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Unformatted text preview: x. Then we can ﬁnd a vector ak ̸= 0 and a number ck ∈ R such that
¯
⟨ak , xk ⟩ ≥ ck ≥ ⟨ak , z ⟩
for all z ∈ C . By dividing both sides by ∥ak ∥ ̸= 0, without loss of generality we
may assume ∥ak ∥ = 1. Since xk → x, the sequence {ck } is bounded. Therefore
¯
we can ﬁnd a convergent subsequence (akl , ckl ) → (a, c). Letting l → ∞, we get
⟨a, x⟩ ≥ c ≥ ⟨a, z ⟩
¯
for any z ∈ C .
Proof of Theorem 2. Let E = C − D = {x − y  x ∈ C, y ∈ D}. Since C, D
are nonempty and convex, so is E . Since C ∩ D = ∅, we have 0 ∈ E . In
/
particular, 0 ∈ int E . By Proposition 4, there exists a ̸= 0 such that ⟨a, 0⟩ =
/
0 ≥ ⟨a, z ⟩ for all z ∈ E . By the deﬁnition of E , we have
⟨a, x − y ⟩ ≤ 0 ⇐⇒ ⟨a, x⟩ ≤ ⟨a, y ⟩
for any x ∈ C and y ∈ D. Letting supx∈C ⟨a, x⟩ ≤ c ≤ inf y∈D ⟨a, y ⟩, it follows
that the hyperplane ⟨a, x⟩ = c separates C and D.
Suppose that C is closed and D is compact. Let us show that E = C − D
is closed. For this purpose, suppose that {zk } ⊂ E and zk → z . Then we can
take {xk } ⊂ C , {yk } ⊂ D such that zk = xk − yk . Since D is compact, there is
a subsequence such that ykl → y ∈ D. Then xkl = ykl + zkl → y + z , but since
C is closed, x = y + z ∈ C . Therefore z = x − y ∈ E , so E is closed.
Since E = C − D is closed and 0 ∈ E , by Proposition 4 there exists a ̸= 0
/
such that ⟨a, 0⟩ = 0 > ⟨a, z ⟩ for all z ∈ E . The rest of the proof is similar. 2014W Econ 172B 3 Operations Research (B) Alexis Akira Toda Cones A set C ⊂ RN is said to be a cone if it contains a ray originating from 0 and
passing through any point of C . Formerly, C is a cone if x ∈ C and α ≥ 0
implies αx ∈ C . An example of a cone is the positive orthant
RN = x = (x1 , . . . , xN ) ∈ RN (∀n)xn ≥ 0 .
+
Another example is the set
K αk ak (∀k )αk ≥ 0 , x= The sum of (ak * ak) given that ak is > 0 k=1 where a1 , . . . , aK are vectors. The latter set is called the cone generated by vectors a1 , . . . , aK , and is denoted by cone[a1 , . . . , aK ]. Clearly RN = cone[e1 , . . . , eN ],
+
where e1 , . . . , eN are unit vectors of RN . A cone generated by vectors are also
called polyhedral cone because it is a polyhedral. A polyhedral cone is a closed
convex cone (easy).
Let C ⊂ RN be any nonempty set. The set
C ∗ = y ∈ RN (∀x ∈ C ) ⟨x, y ⟩ ≤ 0
is called the dual cone of C . Thus the dual cone C ∗ consists of all vectors that
make an obtuse angle with any vector in C .
Proposition 5. Let ∅ = C ⊂ D. Then (i) the dual cone C ∗ is a nonempty,
̸
closed, convex cone, (ii) C ∗ = (...
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This document was uploaded on 02/18/2014 for the course ECON 172b at UCSD.
 Winter '08
 Foster,C

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