$35.00
1. When considering a second order cone constraint, a temptation might be to square it in order
to obhain a classical convex quadratic constraint. This might not always work. Consider the
constraint
2×1 + x2 ≥ kxk2
,
and its squared conterpart:
(2×1 + x2)
2 ≥ kxk
2
2
.
Is the set defined by the first inequality convex? Is the set defined by the second inequality
convex? Draw them both and discuss.
2. We would like to minimize the function f : R
3
7→ R, with values:
f(x) = max
x1 + x2 − min(min(x1 + 2, x2 + 2×1 − 5), x3 − 6),
(x1 − x3)
2 + 2x
2
2
1 − x1
,
with the constraint kxk∞ < 1. Explain precisely how to formulate the problem as an SOCP in
standard form. Solve using GAMS.
3. The returns on n = 4 assets are described by a Gaussian (normal) random vector r ∈ R
4
, having
the following expected value rˆ and covariance matrix Σ:
rˆ =
0.12
0.10
0.07
0.03
, Σ =
0.0064 0.0008 −0.0011 0
0.0008 0.0025 0 0
−0.0011 0 0.0004 0
0 0 0 0
.
The last (fourth) asset corresponds to a risk-free investment. An investor wants to design a
portfolio mix with weights x ∈ R
4
(each weight xi is non-negative , and the sum of the weights
is one) so as to obtain the best expected return rˆ
T x, while guaranteeing that
(a) no single asset weights more that 40%;
(b) the risk-free assests should not weight more that 20%;
(c) no asset should weight less than 5%;
(d) the probability of experiencing a return lower than q = −3% should be no larger that
= 10−4
.
What is the maximal achievable expected return, under the above constraints?
Problem 0 Page 1
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