Topic 4: Multivariate optimisation

Question 1

What are the conditions for a stationary point in a differential multivariate function (3)

Answer 1

-Function z = f(x,y) can have a max/min point in an interior point only if its a stationary point
-f₁‘(x,y) = 0, f₂‘(x,y) = 0
-If we take y as constant in f(x,y), we can think of it as a univariate function when differentiating

Question 2

Why is the FOC necessary only for interior extreme points (2)

Answer 2

-If the maximum is a corner solution, the derivative doesn’t need to exist
-Furthermore, it doesn’t need to be a stationary point if a boundary point

Question 3

How do you find the stationary points for multivariate functions (1)

Answer 3

-Once you partially differentiate, you set the simultaneous equations equal to each other and solve for x and y

Question 4

What does it mean if a set is convex (3,1)

Answer 4

-A set is convex if for all elements of the set, s₁, s₂, as well as λs₁ + (1-λ)s₂ belongs to the set S
-λ₁ + (1-λ)s₂ = weighted average
-A convex set is one where if you draw a straight line (weighted average), all points on the line located between s₁ and s₂ must be part of the set

-Not S is not necessarily a subset of R, but could be Rⁿ if the element could be a vector

Question 5

How can we define the concavity/convexity of a function over a convex set (4)

Answer 5

-Function f = f(x, y), defined over convex set S is concave if λf(X) + (1-λ)f(Y) ≤ f(λX + (1-λ)Y)
-weighted average line < curve at any point
-Note how the bold notation for X and Y means vectors
-If the inequalities are strict we say strictly concave/convex

Question 6

How can we define the concavity/convexity of a function over a convex set with differentiability (4)

Answer 6

-Suppose function f(x, y) is twice differentiable over a convex set S
-f is concave if for all (x, y) in S:
f₁₁’‘(x, y) ≤ 0, f₂₂’‘(x,y) ≤ 0, (f₁₁’‘(x,y))(f₂₂’‘(x,y)) - (f₁₂’‘(x,y))^2 ≥ 0
-Switch the signs of the first 2 equations for convexity
-The expressions above on the right correspond to the determinant of the Hessian matrix

Question 7

What is a saddle point (1)

Answer 7

-A saddle point is a form of stationary point where arbitrarily close to (x₀, y₀), there exist points (x, y) where f(x,y) < f(x₀, y₀), and f(x,y) > f(x₀, y₀)

Question 8

How can we define whether a stationary point is a max, min or saddle with second order conditions (2,3,1)

Answer 8

-Suppose f(x,y) is a function with continuous second order partial derivatives on domain S, and let (x₀, y₀) be an interior stationary point
-Let A = f₁₁’‘(x,y), B = f₁₂’‘(x,y), C = f₂₂’‘(x,y)

-If A<0, AC-B² > 0, then local max
-If A>0, AC-B² > 0, then local min
-If AC-B²<0, then saddle
(These 3 are the second order conditions)

-If AC-B² = 0, then could be any

Question 9

What are the definitions for interior and boundary points of a set (2)

Answer 9

-A point is an interior point if there exists a circle centred at the point which all points on in the circle lie in set S
-A boundary point is if every circle centred at the point where points lie both inside and outside of S

Question 10

What does it mean for a set to be closed, open, bounded and compact (4)

Answer 10

-A set is closed if it contains all boundary points
-A set is open if it doesn’t contain all boundary points
-A set is bounded if it can be contained within a sufficiently large circle (not bounded if x>1 as infinite)
-A set is compact if it is both closed and bounded

Question 11

How do you find extreme points for a compact set (3)

Answer 11

-Find all stationary points in the interior of S
-Find the largest and smallest values of f(x,y) on the boundary
-Compare the first 2

Question 12

What is the euclidean n-dimensional space and euclidean distance (2,2)

Answer 12

-The euclidean n-dimensional space ℝ^n is the set of all possible vectors x = (x1, x2, …) whose elements are real numbers
-For n = 1, 2, 3, ℝ^n is a line, plane and space, but for n > 3 it is a hyperspace

-The euclidean distance in ℝ^n is equal to |(x-y)| = √(x1 - y1)^2, (x2 - y2)^2 …
-An open ball with centre a = (a1, a2 ,, …) and radius r is the set of points such that |(x-a)| <r

Question 13

How can we restate the extreme value theorem and first order condition for the n-variable case (2)

Answer 13

EVT:
-Let f(x) be continuous, on a non-empty, closed and bounded set S in ℝ^n (S is a subset of ℝ^n). There exist points c, d in S where f has a min and a max

FOC:
-Let f(x) be differentiable on set s in ℝ^n, and let c be an interior point. A necessary condition for C to be a max/min is to be a stationary point, satisfying the condition that all the partial first order derivatives = 0