v2009.01.01 - Convex Optimization

4.2. FRAMEWORK 257
4.2.1.1.3 Example. “New” Farkas’ lemma.
In 1995, Lasserre [199,III] presented an example originally offered by
Ben-Israel in 1969 [30, p.378] as evidence of failure in semidefinite Farkas’
Lemma 4.2.1.1.1:
[ ]
A =
∆ svec(A1 ) T
svec(A 2 ) T =
[ 0 1 0
0 0 1
] [ 1
, b =
0
]
(600)
The intersection A ∩ S n + is practically empty because the solution set
{X ≽ 0 | A svec X = b} =
{[
α
1 √2
√1
2
0
]
≽ 0 | α∈ R
}
(601)
is positive semidefinite only asymptotically (α→∞). Yet the dual system
m∑
y i A i ≽0 ⇒ y T b≥0 indicates nonempty intersection; videlicet, for ‖y‖= 1
i=1
y 1
[
0
1 √2
1 √
2
0
]
+ y 2
[ 0 0
0 1
]
[ ] 0
≽ 0 ⇔ y =
1
⇒ y T b = 0 (602)
On the other hand, positive definite Farkas’ Lemma 4.2.1.1.2 shows
A ∩ int S n + is empty; what we need to know for semidefinite programming.
Based on Ben-Israel’s example, Lasserre suggested addition of another
condition to semidefinite Farkas’ Lemma 4.2.1.1.1 to make a “new” lemma.
Ye recommends positive definite Farkas’ Lemma 4.2.1.1.2 instead; which is
simpler and obviates Lasserre’s proposed additional condition.
4.2.1.2 Theorem of the alternative for semidefinite programming
Because these Farkas’ lemmas follow from membership relations, we may
construct alternative systems from them. Applying the method of2.13.2.1.1,
then from positive definite Farkas’ lemma, for example, we get
A ∩ int S n + ≠ ∅
or in the alternative
m∑
y T b ≤ 0, y i A i ≽ 0, y ≠ 0
i=1
(603)