ELEN90026 – INTRODUCTION TO OPTIMISATION ASSIGNMENT 1
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ASSIGNMENT 1
ELEN90026 – INTRODUCTION TO OPTIMISATION
Problem 1. Let f : D → R be twice continuously differentiable . Show that for ∀x ∈ D and ∀z ∈ D the following holds
∇f(z) − ∇f(x) = \0 1 H (x + t(z − x))(z − x)dt,
where H(x) = ∇2 f(x) .
Problem 2. Prove that all isolated local minimizers are strict.
Problem 3. Show that the following sets are convex:
(1) Halfspace: {x|aT x ≤ b}
(2) Ellipsoid: {x|(x − xc)T P−1(x − xc) ≤ 1}, where P ≻ 0 .
(3) Solution set of linear matrix inequality. The condition A(x) = x1A1 + ··· + xnAn ⪯ B
with Ai ⪰ 0 and B ⪰ 0, is called a linear matrix inequality in x.
Problem 4. Show that the following functions are convex:
(1) The function f(x) in (10. 13) [Exercise 10.2 of Numerical Optimiza- tion by Nocedal and Wright]
(2) Exponential function: f(x) = ea北 , for any a ∈ R .
(3) Power function: xa , for x > 0 and any a ≥ 1 .
(4) Negative entropy: xlog x, for x > 0 .
Problem 5. Consider the optimisation problems
• p1(*) = min北 ∈X1 f(x)
• p2(*) = min北 ∈X2 f(x)
• p13(*) = min北 ∈X1∩X3 f(x)
• p23(*) = min北 ∈X2∩X3 f(x)
where X1 ⊆ X2 . Prove the following statements:
(1) p1(*) ≥ p2(*) in other words, enlarging the feasible set cannot worsen the optimal objective .
(2) If p1(*) = p2(*), then it holds that
p13(*) = p1(*) ⇒ p23(*) = p2(*) .
(3) Assume that all problems above attain unique optimal solutions . Un-
der such a hypothesis, if p1(*) = p2(*), then it holds that
p23(*) = p2(*) ⇒ p13(*) = p1(*) .
Problem 6 (Exercise 10.4 of Numerical Optimization by Nocedal and Wright).
(1) Show that p∗ defined in (10.22) is a minimizer of (10. 13) .
(2) Find ∥p∗ ∥ and conclude that this norm is minimized when τi = 0 for all i with σi = 0 .
Problem 7. Let D be a nonempty convex subset of Rn . Let also f = (f1 , . . . ,fm), where fi : D → R, i = 1, . . . ,m, are convex functions, and let g : Rm → R be a function that is convex and monotonically nondecreasing over a convex set that contains the set {f(x)|x ∈ D}, in the sense that for all z and in this set such that z ≤ , g(z) ≤ g() . Show that
(1) The function h defined by h(x) = g (f(x)) is convex over D .
(2) If in addition, m = 1, g is monotonically increasing and f is strictly convex, then h is strictly convex.
Note that there are no assumptions on the differentiability of fi or g .
2022-08-30