Semidefinite Farkas Lemma and Dimensional Rigidity of Bar Frameworks - PowerPoint PPT Presentation

Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks A. Y. Alfakih University of Windsor, ON Canada Workshop on Making Models: Stimulating research in rigidity Theory and Spatial-Visual Reasoning Fields Institute, Aug 2014 Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

Farkas Lemma Given an m × n matrix A and m -vector b , Farkas Lemma states that exactly one of the following two statements holds: there exists x ∈ R n such that Ax = b , x ≥ 0, there exists y ∈ R m such that A T y ≤ 0, b T y > 0. Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

Farkas Lemma Given an m × n matrix A and m -vector b , Farkas Lemma states that exactly one of the following two statements holds: there exists x ∈ R n such that Ax = b , x ≥ 0, there exists y ∈ R m such that A T y ≤ 0, b T y > 0. Farkas Lemma is at the heart of optimization theory. Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

Farkas Lemma Given an m × n matrix A and m -vector b , Farkas Lemma states that exactly one of the following two statements holds: there exists x ∈ R n such that Ax = b , x ≥ 0, there exists y ∈ R m such that A T y ≤ 0, b T y > 0. Farkas Lemma is at the heart of optimization theory. In one direction the proof is easy. In the other direction, the proof is based on the separation theorm. S a Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

Farkas Lemma Given an m × n matrix A and m -vector b , Farkas Lemma states that exactly one of the following two statements holds: there exists x ∈ R n such that Ax = b , x ≥ 0, there exists y ∈ R m such that A T y ≤ 0, b T y > 0. Farkas Lemma is at the heart of optimization theory. In one direction the proof is easy. In the other direction, the proof is based on the separation theorm. S a Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

LP and SDP The linear programming problem: p ∗ = min c T x ( P ) : subject to Ax = b x ≥ 0 Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

LP and SDP The linear programming problem: p ∗ = min d ∗ = max c T x b T y A T y ≤ c ( P ) : subject to Ax = b ( D ) : subject to x ≥ 0 Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

LP and SDP The linear programming problem: p ∗ = min d ∗ = max c T x b T y A T y ≤ c ( P ) : subject to Ax = b ( D ) : subject to x ≥ 0 LP strong duality: If both (P) or (D) are feasible, then both (P) and (D) have optimal solutions x ∗ and y ∗ respectively, and p ∗ = c T x ∗ = d ∗ = b T y ∗ . Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

LP and SDP The linear programming problem: p ∗ = min d ∗ = max c T x b T y A T y ≤ c ( P ) : subject to Ax = b ( D ) : subject to x ≥ 0 LP strong duality: If both (P) or (D) are feasible, then both (P) and (D) have optimal solutions x ∗ and y ∗ respectively, and p ∗ = c T x ∗ = d ∗ = b T y ∗ . LP strict complementarity: If both (P) or (D) are feasible, then ∃ optimal pair x ∗ ≥ 0 and s ∗ = c − A T y ∗ ≥ 0 such that x ∗ i s ∗ i = 0 and x ∗ i + s ∗ i > 0 for all i . Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

LP and SDP The linear programming problem: p ∗ = min d ∗ = max c T x b T y A T y ≤ c ( P ) : subject to Ax = b ( D ) : subject to x ≥ 0 LP strong duality: If both (P) or (D) are feasible, then both (P) and (D) have optimal solutions x ∗ and y ∗ respectively, and p ∗ = c T x ∗ = d ∗ = b T y ∗ . LP strict complementarity: If both (P) or (D) are feasible, then ∃ optimal pair x ∗ ≥ 0 and s ∗ = c − A T y ∗ ≥ 0 such that x ∗ i s ∗ i = 0 and x ∗ i + s ∗ i > 0 for all i . Semidefinite Programming (SDP): p ∗ = min d ∗ = max trace( CX ) � i b i y i trace( A i X ) = b i � i A i y � C s. t. s. t. X � 0 Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

Semidefinite Farkas Lemma A T y ≤ 0 , b T y > 0. Farkas Lemma: Ax = b , x ≥ 0 , Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

Semidefinite Farkas Lemma A T y ≤ 0 , b T y > 0. Farkas Lemma: Ax = b , x ≥ 0 , Potential SD Farkas Lemma: trace( A i X ) = b i , X � 0 , i A i y i � 0 , b T y > 0. � Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

Semidefinite Farkas Lemma A T y ≤ 0 , b T y > 0. Farkas Lemma: Ax = b , x ≥ 0 , False SD Farkas Lemma: trace( A i X ) = b i , X � 0 , i A i y i � 0 , b T y > 0. � Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

Semidefinite Farkas Lemma A T y ≤ 0 , b T y > 0. Farkas Lemma: Ax = b , x ≥ 0 , False SD Farkas Lemma: trace( A i X ) = b i , X � 0 , i A i y i � 0 , b T y > 0. � Correct SD Farkas Lemma: Given symmetric matrices A 1 , . . . , A m and m -vector b . Then exactly one of the following two statements holds: ∃ X ≻ 0: trace ( A i X ) = b i , for i = 1 , . . . , m , ∃ Ω = y 1 A 1 + · · · + y m A m : Ω � 0 , � = 0, � i b i y i ≤ 0. Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

Semidefinite Farkas Lemma A T y ≤ 0 , b T y > 0. Farkas Lemma: Ax = b , x ≥ 0 , False SD Farkas Lemma: trace( A i X ) = b i , X � 0 , i A i y i � 0 , b T y > 0. � Correct SD Farkas Lemma: Given symmetric matrices A 1 , . . . , A m and m -vector b . Then exactly one of the following two statements holds: ∃ X ≻ 0: trace ( A i X ) = b i , for i = 1 , . . . , m , ∃ Ω = y 1 A 1 + · · · + y m A m : Ω � 0 , � = 0, � i b i y i ≤ 0. This Farkas Lemma is used to prove SD strong duality under Slater Condition. Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

Semidefinite Farkas Lemma Cont’d SD Farkas Lemma: Given symmetric matrices A 1 , . . . , A m and m -vector b and assume that there exists X ∗ � 0 such that trace ( A i X ∗ ) = b i for i = 1 , . . . , m . Then exactly one of the following two statements holds: ∃ X ≻ 0: trace ( A i X ) = b i , for i = 1 , . . . , m , ∃ Ω = y 1 A 1 + · · · + y m A m : Ω � 0 , � = 0, trace (Ω X ∗ ) = 0. Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih

Semidefinite Farkas Lemma Cont’d SD Farkas Lemma: Given symmetric matrices A 1 , . . . , A m and m -vector b and assume that there exists X ∗ � 0 such that trace ( A i X ∗ ) = b i for i = 1 , . . . , m . Then exactly one of the following two statements holds: ∃ X ≻ 0: trace ( A i X ) = b i , for i = 1 , . . . , m , ∃ Ω = y 1 A 1 + · · · + y m A m : Ω � 0 , � = 0, trace (Ω X ∗ ) = 0. Let F = { X � 0 :trace ( A i X ) = b i , for i = 1 , . . . , m } . Then F has no interior iff F is contained in the hyperplane: { X : trace (Ω X ) = 0 } . Workshop on Making Models: Stimulating ( University of Windsor, ON Canada ) Semidefinite Farkas’ Lemma and Dimensional Rigidity of Bar Frameworks / 18 A. Y. Alfakih