

Consider the optimization problem

\[\begin{split} \begin{aligned} \text{max}\quad & 6x+5y\\ \text{Subject to}\quad & 2x-3y\le 5\\ & x+4y\le 11\\ & 4x+y\le 15\\ & x,y\ge 0 \end{aligned} \end{split}\]

(a) Find \(\mathbf{A}\), \(\mathbf{b}\), and \(\mathbf{c}\) such that the problem is in standard form.

(b) Sketch the feasible set. Where are the extreme points?

(c) Solve the problem by finding the optimal solution.

---

Worksheet 8-1: Q4

We are planning a football watch party for MSU by making pans of cookies and muffins.

• A pan of muffins sells for \(\$6\).

• A pan of cookies sells for \(\$10\).

• We want at least as many pans of cookies as muffins.

• We have 13 cups of sugar.

• Muffins require 0.5 cup sugar per pan.

• Cookies require 1 cup sugar per pan.

• We want to make at least \(\$100\).

Let \(X\) be muffin pans and \(y\) be cookie pans.

(a) Write the variables and objective function.

(b) Write the constraints as inequalities/equalities.

(c) Recast inequalities as needed and write standard form.

(d) Sketch the feasible region.

(e) Find the minimum of the objective function.