v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 157
A T y ≺ 0
or in the alternative
Ax = 0, x ≽ 0, ‖x‖ 1 = 1
(318)
2.13.3 Optimality condition
The general first-order necessary and sufficient condition for optimality
of solution x ⋆ to a minimization problem ((275p) for example) with
real differentiable convex objective function f(x) : R n →R is [265,3]
(confer2.13.10.1) (Figure 59)
∇f(x ⋆ ) T (x − x ⋆ ) ≥ 0 ∀x ∈ C , x ⋆ ∈ C (319)
where C is a convex feasible set (the set of all variable values satisfying the
problem constraints), and where ∇f(x ⋆ ) is the gradient of f (3.1.8) with
respect to x evaluated at x ⋆ . 2.58
In the unconstrained case, optimality condition (319) reduces to the
classical rule
∇f(x ⋆ ) = 0, x ⋆ ∈ domf (320)
which can be inferred from the following application:
2.13.3.0.1 Example. Equality constrained problem.
Given a real differentiable convex function f(x) : R n →R defined on
domain R n , a fat full-rank matrix C ∈ R p×n , and vector d∈ R p , the convex
optimization problem
minimize f(x)
x
(321)
subject to Cx = d
is characterized by the well-known necessary and sufficient optimality
condition [53,4.2.3]
∇f(x ⋆ ) + C T ν = 0 (322)
where ν ∈ R p is the eminent Lagrange multiplier. [264] Feasible solution x ⋆
is optimal, in other words, if and only if ∇f(x ⋆ ) belongs to R(C T ).
2.58 Direct solution to (319) instead is known as a variation inequality from the calculus
of variations. [215, p.178] [110, p.37]