v2007.09.13 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 149For another example, from membership relation (275) with affinetransformation of dual variable we may write: Given A∈ R n×m and b∈R nb − Ay ∈ K ∗ ⇔ x T (b − Ay)≥ 0 ∀x ∈ K (300)A T x=0, b − Ay ∈ K ∗ ⇒ x T b ≥ 0 ∀x ∈ K (301)From membership relation (300), conversely, suppose we allow any y ∈ R m .Then because −x T Ay is unbounded below, x T (b −Ay)≥0 implies A T x=0:for y ∈ R mIn toto,A T x=0, b − Ay ∈ K ∗ ⇐ x T (b − Ay)≥ 0 ∀x ∈ K (302)b − Ay ∈ K ∗ ⇔ x T b ≥ 0, A T x=0 ∀x ∈ K (303)Vector x belongs to cone K but is also constrained to lie in a subspaceof R n specified by an intersection of hyperplanes through the origin{x∈ R n |A T x=0}. From this, alternative systems of generalized inequalitywith respect to pointed closed convex cones K and K ∗Ay ≼K ∗or in the alternativeb(304)x T b < 0, A T x=0, x ≽K0derived from (303) simply by taking the complementary sense of theinequality in x T b . These two systems are alternatives; if one system hasa solution, then the other does not. 2.48 [228, p.201]By invoking a strict membership relation between proper cones (282),we can construct a more exotic alternative strengthened by demand for aninterior point;2.48 If solutions at ±∞ are disallowed, then the alternative systems become insteadmutually exclusive with respect to nonpolyhedral cones. Simultaneous infeasibility ofthe two systems is not precluded by mutual exclusivity; called a weak alternative.Ye provides an example illustrating simultaneous [ ] infeasibility[ with respect ] to the positivesemidefinite cone: x∈ S 2 1 00 1, y ∈ R , A = , and b = where x0 01 0T b means〈x , b〉 . A better strategy than disallowing solutions at ±∞ is to demand an interiorpoint as in (306) or Lemma 4.2.1.1.2. Then question of simultaneous infeasibility is moot.