v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 173From this, alternative systems of generalized inequality [61, pages:50,54,262]Ay ≺K ∗or in the alternativeb(350)x T b ≤ 0, A T x=0, x ≽K0, x ≠ 0derived from (349) by taking complementary sense of the inequality in x T b .And from this, alternative systems with respect to the nonnegativeorthant attributed to Gordan in 1873: [160] [55,2.2] substituting A ←A Tand setting b = 0A T y ≺ 0or in the alternative(351)Ax = 0, x ≽ 0, ‖x‖ 1 = 1Ben-Israel collects related results from Farkas, Motzkin, Gordan, and Stiemkein Motzkin transposition theorem. [33]2.13.3 Optimality condition(confer2.13.10.1) The general first-order necessary and sufficient conditionfor optimality of solution x ⋆ to a minimization problem ((302p) for example)with real differentiable convex objective function f(x) : R n →R is [306,3]∇f(x ⋆ ) T (x − x ⋆ ) ≥ 0 ∀x ∈ C , x ⋆ ∈ C (352)where C is a convex feasible set, 2.65 and where ∇f(x ⋆ ) is the gradient (3.6) off with respect to x evaluated at x ⋆ . In words, negative gradient is normal toa hyperplane supporting the feasible set at a point of optimality. (Figure 67)Direct solution to variation inequality (352), instead of the correspondingminimization, spawned from calculus of variations. [250, p.178] [132, p.37]One solution method solves an equivalent fixed point-of-projection problemx = P C (x − ∇f(x)) (353)2.65 presumably nonempty set of all variable values satisfying all given problem constraints;the set of feasible solutions.