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EC123 - Maths > Optimisation > Flashcards

Flashcards in Optimisation Deck (30)
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1
Q

What does ‘only if’ mean?

A

Necessary but not sufficient

2
Q

What does ‘if’ mean?

A

Sufficient but not necessary

3
Q

What does ‘iff’ mean?

A

Both necessary and sufficient

4
Q

Describe the interval (a,b)

A

It is an open interval, that does not include a or b

5
Q

Describe the interval [a,b]

A

closed bound interval aka. including a and b

6
Q

What is a strict inequality?

A

<>

7
Q

What is a weak inequality?

A

≤≥

8
Q

How do you denote a maximising solution?

A

*

9
Q

What does a convex function look like?

A

U

10
Q

What does a concave function look like?

A

11
Q

How can you determine convexity or concavity of a 1-variable function?

A
  1. Using second order derivatives
    Convex (min point) → f’‘(x)≥0
    Concave (max point) → f’‘(x)≤0
  2. Line segment
    If the line segment joining any 2 points on the graph f(x) is below (above) the graph, or on the graph∴f(x) is concave (convex) if for any two x1,x2 , and t∈[0,1]

tf(x1)+(1-t)f(x2 )≤f(tx1+(1-t)x2)

If the inequality holds as ≥ the function in convex

12
Q

How can you determine convexity or concavity of a 2-variable function?

A
  1. ABC:
    If A<0 and AC-B2>0 → concave (max)

If A>0 and AC-B2>0 → convex (min)

  1. Line segment
    A function f(x_1,x_2) defined over a convex set S is concave if…

λf(x)+(1-λ)f(y)≤f(λx+(1-λ)y) If the inequality holds as ≥ the function in convex

13
Q

What is a local extreme point?

A

An extreme point over a given domain of the function

14
Q

What is a global extreme point?

A

An extreme point for the entire function

15
Q

What are the sufficient conditions for extreme points?

A

If f(x) or f(x,y) or L(x,y,λ) is a convex/concave continuous function over an interval I/concave set S, and c is an interior stationary point, then c in a min/max point for f(x)/f(x,y) in I/S

16
Q

What are the first order conditions for 1 and 2 variable functions?

A

1-varible → f(x) is differentiable over an interval I and that c is an interior point of I. Then c is an extreme point in I only if f’(x)=0

2-varibles → f(x,y) is differentiable over an interval I and that (x0,y0) is an interior point of I. Then c is an extreme point in I only if f’1 (x0,y0)=0 and f’2(x0,y0)=0

17
Q

Can you maximise a transform function?

A

Maximising/minimising a transformed function F(f(x)) is equivalent to maximising f(x), where F is strictly an increasing function

18
Q

What is the method of find + determining stationary points for 1-variable functions?

A

Finding an extreme point:

  1. Check interior extreme points using FOC → f’(x)=0
  2. Check end point values, and where f(x) isn’t defined for non-interior extreme points

Defining a stationary point, c:
1. Sufficient conditions → show if concave/convex to show point is a max/min

  1. Let f^j(x) be the jth derivative and n the smallest no. such that f^n (x)≠0. Then c is:
    - A local max. if n is even and f^n (c)<0
    - A local min. if n is even and f^n (c)>0
    - An inflection point if n is odd
19
Q

What is the method of find + determining stationary points for 2-variable functions?

A

Finding an extreme point:

  1. Check interior extreme pointing using FOC) → f’1 (x0,y0)=0 and f’2(x0,y0)=0
  2. Evaluate (solve simultaneous) the function at stationary points and compare
  3. Check end point values, and where f(x,y) isn’t defined to see if extreme point isn’t interior

Defining a stationary point, (x0,y0):
1. Determine if graph is concave/convex → shows if point is a max/min
2. Evaluate the function at stationary points and compare
Let: A=f’‘11 B=f’‘22 C=f’‘12 Then (x_0,y_0)is:
If A<0: AC-B2>0 → strict max AC-B2<0 → saddle point
If A>0: AC-B2>0 → strict min AC-B2=0 → test is inconclusive

20
Q

Define a saddle point

A

Saddle point (x0,y0) is a stationary point with the property that arbitrarily close to (x0,y0 ) there exists point (x,y) with f(x,y)

21
Q

What is a binding constraint?

A

A constraint that stops you from achieving the actual max

22
Q

How would you solve a non-linear constraint?

A

Solve normally, if constraints are violated + look for corner solutions

23
Q

How would you solve a non-negativity constraint?

A

Solve normally if x^≥0 and y^≥0 you are done. If x^<0 and y^<0 the constraints are violated + you must look for corner solutions

24
Q

What are the 2 methods of solving linear constrained optimisation?

A
  1. Substituting the constraint → sub constraint in
  2. Lagrange multiplier method → L(x,y,λ)=f(x,y)-λ[g(x,y)-c]
    maxf(x,y) is subject to g(x,y)=c,& λ is an unknown constant called the Lagrange multiplier
    → Find FOCs of the Lagrange equation; use simulations equations to find x, y and λ
    → Sufficient + necessary conditions are the same, find if L(x,y,λ) is concave etc.
    → For n variable cases L(x1…xn,λ1…λn )=f(x1…xn )-∑[(j=1)^m]λj [gj(x1…xn )-cj]
25
Q

What is the interpretation of the Lagrange multiplier, λ?

A
x^* and y^*  solve maxf(x,y) subject to g(x,y)=c  →  f* (c)=f* (x*(c),y*(c))    
then  df(c)= λdc⟺ df*(c) / dc=λ   

∴ λ represents the rate at which the value of f(x,y) changes when the constant c changes
(λ is called the shadow price / marginal value of the resource represented by the constraint)

26
Q

State the extreme value theory

A

Extreme value theorem (EVT) → if f is continuous over a closed bounded interval [x0,x1], then there exists a point min. point c in [x0,x1] and a max. point c’ so that f(c)≤f(x)≤f(c^’)

27
Q

States the mean value theorem

A

Mean value theorem (MVT) → if f is continuous over [x0,x1] and differentiable in (x0,x1), then there exists at least 1 interior point c with in (x0,x1) such that f’(c)=(f(x1)-f(x0))/(x1-x0 )

(aka. A point where the gradient of the sectant line = the gradient of the function at that point) → If f’(c)=0 there is a stationary point (Rolle’s theorem FOC)

28
Q

Define elasticity

A

A unit-free measure of the Δ in the function’s value when one of its arguments changes

29
Q

What is the equation for elasticity of f(x,y) in regards to x?

A
El(x)f= %∆f(x,y) / %∆x 
= ∆f(x,y) / f(x,y) ÷ ∆x/x
= ∆f(x,y) / f(x,y)∙x/∆x
= x/f(x,y)∙∂f(x,y)/∂x
=x/f∙∆f/∆x
30
Q

What is the equation for the function y=f(x,z) partial elasticities of f with respects to x & y?

A

El(x)f=x/f(x,y)∙∂f(x,y)/∂x = ∂lnf/∂lnx

El(y)f=y/(f(x,y))∙∂f(x,y)/∂y = ∂lnf/∂lnz