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 61/159 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. 62/159 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: 63/159 (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. 64/159 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. 65/159 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. 66/159 Yannis Paschalidis, Boston University SE524/EC524: Lecture 10