Optimisation Flashcards
What does ‘only if’ mean?
Necessary but not sufficient
What does ‘if’ mean?
Sufficient but not necessary
What does ‘iff’ mean?
Both necessary and sufficient
Describe the interval (a,b)
It is an open interval, that does not include a or b
Describe the interval [a,b]
closed bound interval aka. including a and b
What is a strict inequality?
<>
What is a weak inequality?
≤≥
How do you denote a maximising solution?
*
What does a convex function look like?
U
What does a concave function look like?
∩
How can you determine convexity or concavity of a 1-variable function?
- Using second order derivatives
Convex (min point) → f’‘(x)≥0
Concave (max point) → f’‘(x)≤0 - 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
How can you determine convexity or concavity of a 2-variable function?
- ABC:
If A<0 and AC-B2>0 → concave (max)
If A>0 and AC-B2>0 → convex (min)
- 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
What is a local extreme point?
An extreme point over a given domain of the function
What is a global extreme point?
An extreme point for the entire function
What are the sufficient conditions for extreme points?
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
What are the first order conditions for 1 and 2 variable functions?
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
Can you maximise a transform function?
Maximising/minimising a transformed function F(f(x)) is equivalent to maximising f(x), where F is strictly an increasing function
What is the method of find + determining stationary points for 1-variable functions?
Finding an extreme point:
- Check interior extreme points using FOC → f’(x)=0
- 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
- 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
What is the method of find + determining stationary points for 2-variable functions?
Finding an extreme point:
- Check interior extreme pointing using FOC) → f’1 (x0,y0)=0 and f’2(x0,y0)=0
- Evaluate (solve simultaneous) the function at stationary points and compare
- 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
Define a saddle point
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)
What is a binding constraint?
A constraint that stops you from achieving the actual max
How would you solve a non-linear constraint?
Solve normally, if constraints are violated + look for corner solutions
How would you solve a non-negativity constraint?
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
What are the 2 methods of solving linear constrained optimisation?
- Substituting the constraint → sub constraint in
- 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]
What is the interpretation of the Lagrange multiplier, λ?
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)
State the extreme value theory
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^’)
States the mean value theorem
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)
Define elasticity
A unit-free measure of the Δ in the function’s value when one of its arguments changes
What is the equation for elasticity of f(x,y) in regards to x?
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
What is the equation for the function y=f(x,z) partial elasticities of f with respects to x & y?
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
