v2007.09.13 - Convex Optimization

148 CHAPTER 2. CONVEX GEOMETRYBy alternative is meant: these two systems are incompatible; one system isfeasible while the other is not.2.13.2.1.1 Example. Theorem of the alternative for linear inequality.Myriad alternative systems of linear inequality can be explained in terms ofpointed closed convex cones and their duals.Beginning from the simplest Cartesian dual generalized inequalities (277)(with respect to the nonnegative orthant R m + ),y ≽ 0 ⇔ x T y ≥ 0 for all x ≽ 0 (293)Given A∈ R n×m , we make vector substitution A T y ← yA T y ≽ 0 ⇔ x T A T y ≥ 0 for all x ≽ 0 (294)Introducing a new vector by calculating b ∆ = Ax we getA T y ≽ 0 ⇔ b T y ≥ 0, b = Ax for all x ≽ 0 (295)By complementing sense of the scalar inequality:A T y ≽ 0or in the alternativeb T y < 0, ∃ b = Ax, x ≽ 0(296)If one system has a solution, then the other does not; define aconvex cone K={y ∆ | A T y ≽0} , then y ∈ K or in the alternative y /∈ K .Scalar inequality b T y