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.