Lecture 2

Question 1

The first derivative

Answer 1

gradient of f
𝜵𝑓(𝒙)

Question 2

Directional derivative

Answer 2

𝜵𝑓(𝒙)^𝑇*𝒑
𝒑 = direction

Question 3

second derivative

Answer 3

Hessian Matrix
𝑯(𝒙) = 𝜵^2 𝑓(𝒙)

Question 4

Necessary condition

Answer 4

Statement A is a necessary condition for statement B if (and only if) the falsity of A
guarantees the falsity of B. In math notation: not𝐴⇒not𝐵

Question 5

Sufficient condition

Answer 5

Statement A is a sufficient condition for statement B , if (and only if) the truth of A
guarantees the truth of B. In math notation: 𝐴⇒𝐵

Question 6

Optimality Conditions:
1st Order Necessary
2nd Order Necessary
2nd Order Sufficient

Answer 6

1st Order Necessary : If 𝒙∗is a local minimum then 𝜵𝑓𝒙∗=𝟎

2nd Order Necessary: If 𝒙∗is a local minimum then 𝜵𝑓𝒙∗=𝟎and 𝑯𝒙∗is positive semi definite

2nd Order Sufficient : If 𝜵𝑓𝒙∗=𝟎and 𝑯𝒙∗is positive definite then 𝒙∗is a local minimum

Question 7

𝑓(𝒙) is
1. strictly convex
2. convex
3. strictly concave
4. concave
5. neither convex nor concave

Answer 7

𝑯(𝒙) is
1. positive definite
2. positive semi definite
3. negative definite
4. negative semi definite
5. -

Question 8

Calculating Eigenvalues

Answer 8

det(𝑨-𝜆𝑰)=0