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v2006.03.09 - Convex Optimization

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6.2. FRAMEWORK 3656.2.1.1.3 Example. “New” Farkas’ lemma.In 1995, Lasserre [141,III] presented an example originally offered byBen-Israel in 1969 [23, p.378] as evidence of failure in semidefinite Farkas’Lemma 6.2.1.1.1:[ ]A =∆ svec(A1 ) Tsvec(A 2 ) T =[ 0 1 00 0 1] [ 1, b =0](860)The intersection A ∩ S n + is practically empty because the solution set{X ≽ 0 | A svec X = b} ={[α1 √2√120]≽ 0 | α∈ R}(861)is positive semidefinite only asymptotically (α→∞). Yet the dual systemm∑y i A i ≽0 ⇒ y T b≥0 indicates nonempty intersection; videlicet, for ‖y‖= 1i=1y 1[01 √21 √20]+ y 2[ 0 00 1][ ] 0≽ 0 ⇔ y =1⇒ y T b = 0 (862)On the other hand, positive definite Farkas’ Lemma 6.2.1.1.2 showsA ∩ int S n + is empty; what we need to know for semidefinite programming.Based on Ben-Israel’s example, Lasserre suggested addition of anothercondition to semidefinite Farkas’ Lemma 6.2.1.1.1 to make a “new” lemma.Ye recommends positive definite Farkas’ Lemma 6.2.1.1.2 instead; which issimpler and obviates Lasserre’s proposed additional condition. 6.2.1.2 Theorem of the alternative for semidefinite programmingBecause these Farkas’ lemmas follow from membership relations, we mayconstruct alternative systems from them. From positive definite Farkas’lemma and using the method of2.13.2.1.1, we getA ∩ int S n + ≠ ∅or in the alternativem∑y T b ≤ 0, y i A i ≽ 0, y ≠ 0i=1(863)

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