課程名稱︰ 機器學習特論
課程性質︰ 選修
課程教師︰ 林智仁
開課學院： 電資院
開課系所︰ 資工系
考試日期（年月日）︰ 2017/3/28
考試時限（分鐘）： 2.5hr
試題 :
Please give details of your answer. A direct answer without explanation is
not counted.
Your Answers must be in English.
You can bring notes and text book. Other books or electronic devices are
not allowed.
(數學式用latex格式表示)
Problem 1 (10 pts)
1. Consider the folloing function
f(x) = x^3, x \in R
(a) (5 pts) Is f convex?
(b) (5 pts) Is f quasi-convex?
You cannot answer this question by drawing figures. You must give
mathmatical proofs.
Problem 2 (15 pts)
2. Assume f_1,f_2 are convex functions. Consider
f(x) = min{f_1(x),f_2(x)}
(a) (7 pts) Is f convex?
(b) (8 pts) Is f concave?
Poblem 3 (15 pts)
3. In slide 2-19 we proved the existance of a separating hyperplane.
In the proof, we use
frac(d,dt)||d- c + t(u -d)|| < 0
to argue the existance of a small t \in (0,1) such that
||d - c + t(u-d)|| \leq ||d-c||
Instead of using (1), can you specifically derive a t^* > 0 so that
||d - c + t(u-d)|| \leq ||d-c||, \forall t \in [0.t^*] ?
Problem 4 (30 pts)
4. Let
g(x): R^n \rightarrow R
(a) (10 pts) Assume g(x) is convex, Is
f(x) = g(x)^2
convex or not ?
(b) (20 pts) Assume g(x) is strictly convex and
g(x) \geq 0, \forall x \in R ^n
Is
f(x) = g(x)^2
strictly convwx or not?
Problem 5 (15 pts)
5. Consider
g(x_1,x_2) = x_1 x_2, where x_1 > 0, x_2 > 0
Use the first-order condition to checke if this function is quasi-convex
or not.
Problem 6 (15 pts)
6. Consider
f(x) = 2e^x + e^{-x}
What is the conjugate function of f(x) ?