Non-linear Optimisation (FEB22006X)

Is the max function of two strictly convex functions a strictly convex function itself?

prove that any locally optimal solution to the convex optimization problem inf{ f (x ) : ci (x ) = 0 ∀i ∈ E, ci (x ) ≥ 0 ∀i ∈ I} is a (globally) optimal solution

9. Let f and g be convex functions from Rn to R. Prove that the following
are convex functions:
(a) x 7 → α f (x ) + β g(x ) for α, β ≥ 0.
(b) x 7 → f (Ax + b) for A ∈ Rn×m and b ∈ Rn.
(c) x 7 → max{ f (x ), g(x )}.
2 ©Erasmus University Rotterdam, 2023
(d) x 7 → ϕ( f (x )) where ϕ : R → R is convex and non-decreasing on
the range of f 

8. Prove that the following are convex sets:
(a) S ∩ T , where S, T ⊆ Rn are convex sets.
(b) The set of solutions to a system of linear equations.
(c) The set of global minimizers of a convex function.
(d) The ellipsoid {x ∈ Rn : x >Ax ≤ 1} for some symmetric, positive
definite A ∈ Rn×n.
Hint: use a spectral decomposition to write A = P D1/2 D1/2 P> for
some matrices P and D1/2, and apply the triangle inequality.

he Sum rule says the derivative of a sum of functions is the sum of their derivatives