v2009.01.01 - Convex Optimization

156 CHAPTER 2. CONVEX GEOMETRY
Ay ≼
K ∗
or in the alternative
b
(315)
x T b < 0, A T x=0, x ≽
derived from (314) simply by taking the complementary sense of the
inequality in x T b . These two systems are alternatives; if one system has
a solution, then the other does not. 2.57 [266, p.201]
By invoking a strict membership relation between proper cones (294),
we can construct a more exotic alternative strengthened by demand for an
interior point;
b − Ay ≻ 0 ⇔ x T b > 0, A T x=0 ∀x ≽ 0, x ≠ 0 (316)
K ∗ K
From this, alternative systems of generalized inequality [53, pages:50,54,262]
K
0
Ay ≺
K ∗
or in the alternative
b
(317)
x T b ≤ 0, A T x=0, x ≽
K
0, x ≠ 0
derived from (316) by taking the complementary sense of the inequality
in x T b .
And from this, alternative systems with respect to the nonnegative
orthant attributed to Gordan in 1873: [135] [48,2.2] substituting A ←A T
and setting b = 0
2.57 If solutions at ±∞ are disallowed, then the alternative systems become instead
mutually exclusive with respect to nonpolyhedral cones. Simultaneous infeasibility of
the two systems is not precluded by mutual exclusivity; called a weak alternative.
Ye provides an example illustrating simultaneous [ ] infeasibility[ with respect ] to the positive
semidefinite cone: x∈ S 2 1 0
0 1
, y ∈ R , A = , and b = where x
0 0
1 0
T b means
〈x , b〉 . A better strategy than disallowing solutions at ±∞ is to demand an interior
point as in (317) or Lemma 4.2.1.1.2. Then question of simultaneous infeasibility is moot.