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THE CHINESE UNIVERSITY OF HONG KONG
Department of Systems Engineering and Engineering Management
SEEM5121 Numerical Optimization
Assignment 1
Due time and date: Feb 6, 2015, at the beginning of the class
Question 1.
Book [1] Exercise 2.12 (d,e,g)
Question 2.
Logarithmic barrier for the second-order cone. The function f (x, t) = − log(t2 − x> x),
with dom f = {(x, t) ∈ Rn × R|t > kxk2 } (i.e., the second-order cone), is convex. (The
function f is called the logarithmic barrier function for the second-order cone.)
• (a) Explain why t − (1/t)u> u is a concave function on dom f . (Hint: Use convexity
of the quadratic over linear function).
• (b) From this, show that − log(t − (1/t)u> u) is a convex function on dom f .
• (c) From this, show that f is convex.
Question 3.
Show that the following functions f : Rn → R are convex.
• (a) The difference between the maximum and minimum value of a polynomial on
a given interval, as a function of its coefficients:
where p(t) = x1 + x2 t + x3 t2 + . . . + xn tn−1 .
f (x) = sup p(t) − inf p(t),
t∈[a,b]
t∈[a,b]
a, b are real constants with a < b.
• (b) The “exponential barrier” of a set of inequalities:
f (x) =
m
X
e−1/fi (x) ,
dom f = {x | fi (x) < 0, i = 1, . . . , m}.
i=1
The functions fi are convex.
• (c) The function
g(y + αx) − g(y)
α>0
α
if g is convex and y ∈ dom g. (It can be shown that this is the directional derivative
of g at y in the direction x.)
f (x) = inf
Question 4.
For each of the following functions on Rn , explain how to calculate a subgradient at a
given x.
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(a) f (x) = sup0≤t≤1 p(t), where p(t) = x1 + x2 t + . . . + xn tn−1 .
(b) f (x) = x[1] + x[2] + . . . + x[k] , where x[i] denotes the i-th largest element of x.
(c) f (x) = kAx − bk2 + kxk2 where A ∈ Rm×n .
(d) f (x) = λmax (W + diag(x)) where W ∈ Sn , and diag(x) denotes the diagonal matrix
whose diagonal vector is x.
(e) f (x) = inf y kAy − xk∞ where A ∈ Rn×m .
(f) f (x) = supAy≤b x> y, where A ∈ Rm×n and the polyhedron defined by Ay ≤ b is
nonempty and bounded.
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