Week 10

Question 1

What are some of the properties of convex functions?

Answer 1

1. Local minima = global minima
2. global minima need not be unique
3. set of all global minima of a convex function is a convex set.

Question 2

What are the necessary and sufficient conditions for optimality of convex functions?

Answer 2

Necessary Condition: let f be a differential, convex function from Rd->R, xbelongs to Rd is a global minimum of f if and only if gradient (f(x)) =0

Sufficient condition: If there exists x* such that gradient (f(x*))=0

Question 3

What is the property of convex functions?

Answer 3

If f and g are convex functions and if h(x)=f(x)+g(x) , then h(x) is convex.

i.e., sum of convex functions are convex

Question 4

What is the propery regarding composition of convex functions?

Answer 4

let f be a convex and non decreasing function and g be a convex function. If h(x)= fog(x) then h(x) is also convex.

Note: In general, if f and g are convex, then h=fog may not be convex

Question 5

What is the property of convex functions on composition with linear function?

Answer 5

Let f(R->R) be a convex function and g(Rd->R) be a linear function and if h=fog then h is a convex function.

Note: In general, if f and g are convex, then h=fog may not be convex