Math 301
Winter 2015
Homework 1
Due Wednesday, January 28
1. Prove the following statements regarding dual cones. All sets are in a Euclidean space. (Most
parts have very short answers.)
(a) Consider any set S. Its “dual cone”
S ∗ = {λ | λT x ≥ 0 for all x ∈ S}
is a closed convex cone.
(b) The bi-dual cone S, which we write as S ∗∗ = (S ∗ )∗ , satisfies S ⊆ S ∗∗ .
(c) Let C is a closed convex cone. If C ⊇ S then C ⊇ S ∗∗ . Hint. Assume for contradiction
that there is a point y in S ∗∗ but not in C. Use the separating hyperplane theorem
between {y} and C and establish a contradiction.
(d) If S 6= ∅ then S ∗∗ is the smallest closed convex cone containing S, i.e., it is the intersection
of all closed convex cones containing S.
(e) A nonempty closed convex cone C is reflexive, i.e., C ∗∗ = C.
(f) If a set S is solid (i.e., has nonempty interior), S ∗ is pointed.
(g) If a closed convex cone C is not solid, then C ∗ is not pointed.
(h) If K is a proper cone, i.e, closed, convex, solid, and pointed, then K ∗ is proper.
(i) For any proper cone K, the relation, K defined by x K y if and only if x − y ∈ K, is
a partial order.
(j) If x K y then αx K αy for any α ≥ 0.
(k) If x K 0 and y K 0 then x + y K 0.
(l) We write x ≺K y if and only if x − y ∈ int(K). If λ K ∗ 0 then λT x > 0 for all nonzero
x K 0.
(m) With a non-pointed convex cone, we cannot define a partial order as in part (i) since
the resulting relation is not antisymmetric.
(n) Let C1 ⊆ Rn and C2 ∈ Rm be nonempty cones and write
C1 × C2 = {(x1 , x2 ) | x1 ∈ C1 , x2 ∈ C2 }.
Then (C1 × C2 )∗ = C1∗ × C2∗ .
2. Cast the following optimization problem
minimize Tr(CX)
subject to QT X + XQ 0
X 0,
where X ∈ Sn is the optimization variable and Q and C are in Rn×n , into the SDP form
given in page 20 of the cone programming slides.
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3. Let a1 , a2 , . . . , aN be N points in Rn , and consider the problem
minimize max1≤i≤N kx − ai k2 ,
where x is the optimization variable. That is, we want to minimize the largest distance. (If
the ai ’s are village locations, the solution x? might be the place we would want to build a fire
station.)
(a) Pose the problem as a SOCP.
(b) Construct the dual problem.
(c) What is the geometric interpretation of the dual? Are the primal and dual problems
strictly feasible? Are they solvable? Does strong duality hold? What is the meaning of
complementary slackness?
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