IntroductionOptimization
Math Boot Camp Part IIUIC Economics Department
Erik Hembre
August 21st, 2015
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Welcome!
Let’s do some quick introductions.Just a few comments/suggestions about your first year:
Get on top of things!It will be difficult. That’s ok.Push yourself. High returns on investment.Find a good study group. Learn how to learn by yourself, withother, and to others.Work on your weaknesses. Focus on your interests.Grades << Knowledge
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Welcome!
Let’s do some quick introductions.
Just a few comments/suggestions about your first year:
Get on top of things!It will be difficult. That’s ok.Push yourself. High returns on investment.Find a good study group. Learn how to learn by yourself, withother, and to others.Work on your weaknesses. Focus on your interests.Grades << Knowledge
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Welcome!
Let’s do some quick introductions.Just a few comments/suggestions about your first year:
Get on top of things!It will be difficult. That’s ok.Push yourself. High returns on investment.Find a good study group. Learn how to learn by yourself, withother, and to others.Work on your weaknesses. Focus on your interests.Grades << Knowledge
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Welcome!
Let’s do some quick introductions.Just a few comments/suggestions about your first year:
Get on top of things!
It will be difficult. That’s ok.Push yourself. High returns on investment.Find a good study group. Learn how to learn by yourself, withother, and to others.Work on your weaknesses. Focus on your interests.Grades << Knowledge
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Welcome!
Let’s do some quick introductions.Just a few comments/suggestions about your first year:
Get on top of things!It will be difficult. That’s ok.
Push yourself. High returns on investment.Find a good study group. Learn how to learn by yourself, withother, and to others.Work on your weaknesses. Focus on your interests.Grades << Knowledge
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Welcome!
Let’s do some quick introductions.Just a few comments/suggestions about your first year:
Get on top of things!It will be difficult. That’s ok.Push yourself. High returns on investment.
Find a good study group. Learn how to learn by yourself, withother, and to others.Work on your weaknesses. Focus on your interests.Grades << Knowledge
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Welcome!
Let’s do some quick introductions.Just a few comments/suggestions about your first year:
Get on top of things!It will be difficult. That’s ok.Push yourself. High returns on investment.Find a good study group. Learn how to learn by yourself, withother, and to others.
Work on your weaknesses. Focus on your interests.Grades << Knowledge
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Welcome!
Let’s do some quick introductions.Just a few comments/suggestions about your first year:
Get on top of things!It will be difficult. That’s ok.Push yourself. High returns on investment.Find a good study group. Learn how to learn by yourself, withother, and to others.Work on your weaknesses. Focus on your interests.
Grades << Knowledge
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Welcome!
Let’s do some quick introductions.Just a few comments/suggestions about your first year:
Get on top of things!It will be difficult. That’s ok.Push yourself. High returns on investment.Find a good study group. Learn how to learn by yourself, withother, and to others.Work on your weaknesses. Focus on your interests.Grades << Knowledge
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Today:
The purpose of today is a brief review some relevantmathematics concepts.
Again: REVIEW. Its ok if you don’t remember or haven’tlearned the material.Constrained/Unconstrained Optimization.Highly recommend: Simon and Blume: Mathematics forEconomists
Optimization: Ch: 16-19Also Consider: Ch: 1-4, 12-14, 20-22. Appendix A1.
Also consider: Real Analysis: A First Course (by RussellGordon). For an introduction to real analysis.
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Today:
The purpose of today is a brief review some relevantmathematics concepts.Again: REVIEW. Its ok if you don’t remember or haven’tlearned the material.
Constrained/Unconstrained Optimization.Highly recommend: Simon and Blume: Mathematics forEconomists
Optimization: Ch: 16-19Also Consider: Ch: 1-4, 12-14, 20-22. Appendix A1.
Also consider: Real Analysis: A First Course (by RussellGordon). For an introduction to real analysis.
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Today:
The purpose of today is a brief review some relevantmathematics concepts.Again: REVIEW. Its ok if you don’t remember or haven’tlearned the material.Constrained/Unconstrained Optimization.
Highly recommend: Simon and Blume: Mathematics forEconomists
Optimization: Ch: 16-19Also Consider: Ch: 1-4, 12-14, 20-22. Appendix A1.
Also consider: Real Analysis: A First Course (by RussellGordon). For an introduction to real analysis.
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Today:
The purpose of today is a brief review some relevantmathematics concepts.Again: REVIEW. Its ok if you don’t remember or haven’tlearned the material.Constrained/Unconstrained Optimization.Highly recommend: Simon and Blume: Mathematics forEconomists
Optimization: Ch: 16-19Also Consider: Ch: 1-4, 12-14, 20-22. Appendix A1.
Also consider: Real Analysis: A First Course (by RussellGordon). For an introduction to real analysis.
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Applications for Optimization problems are everywhere ineconomics:
Which set of goods to buy?How much/what ratio of inputs for production?How to allocate time between leisure and work?
Economics is all about solving decision problems. This isinherently linked to optimization. Typically there is a scareresource which constrains the choice set.
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Introduction
Applications for Optimization problems are everywhere ineconomics:
Which set of goods to buy?How much/what ratio of inputs for production?How to allocate time between leisure and work?
Economics is all about solving decision problems. This isinherently linked to optimization. Typically there is a scareresource which constrains the choice set.
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General Problem:Maximize f (x) : x ∈ C where C is a non-empty subset of dom(f ).
Definition
A point x∗ is a maximum of f on C if f (x∗) ≥ f (x)∀x ∈ C .
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Definition
x∗ is a critical point of f if f ′(x) = 0
The first-order condition (FOC) for x∗ to be a maximum (orminimum) of a function f (x) is that x∗ is a critical point.
x∗ must be on the interior of dom(f ).
The same FOC holds for a function F of n variables.
If ∂F∂xi
= 0∀i ∈ n, then x∗ satisfies the FOC for F .
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To determine whether a critical value x∗ is a local minimumor local maximum (or neither), we use conditions on thesecond derivative on function f .
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If Hess(f ) is a negative definite matrix at x∗, then x∗ is alocal maximum value for f .
In the one-dimensional case, this is the same as if f ′′(x∗) < 0.
Definition
A matrix is positive definite if xTAx > 0∀x 6= 0.
Definition
A matrix is negative definite if xTAx < 0∀x 6= 0.
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If Hess(f ) is indefinite at x∗, then x∗ is a “saddle point”, meaninga maximum going in some directions, and a minimum going inothers.
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Let’s do a couple quick examples:
f (x) = −x2
f (x) = 3x + ln(x)
f (x) = 3− 4x + 2x2
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What about in the two dimensional case?
Consider a 2x2 symmetric matrix: A =
[a bb c
]
A is positive definite if a > 0 and detA > 0⇒ ac − b2 > 0
A is negative definite if a < 0 and ac − b2 > 0.
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What about in the two dimensional case?
Consider a 2x2 symmetric matrix: A =
[a bb c
]A is positive definite if a > 0 and detA > 0⇒ ac − b2 > 0
A is negative definite if a < 0 and ac − b2 > 0.
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Again, lets start with an easier example:
f (x , y) = −x2 − y2
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Lets solve: f (x) = x4 + x2 − 6xy + 3y2
First lets find the critical values:
∂f
∂x= 4x3 + 2x − 6y = 0
∂f
∂y= −6x + 6y = 0
fy = 0⇒ x = y , and fx = 0⇒ 4x3 = 4x ⇒ x3 = x .Critical values: x=-1,0,1. Plug back in to get (-1,-1), (0,0), (1,1).
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Lets solve: f (x) = x4 + x2 − 6xy + 3y2
First lets find the critical values:
∂f
∂x= 4x3 + 2x − 6y = 0
∂f
∂y= −6x + 6y = 0
fy = 0⇒ x = y , and fx = 0⇒ 4x3 = 4x ⇒ x3 = x .Critical values: x=-1,0,1. Plug back in to get (-1,-1), (0,0), (1,1).
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Lets solve: f (x) = x4 + x2 − 6xy + 3y2
First lets find the critical values:
∂f
∂x= 4x3 + 2x − 6y = 0
∂f
∂y= −6x + 6y = 0
fy = 0⇒ x = y , and fx = 0⇒ 4x3 = 4x ⇒ x3 = x .
Critical values: x=-1,0,1. Plug back in to get (-1,-1), (0,0), (1,1).
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Lets solve: f (x) = x4 + x2 − 6xy + 3y2
First lets find the critical values:
∂f
∂x= 4x3 + 2x − 6y = 0
∂f
∂y= −6x + 6y = 0
fy = 0⇒ x = y , and fx = 0⇒ 4x3 = 4x ⇒ x3 = x .Critical values: x=-1,0,1. Plug back in to get (-1,-1), (0,0), (1,1).
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Hessian: Hess =
[12x2 − 2x− 6−6 6
]
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Unconstrained Optimization
Hessian: Hess =
[12x2 − 2x− 6−6 6
]
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Weierstrauss Theorem
Will every function have a maximum?
Theorem (Weierstrauss Theorem)
Let the function f : Rn → R be continuous. Let C be anon-empty, closed, and bounded subset of dom(f ). Then, ∃ amaximum for f in C.
Note that this is simply a sufficient condition (but verygeneral).
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Weierstrauss Theorem
Will every function have a maximum?
Theorem (Weierstrauss Theorem)
Let the function f : Rn → R be continuous. Let C be anon-empty, closed, and bounded subset of dom(f ). Then, ∃ amaximum for f in C.
Note that this is simply a sufficient condition (but verygeneral).
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Definition
A set E is non-empty if it has at least one element.
Definition
A set E is open if ∀ x ∈ E∃B(x) such that B(x) ⊂ E
Definition
A set E is closed if its complement is open.
Definition
A set E is bounded if ∃k such that k ≥ s∀s ∈ E .
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Which of these graphs has a maximum point? Whichcondition is not satisfied if not?
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• • xa b
• •
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(1)
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.
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(2)
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x◦ •a b
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(3)
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• xa
........................................................................
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....................
(4)
•
..........
.
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Unconstrained OptimizationConstrained Optimization
Optimization
Which of these graphs has a maximum point? Whichcondition is not satisfied if not?
...................................................................................................................................................................................................................................................................................................................................................................................
...............
...............
...............
...............
...............
...............
...............
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....
• • xa b
• •
x∗
(1)
..................................................................................................................................................................................................................................................................................................................................................
.
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• • xa bd
................................................................................................................................◦
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•
(2)
•................
•
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x◦ •a b
.....................................................................................................................................................................................................................................................................................................................
◦
•
(3)
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....
• xa
........................................................................
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....................
(4)
•
..........
.
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1) yes
2) no (not continuous)
3) no (not closed)
4) no (not bounded)
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1) yes
2) no (not continuous)
3) no (not closed)
4) no (not bounded)
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1) yes
2) no (not continuous)
3) no (not closed)
4) no (not bounded)
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1) yes
2) no (not continuous)
3) no (not closed)
4) no (not bounded)
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For each of the following functions, find the critical points andclassify them as local max, local min, saddle point, or can’t tell(from S&B Ex. 17.1, 17.2):
1 x2 − 6xy + 2y2 + 10x + 2y − 5
2 xy2 + x3y − xy
3 3x4 + 3x2y − y3
4 x2 + 6xy + y2 − 3yz + 4z2 − 10x − 5y − 21z
5 (x2 + 2y2 + 3z2)e−(x2+y2+z2)
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Now we move to constrained optimization problems, wherethe choice set becomes limited. These are much morecommon (and interesting) in economics.
Since we will generally assume utility functions are increasingfunctions, with no constraint there will be no maximum...
These can come in a variety of forms:x ≥ 0, x + y = 25, x2 − 3y + z ≤ 5, . . ..
One simple way to solve a constrained maximization problemis to solve it as if it were unconstrained and see if the answersatisfies all the constraints.
If it does, then you are done.If not, you must try something else.
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Unconstrained OptimizationConstrained Optimization
Constrained Optimization
Now we move to constrained optimization problems, wherethe choice set becomes limited. These are much morecommon (and interesting) in economics.
Since we will generally assume utility functions are increasingfunctions, with no constraint there will be no maximum...
These can come in a variety of forms:x ≥ 0, x + y = 25, x2 − 3y + z ≤ 5, . . ..
One simple way to solve a constrained maximization problemis to solve it as if it were unconstrained and see if the answersatisfies all the constraints.
If it does, then you are done.If not, you must try something else.
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Unconstrained OptimizationConstrained Optimization
Constrained Optimization
Now we move to constrained optimization problems, wherethe choice set becomes limited. These are much morecommon (and interesting) in economics.
Since we will generally assume utility functions are increasingfunctions, with no constraint there will be no maximum...
These can come in a variety of forms:x ≥ 0, x + y = 25, x2 − 3y + z ≤ 5, . . ..
One simple way to solve a constrained maximization problemis to solve it as if it were unconstrained and see if the answersatisfies all the constraints.
If it does, then you are done.If not, you must try something else.
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How to solve an optimization problem subject to a givenconstraint?
maxx ,y
f (x , y) such that x + y = Z̄
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A convenient way to solve such problems is to use theLagrangian function:
L(x , y , λ) ≡ f (x , y)− λ(Z̄ − x − y)
We re-arranged the constraint to equal zero (in the case of anequality), then add it into the problem, multiplied by λ.
λ is known as the Lagrange multiplier.
Now we have transformed the constrained problem into anunconstrained problem.
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So again lets start with an easier example:
f (x) = −x2 such that x = −4 (1)
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Now moving into two variables:
Maximize f (x1, x2) = x1x2 such that x1 + 4x2 = 16
Begin by forming the Lagrangian: L = x1x2− λ(x1 + 4x2− 16)
Then set partial derivatives to 0:
∂L
∂x1= x2 − λ = 0
∂L
∂x2= x1 − 4λ = 0
∂L
∂λ= −(x1 + 4x2 − 16) = 0
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Now moving into two variables:
Maximize f (x1, x2) = x1x2 such that x1 + 4x2 = 16
Begin by forming the Lagrangian: L = x1x2− λ(x1 + 4x2− 16)
Then set partial derivatives to 0:
∂L
∂x1= x2 − λ = 0
∂L
∂x2= x1 − 4λ = 0
∂L
∂λ= −(x1 + 4x2 − 16) = 0
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Now moving into two variables:
Maximize f (x1, x2) = x1x2 such that x1 + 4x2 = 16
Begin by forming the Lagrangian: L = x1x2− λ(x1 + 4x2− 16)
Then set partial derivatives to 0:
∂L
∂x1= x2 − λ = 0
∂L
∂x2= x1 − 4λ = 0
∂L
∂λ= −(x1 + 4x2 − 16) = 0
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Then solve the system of equations. 3 equations, 3 unknowns:
λ = x2 = 14x1
x1 = 4x2
(4x2) + 4x2 = 16
x2 = 2
Substituting, we then find the answer to be: x1 = 8, x2 = 2, λ = 2.
λ can have some interpretation, though we won’t talk about ittoday.
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Then solve the system of equations. 3 equations, 3 unknowns:
λ = x2 = 14x1
x1 = 4x2
(4x2) + 4x2 = 16
x2 = 2
Substituting, we then find the answer to be: x1 = 8, x2 = 2, λ = 2.λ can have some interpretation, though we won’t talk about ittoday.
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Note that the previous example could have been the answer tothe following set up:
Suppose a consumer is deciding how much peanut butter (P)and jelly (J) to purchase.She has $16 to spend. Peanut Butter costs $1/jar and Jellycosts $4/jar.Her utility from consuming Peanut Butter and Jelly isu(P, J) = P ∗ J.How much of each should she purchase?
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If given several equality constraints, the process is similar:
maxx ,y ,z
f (x , y , z) such that h(x , y) = A, g(y , z) = B
Again form a Lagrangian, but add separate multipliers for eachconstraint:
L(x , y , z , λ1, λ2) ≡ f (x , y , z)− λ1(A− h(x , y))− λ2(B − g(y , z))
Now we solve a 5-variable optimization problem (Luckily we knowhow to solve for N variables)
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If given several equality constraints, the process is similar:
maxx ,y ,z
f (x , y , z) such that h(x , y) = A, g(y , z) = B
Again form a Lagrangian, but add separate multipliers for eachconstraint:
L(x , y , z , λ1, λ2) ≡ f (x , y , z)− λ1(A− h(x , y))− λ2(B − g(y , z))
Now we solve a 5-variable optimization problem (Luckily we knowhow to solve for N variables)
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Unconstrained OptimizationConstrained Optimization
Constrained Optimization
If given several equality constraints, the process is similar:
maxx ,y ,z
f (x , y , z) such that h(x , y) = A, g(y , z) = B
Again form a Lagrangian, but add separate multipliers for eachconstraint:
L(x , y , z , λ1, λ2) ≡ f (x , y , z)− λ1(A− h(x , y))− λ2(B − g(y , z))
Now we solve a 5-variable optimization problem (Luckily we knowhow to solve for N variables)
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Some more problems taken from S& B:
Maximize f (x1, x2) = x21x2 such that 2x2
1 + x22 = 3.
Maximize f (x , y , z) = yz + xz such that y2 + z2 = 1; xz = 3.
Maximize U(x1, x2) = kxa1x1−a
2 such that p1x1 + p2x2 = I .
Maximize f (x , y , z) = x2y2z2 such that x2 + y2 + z2 = c .
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Inequality constraints become more complicated. But alsomore common:
You don’t have to spend ALL your money...You can’t hire negative people...
We will cover an example here. I encourage you to read S& BChapter 18 for more.
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An inequality gives us two possibilities.
Either the inequality is binding, which results in the sameproblem that we had with and equality constraint.Or the inequality is non-binding, which means we should beable to find the answer as an unconstrained problem.
What to do? Basically we will add in extra constraints on theFOC to make sure we allow for either possibility.
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Consider the problem of maximizing:
f (x , y) = xy such that g(x , y) = x2 + y2 ≤ 1 (2)
Solve for the critical values of f :
This only one occurs at the origin: (x∗, y∗) = (0, 0), which isfar away from the constraint (and is a minimum).
Erik Hembre Math Boot Camp Part II UIC Economics Department
IntroductionOptimization
Unconstrained OptimizationConstrained Optimization
Constrained Optimization
Consider the problem of maximizing:
f (x , y) = xy such that g(x , y) = x2 + y2 ≤ 1 (2)
Solve for the critical values of f :
This only one occurs at the origin: (x∗, y∗) = (0, 0), which isfar away from the constraint (and is a minimum).
Erik Hembre Math Boot Camp Part II UIC Economics Department