Lecture 11- Constrained Optimisation: The Lagrange Multiplier Method

Question 1

What is an unconstrained optimisation?

Answer 1

When there is no constraint on what x and y can be- x and y can be any real values
… normal method of maximising/minimising a function

Question 2

What is a constrained/restricted optimisation?

Answer 2

When there is a constraint on what x and y can be- they can only be certain values e.g. the constraint might be x+y=40- here x and y can only be certain values
… maximising/minimising a function with a constraint

Question 3

What would be a typical constrained optimisation question?

Answer 3

Find the optimum value of z = f (x , y ) subject to g (x , y ) = c

For example in practice this could be the following in practice:
Find the optimum value of z = x^2 − xy + 2y^2 subject to the constraint y + x = 40

Question 4

What would you name each part of a typical constrained optimisation question?- identify the names given to each part of the question

Answer 4

Find the optimum value of z = f (x , y ) subject to g (x , y ) = c

For the question above z = f (x , y ) is just a function of x and y which is equal to z and this called the objective function (as you are trying to find the optimum of this function)

g (x , y ) = c is a function of x and y which is equal to c where tue function is called the functional constraint and the c is simply the constraint (not constant in this case)

Also the optimum value calculated for z is called the constrained optimum

Question 5

What is the Lagrange multiplier method?

Answer 5

Method used to solve constrained optimisation problems

Question 6

What is the general Langrange equation and identify the Langrange multiplier in the equation?
Also explain how you got each part of the equation

Answer 6

L = f (x,y) - lambda (g (x,y) - c)
… lambda is the Langrange multiplier

f (x,y) is taken directly from the question
g (x,y) - c is simply the functional constraint but everything is moved over to the left side

Question 7

What are the 4 steps of the Lagrange multiplier method?

Answer 7

1) Form the Lagrange equation from the information given in the question
2) Find the first order condition (FOC) by finding the partial derivative of L with respect to x, the with respect to y and lastly with respect to lambda (REMEMBER the partial derivative of L with respect to lambda will just be the functional constraint I.e g (x,y) - c which is equal to g (x,y) = c as there is no lambda in the objective function- 1st part of Lagrange equation and … lambda itself will just equal 1 when differentiated … it is just 1 multiplied by the functional constraint which is equal to the functional constraint)
3) Solve the 3 first order condition (FOC) partial derivative equations simultaneously to find a value for x, y and lambda which is denoted as x, y and lambda*
4) Substitute the first order values you found for x and y (x* and y) into the initial objective function (f (x,y) ) to find z as an actual value
This value is the optimum value of z

Question 8

What does the lambda signify?

Answer 8

Lambda is the marginal change in z due to a unitary change in the budget constraint
… this basically means if c was to increase by 1, z would increase by lambda* (the calculated value of lambda)