AMA 505代写、代写c/c++编程设计

日期：2022-11-16 07:30

AMA 505 Assignment 2 1st Sem, 2022 – 2023

Due date: November 30, 2022, 10:30 pm. No late submission will be accepted.

Please submit the assignment online.

Show your steps clearly. A mere numerical answer will receive no scores.

1. (a) Consider the following optimization problem.

Minimize

x∈IR2

x31 + x

3

2

Subject to x21 + 4x

2

2 ≤ 8,

2x2 ≥ x21.

i. (5 points) Show that the MFCQ holds at every feasible point.

ii. (30 points) Write down the KKT conditions and find all the stationary points.

(b) (5 points) Let h(y) =

∑m

i=1(y

4

i + e

2yi ? 1) and A ∈ IR(m+1)×n has full row rank, where 1 <

m < n. Let B ∈ IRm×n be the matrix formed from the first m rows of A, and let aT denote the

last row of A. Let δ > 0 and consider the set

C := {x ∈ IRn : h(Bx) ≤ δ, aTx = 1}.

Show that the MFCQ holds at every point in C.

2. Consider the following optimization problem, where n ≥ 2022:

Minimize

x∈IRn

n∑

i=1

x2i

Subject to

n∑

i=2

xi ≥ 2.

(a) (15 points) For each c > 0, define

qc(x) :=

n∑

i=1

x2i +

c

2

(

2?

n∑

i=2

xi

)2

+

Argue that qc is convex and find the global minimizer of qc.

(b) (15 points) For each μ > 0, define

fμ(x) :=

n∑

i=1

x2i ? μ ln

(

n∑

i=2

xi ? 2

)

.

Argue that fμ is convex and find the global minimizer of fμ. You may use without proof the

fact that the function t 7→ ? ln(t? 2) is convex (as an extended real-valued function).

3. (a) For each of the following optimization problems, write a CVX code that solves it, if possible.

Also write down the optimal value returned by CVX (corrected to 4 decimal places).

i. (10 points)

Minimize |3x1 + 4x3 ? 3|+ |x1 + x2 + x3 + 6|

Subject to 2x21 + 6x

2

2 + 10x

2

3 + x1x3 ≤ 5,

max{|x1|, x2, x3} ≤ 3.

ii. (10 points)

Minimize |x1 ? x2 ? x3 + 1|+ (x1 ? 2x2 + 3x3 + 1)6

Subject to (x22 + x

2

3 + 1)

3 ≤ 2020,[

5x2 x1 + x3

x1 + x3 x3

]

2I.

(b) (10 points) Explain whether the following optimization problems can be reformulated equiva-

lently as an SDP problem.

Minimize |x1|+ (x1 ? 3x2)2

Subject to (x21 + x

2

2 + 1)

2 ≤ x3.