# Worksheet 8-1: Convex Optimization

Download: [CMSE382-WS8_1.pdf](CMSE382-WS8_1.pdf)

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## 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)

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

(b)

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

(c)

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

(d)

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

(e)

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

---

## 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

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

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

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

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

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

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

---

## Worksheet 8-1: Q3

Consider the optimization problem

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

(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.