Assignment 1
to be completed by November 10, 2014
Exercise 1 Prove the following statement :
Let M be a nonempty and closed convex set in Rn , and let x be a point outside M
(x 6∈ M ). Consider the optimization program
min{|x − y| | y ∈ M }.
The program is solvable and has a unique solution y ∗ , and the linear form aT h, a = x−y ∗ ,
strongly separates x and M : supy∈M aT y = aT y ∗ = aT x − |a|2 .
Exercise 2 Prove the Homogeneous Farkas Lemma :
System
aT x < 0, aTi x ≥ 0, i = 1, ..., m,
(1)
has no solutions if and only if the vector a is a linear combination with nonnegative
coefficients of the vectors a1 , ..., am :
{ (1) is infeasible} ⇔ {∃λ ≥ 0 : a =
m
X
λ i ai }
i=1
Exercise 3 Solve the problem
n
X
ai
i=1 xi
→ min | x ∈ Q = {x ∈ Rn | x > 0,
n
X
x4i ≤ 1}.
i=1
where ai ≥ 0, i = 1, ..., n. What is the optimal value of the problem ? What if at least one
of ai were negative ?
Exercise 4 Let S(3) be the space of real symmetric matrices of size 3 × 3, and S+ (3) ⊂
S(3) the set of positive semidefinite matrices of size 3 × 3.
a) Verify that S+ (3) is a closed cone.
For x, y ∈ S+ (3) we define the scalar product
hx, yi = Trace(xT y) = Trace(xy) =
3
X
k,l=1
1
xkl ykl .
b) Let


−2 0 

 3

z=
 −2

0

1
.
0 

0
1


We can easily verify that z is not positive semidefinite – its eigenvalues are −0.2361, 1
and 4.2361.
Find a hyperplane in S(3) which separates z and S+ (3) ( a non-zero matrix a ∈ S(3)
and a scalar β ∈ R such that ha, xi < β for all x ∈ S+ (3), and ha, zi = β.
Exercise 5 Function f is called log-convex if φ(x) = ln f (x) is convex.
a)Show that the function
f (u, v) = ln(exp{u} + exp{v})
is convex on R2 .
b) Prove that if g and h are two positive functions on Rn such that γ(x) = ln h(x) et
η(x) = ln h(x) are convex, then the function
ln(g(x) + h(x))
is convex on Rn (in other words, sum of log-convex functions is log-convex)
b) Find the values of p and q for which
f (u, v) − pu − qv
attains its minimum on R2 and find this minimum.
Exercise 6 We consider the convex piecewise-linear minimization problem
min
x
max (aT x
i=1,...,m i
+ bi ) ,
(2)
with variable x ∈ Rn .
1. Derive a dual problem, based on the Lagrange dual of the equivalent ( ?) problem
min
x
max yi , subject to aTi x + bi = yi , i = 1, ..., m ,
i=1,...,m
with variables x ∈ Rn , y ∈ Rm .
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2. Formulate the piecewise-linear minimization problem (2) as an LP, and form the
dual of the LP. Relate the LP dual to the dual obtained in 1).
Suppose we approximate the objective function in (2) by the smooth function
fγ (x) = γ ln
(m
X
exp γ
−1
(aTi x
+ bi )
)
,
i=1
and solve the unconstrained Geometric Program
min fγ (x)
(3)
x
3. Write down the problem dual to (3) (introduce new variable y = Ax + b as in 1))
Let Optmax and Optlog be the optimal values of (2) and (3), respectively.
4. Show that
0 ≤ Optlog − Optmax ≤ γ ln m.
Exercise 7 Compute the gradient with respect to x of following functions :
a) f (x) = bT x + c, where x ∈ Rn , b ∈ Rn , A ∈ Rn×n , c ∈ R,
b) f (x) = xT Ax/2 + bT x + c, where x ∈ Rn , b ∈ Rn , A ∈ Rn×n , c ∈ R,
c) f (x) = g(Ax + b), where x ∈ Rn , g : Rm → R, A ∈ Rm×n , b ∈ Rm ,
d) f (x) = cT x−1 c, where x ∈ S(n) (a symmetric n × n matrix), c ∈ Rn .
Hint : you may use the matrix identity : (I + A)−1 = I − (A−1 + I)−1 .
Exercise 8 Let S(n) be the space of real symmetric matrices of size n × n, and S+ (n) ⊂
S(n) the set of positive semidefinite matrices of size n × n. For x, y ∈ S+ (n) we define the
scalar product
hx, yi = Trace(xT y) = Trace(xy) =
n
X
xkl ykl .
k,l=1
a) Show that S+ (n) is a closed cone.
b) Let S++ (n) be the set of positive definite symmetric n × n matrices. Consider the
function f (x) = ln(det x−1 ) : S++ (n) → R. Let
φ(t) = ln(det(x + th)−1 ), x ∈ S++ (n), h ∈ S(n).
1) verify that φ is defined on an open interval Ix,h , containing the origin,
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2) show that
φ(t) = φ(0) −
n
X
ln(1 + tλi ), ∀t ∈ Ix,h ,
i=1
where λi = λi (x−1/2 hx−1/2 ) are the eigenvalues of the matrix x−1/2 hx−1/2 ,
3) Use 1) and 2) to prove that f (x) is strictly convex on S++ (n).
4) Compute the gradient ∇f (x) at x ∈ S++ (n).
Exercise 9 Let
f (x) = min{k x − y k22 , Ay ≤ b},
with x ∈ Rn , A ∈ Rm×n , et b ∈ Rm (in other words, f (x) is the squared Euclidean distance
from x to the polyhedra Y = {y ∈ Rn | Ay ≤ b}). We assume that the optimization
problem above is feasible.
1. Write down the Lagrange dual to the optimization problem in the definition of f ;
2. show that f is convex ;
3. Assuming that Y is a bounded polyhedral set, write down the expression of the
subdifferential ∂f (x) of f at x.
Exercise 10
Let 0 < p < 1. Show that f (x1 , x2 ) = −xp1 x21−p is convex on {(x, y)| x, y > 0}.
Let now n be a positive integer and 0 < α ≤ 1/n. Prove that the function
f (x) = −(x1 x2 ...xn )α
is convex on the domain {x ∈ Rn | x1 , ..., xn > 0}.
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