[SOLVED] Si152 homework 1

Problem i. Write the gradient and Heissan matrix of the following formula. [10pts]
x
TAx + b
Tx + c (A ∈ Rn∗n
, b ∈ Rn
, c ∈ R)Problem ii. Write the gradient and Heissan matrix of the following formula. [10pts]
∥Ax − b∥
2
2
(A ∈ Rm∗n
, b ∈ Rm)Problem iii. Convert the following problem to linear programming. [10pts]
min
x∈Rn
∥Ax − b∥1 + ∥x∥∞ (A ∈ Rm∗n
, b ∈ Rm)Problem vi. Proof the convergence rates of the following point sequences. [30pts]
x
k =
1
k
x
k =
1
k!
x
k =
1
2
2
k
(Hint: Given two iterates x
k+1 and x
k
, and its limit point x
∗
, there exists real number q > 0,
satisfies
lim
k→∞x
k+1 − x
∗∥xk − x∗∥
= q
if 0 < q < 1, then the point sequence Q-linear convergence; if q = 1, then the point sequence
Q-sublinear convergence; if q = 0, then the point sequence Q-superlinear convergence)Problem v. Select the Haverly Pool Problem or the Horse Racing Problem in the courseware, compile the program using AMPL model language and submit it to https://neos-server.
org/neos/solvers/index.html.(Hint: both AMPL solver and NEOS solver can be used,
please indicate the type of solver used in the submitted job, show the solution results (eg:
screenshots attached to the PDF file), and submit the source code together with the submitted job, please package as .zip file, including your PDF and source code.) [40pts]