v2007.09.13 - Convex Optimization

150 CHAPTER 2. CONVEX GEOMETRYb − Ay ≻0 ⇔ x T b > 0, A T x=0 ∀x ≽K ∗ K0, x ≠ 0 (305)From this, alternative systems of generalized inequality [46, pages:50,54,262]Ay ≺K ∗or in the alternativeb(306)x T b ≤ 0, A T x=0, x ≽K0, x ≠ 0derived from (305) by taking the complementary sense of the inequalityin x T b .And from this, alternative systems with respect to the nonnegativeorthant attributed to Gordan in 1873: [110] [41,2.2] substituting A ←A Tand setting b = 0A T y ≺ 0or in the alternativeAx = 0, x ≽ 0, ‖x‖ 1 = 1(307)2.13.3 Optimality conditionThe general first-order necessary and sufficient condition for optimalityof solution x ⋆ to a minimization problem ((263p) for example) withreal differentiable convex objective function f(x) : R n →R is [227,3](confer2.13.10.1) (Figure 53)∇f(x ⋆ ) T (x − x ⋆ ) ≥ 0 ∀x ∈ C , x ⋆ ∈ C (308)where C is the feasible set, a convex set of all variable values satisfying theproblem constraints, and where ∇f(x ⋆ ) is the gradient of f (3.1.8) withrespect to x evaluated at x ⋆ .