### Lecture 10 - Yannis Paschalidis

SE524/EC524 Optimization Theory and Methods
Yannis Paschalidis
[email protected], http://ionia.bu.edu/
Department of Electrical and Computer Engineering,
Division of Systems Engineering,
and Center for Information and Systems Engineering,
Boston University
Farkas’ lemma Cones Resolution thm.
Lecture 10: Outline
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1
Farkas’ lemma.
2
Application: Asset Pricing in an arbitrage-free environment.
3
Cones and extreme rays.
4
Unboundness conditions.
5
Resolution Theorem.
Yannis Paschalidis, Boston University
SE524/EC524: Lecture 10
Farkas’ lemma Cones Resolution thm.
Farkas’ lemma
Theorem
(Farkas’ lemma) Exactly one of the following two alternatives hold:
(a) ∃x ≥ 0 s.t. Ax = b.
(b)
∃p s.t. p′ A ≥ 0′ and p′ b < 0.
Corollary
Assume that any p satisfying p′ Ai ≥ 0, also satisfies p′ b ≥ 0. Then b can
be written as a nonnegative lin. combination of A1 , . . . , An .
Theorem
Suppose Ax ≤ b has at least one feasible solution. Let d scalar. Then the
following are equivalent:
(a)
∀ feasible sols. of Ax ≤ b we have c′ x ≤ d.
(b)
∃p ≥ 0 s.t. p′ A = c′ and p′ b ≤ d.
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Yannis Paschalidis, Boston University
SE524/EC524: Lecture 10
Farkas’ lemma Cones Resolution thm.
Cones
Definition
A set C ⊂ Rn is a cone if λx ∈ C ∀λ ≥ 0 and ∀x ∈ C .
Definition
Polyhedral cone: {x ∈ Rn | Ax ≥ 0}. If 0 is an extreme point we
have a pointed polyhedral cone.
Theorem
Let polyhedral cone C ⊂ Rn s.t. C = {x | a′i x ≥ 0}. Then the
following are equivalent:
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(a)
0 is an extreme point of C .
(b)
C does not contain a line.
(c)
∃ n lin. ind. vectors in a′1 , . . . , a′m .
Yannis Paschalidis, Boston University
SE524/EC524: Lecture 10
Farkas’ lemma Cones Resolution thm.
Recession cones and extreme rays
Let nonempty polyhedron P = {x | Ax ≥ b}.
Definition
Recession cone at y ∈ P:
{d | A(y + λd) ≥ b ∀λ ≥ 0} ⇒ {d | Ad ≥ 0}
d in recession cone are called rays of polyhedron.
Extreme rays of polyhedral cone C : x ∈ C s.t. n − 1 lin. ind.
constraints are active at x.
Extreme rays of recession cone of P are called extreme rays of P.
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Yannis Paschalidis, Boston University
SE524/EC524: Lecture 10
Farkas’ lemma Cones Resolution thm.
Unboundness Conditions
Theorem
Consider min c′ x over pointed C = {x | a′i x ≥ 0}. Cost = −∞ iff ∃
extreme ray d with c′ d < 0.
Theorem
Consider min c′ x over Ax ≥ b. Assume at least one extreme point
exists. Cost = −∞ iff ∃ extreme ray d with c′ d < 0.
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Yannis Paschalidis, Boston University
SE524/EC524: Lecture 10
Farkas’ lemma Cones Resolution thm.
Resolution Theorem
Theorem
Let P = {x ∈ Rn | Ax ≥ b} =
6 ∅ with at least one extreme point.
Let x1 , . . . , xk be the extreme points and w1 , . . . , wr a complete set
of rays. Then
△
Q=
X
k
i=1
λi xi +
r
X
θj wj | λi ≥ 0, θj ≥ 0,
j=1
k
X
i=1
λi = 1 = P
Converse is also true: Every set of the form of Q (finitely
generated) is a polyhedron.
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Yannis Paschalidis, Boston University
SE524/EC524: Lecture 10