v2010.10.26 - Convex Optimization

172 CHAPTER 2. CONVEX GEOMETRYFrom membership relation (344), 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 (346)b − Ay ∈ K ∗ ⇔ x T b ≥ 0, A T x=0 ∀x ∈ K (347)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(348)x T b < 0, A T x=0, x ≽K0derived from (347) 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. 2.64 [307, p.201]By invoking a strict membership relation between proper cones (326),we can construct a more exotic alternative strengthened by demand for aninterior point;b − Ay ≻0 ⇔ x T b > 0, A T x=0 ∀x ≽K ∗ K0, x ≠ 0 (349)2.64 If solutions at ±∞ are disallowed, then the alternative systems become insteadmutually exclusive with respect to nonpolyhedral cones. Simultaneous infeasibility of thetwo systems is not precluded by mutual exclusivity; called a weak alternative. Ye providesan example illustrating simultaneous [ ] infeasibility with [ respect ] to the positive semidefinitecone: 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 interior point asin (350) or Lemma 4.2.1.1.2. Then question of simultaneous infeasibility is moot.