8 Multivariable Optimisation Flashcards
For function f(x,y) what is the necessary first order condition for an interior extreme point
f’x(x,y)=0
And
f’y(x,y)=0
The function f(x,y) is concave for all (x,y) if…
f’’xx(x,y)<=0
f’’yy(x,y)<=0
f’’xx(x,y)f’’yy(x,y)- (f’’xy(x,y))^2 >=0
If f(x,y) is concave what can be said about a stationary point in this function
It is a global max
The function f(x,y) is convex for all (x,y) if…
f’’xx(x,y)>=0
f’’yy(x,y)>=0
f’’xx(x,y)f’’yy(x,y)- (f’’xy(x,y))^2 >=0
If f(x,y) is convex then what can be said about a stationary point on it
It is a global min
How do we know if a stationary point (x,y) in the function f(x,y) is a local max?
f’’xx(x,y)<0
f’’yy(x,y)<0
f’’xx(x,y)f’’yy(x,y)- (f’’xy(x,y))^2 >0
How do we know if a stationary point (x,y) in the function f(x,y) is a local min?
f’’xx(x,y)>0
f’’yy(x,y)>0
f’’xx(x,y)f’’yy(x,y)- (f’’xy(x,y))^2 >0
How do we identify a saddle point?
If
f’’xx(x,y)f’’yy(x,y)- (f’’xy(x,y))^2 <0
Then (x,y) is a saddle point
When is it not possible to classify an extreme point?
If
f’’xx(x,y)f’’yy(x,y)- (f’’xy(x,y))^2 =0
When is F(f(x)) maximised if F is a strictly increasing function
When f(x)is maximised
What is the pure existence theorem?
Suppose f(x1,…,xn) is a continuous function over a closed and bounded set s. Then there exists in s
- a point where f has a global max
- a point where f has a global min
How does the substitution method work
Substitute the constraint into the objective function
What is the Lagrange multiplier method
A method which finds the only possible solutions of the problem that maximise or minimise f(x,y) subject to g(x,y)=c
What does lamda represent in the Lagrange multiplier?
The rate at which the optimal value of the objective function changes with respect to changes in the constant c
What are the sufficient conditions for solutions to the lagrangian method
- if the Lagrangian is concave, the solution to the FOC gives the constrained max
- if the Lagrangian is convex, the solution to the FOC gives the constrained min
