2. Course prerequisites
First three units: math content around Algebra 1
level, analytical skills approaching Calculus.
Students at the Pre-Calculus level should feel
comfortable. Talented students in Algebra 1 can
certainly give it a shot.
Last two units: Calculus required know how to
take derivatives and be familiar with their
implications for finding maxima and minima.
Computer programming skills will be taught from
the ground up. Previous experience is not
necessary.
3. Equipment Needed
For much of the first unit, a scientific calculator
is sufficient, though a graphing calculator will
make your life easier.
Towards the end of the first unit, when we get
into coding, a computer able to download and
install software (specifically, the programming
language Julia) is necessary. Julia is written for
Mac, Windows and Linux systems.
4. So what is mathematical
optimization, anyway?
Optimization comes from the same root as
optimal, which means best. When you
optimize something, you are making it best.
5. So what is mathematical
optimization, anyway?
Optimization comes from the same root as
optimal, which means best. When you
optimize something, you are making it best.
But best can vary. If youre a football player,
you might want to maximize your running
yards, and also minimize your fumbles. Both
maximizing and minimizing are types of
optimization problems.
6. Mathematical Optimization in the
Real World
Mathematical Optimization is a branch of
applied mathematics which is useful in many
different fields. Here are a few examples:
7. Mathematical Optimization in the
Real World
Mathematical Optimization is a branch of
applied mathematics which is useful in many
different fields. Here are a few examples:
Manufacturing
Production
Inventory control
Transportation
Scheduling
Networks
Finance
Engineering
Mechanics
Economics
Control engineering
Marketing
Policy Modeling
8. Optimization Vocabulary
Your basic optimization problem consists of
The objective function, f(x), which is the output
youre trying to maximize or minimize.
9. Optimization Vocabulary
Your basic optimization problem consists of
The objective function, f(x), which is the output
youre trying to maximize or minimize.
Variables, x1 x2 x3 and so on, which are the inputs
things you can control. They are abbreviated xn to
refer to individuals or x to refer to them as a group.
10. Optimization Vocabulary
Your basic optimization problem consists of
The objective function, f(x), which is the output
youre trying to maximize or minimize.
Variables, x1 x2 x3 and so on, which are the inputs
things you can control. They are abbreviated xn to
refer to individuals or x to refer to them as a group.
Constraints, which are equations that place limits
on how big or small some variables can get.
Equality constraints are usually noted hn(x) and
inequality constraints are noted gn(x).
11. Optimization Vocabulary
A football coach is planning practices for his running backs.
His main goal is to maximize running yards this will
become his objective function.
He can make his athletes spend practice time in the weight
room; running sprints; or practicing ball protection. The
amount of time spent on each is a variable.
However, there are limits to the total amount of time he
has. Also, if he completely sacrifices ball protection he may
see running yards go up, but also fumbles, so he may place
an upper limit on the amount of fumbles he considers
acceptable. These are constraints.
Note that the variables influence the objective function and
the constraints place limits on the domain of the variables.
12. Types of Optimization Problems
Some problems have constraints and some do
not.
unlimited limited
13. Types of Optimization Problems
Some problems have constraints and some do not.
There can be one variable or many.
x1
x3
x2
x6
x
8
x5
x4
x7
14. Types of Optimization Problems
Some problems have constraints and some do not.
There can be one variable or many.
Variables can be discrete (for example, only
have integer values) or continuous.
15. Types of Optimization Problems
Some problems have constraints and some do not.
There can be one variable or many.
Variables can be discrete (for example, only have integer
values) or continuous.
Some problems are static (do not change over
time) while some are dynamic (continual
adjustments must be made as changes occur).
16. Types of Optimization Problems
Some problems have constraints and some do not.
There can be one variable or many.
Variables can be discrete (for example, only have integer
values) or continuous.
Some problems are static (do not change over time) while
some are dynamic (continual adjustments must be made as
changes occur).
Systems can be deterministic (specific causes
produce specific effects) or stochastic (involve
randomness/ probability).
17. Types of Optimization Problems
Some problems have constraints and some do not.
There can be one variable or many.
Variables can be discrete (for example, only have integer
values) or continuous.
Some problems are static (do not change over time) while
some are dynamic (continual adjustments must be made as
changes occur).
Systems can be deterministic (specific causes produce specific
effects) or stochastic (involve randomness/ probability).
Equations can be linear (graph to lines) or
nonlinear (graph to curves)
18. Why Mathematical Optimization is
Important
Mathematical Optimization works better than
traditional guess-and-check methods
M. O. is a lot less expensive than building and
testing
In the modern world, pennies matter,
microseconds matter, microns matter.
19. Why Mathematical Optimization is
worth learning
Q: Which of these things is not like the others?
a) A degree in engineering
b) A degree in chemistry
c) A degree in pure mathematics
d) A large pepperoni pizza
20. Why Mathematical Optimization is
worth learning
Q: Which of these things is not like the others?
a) A degree in engineering
b) A degree in chemistry
c) A degree in pure mathematics
d) A large pepperoni pizza
(With the others, you can feed a family of four)
21. Why Mathematical Optimization is
worth learning
Joking aside, if youre interested in a career in
mathematics (outside of teaching or academia),
your best bet is applied mathematics with
computers. Mathematical optimization is a
powerful career option within applied math.
If youre not interested in a career in
mathematics, you will probably run into
optimization problems anyway.
22. Course Outline
Unit 1: Introductions and Skills
Optimization, vectors, iteration and recursion, foundational
programming skills
Unit 2: Non-calculus methods without
constraints Methods in two dimensions using
computers; extension to methods in three or more
dimensions
Unit 3: Non-calculus methods with constraints
Linear programming
Unit 4: Calculus methods without constraints
Newtons method and review of derivative meaning;
derivatives in 3D and above with implications for optimization
Unit 5: Calculus methods with constraints
Penalty functions; overview of other methods; Lagrange
multipliers
23. Practice Question 1
Group the following into what might be maximized,
minimized or cannot be optimized.
1. When choosing a new phone and plan, you might
consider: minutes of talk time per month; how
much is charged for overages; whether extra
minutes roll over; amount of data allowed; cost per
month; amount of storage/memory; how many
phones are available; brands/types of available
phones; cost of the phone; amount of energy used;
time it takes to download apps or music; whether or
not you get signal in your home.
24. Practice Question 2
2. An airplane designer is trying to build the
most fuel-efficient airplane possible. Write one
factor as an objective (Minimize/maximize
) and the rest as constraints (
c1, or or =). Delete any non-numerical factors:
speed, fuel consumption, range, noise, weight,
type of propulsion, cost, ease of use, amount of
lift, amount of drag, sonic boom volume,
payload (how much it can carry).
25. Practice Questions 3-5
For each of the following tasks, write an objective
function (maximize ) and at least two
constraints (subject to c1, or or =)
3.A student must create a poster project for a
class.
4.A shipping company must deliver packages to
customers.
5.A grocery store must decide how to organize
the store layout.
