v2009.01.01 - Convex Optimization

4.2. FRAMEWORK 259
4.2.2 Duals
The dual objective function from (584D) evaluated at any feasible point
represents a lower bound on the primal optimal objective value from (584P).
We can see this by direct substitution: Assume the feasible sets A ∩ S n + and
C ∗ are nonempty. Then it is always true:
〈 ∑
i
〈C , X〉 ≥ 〈b, y〉
〉
y i A i + S , X ≥ [ 〈A 1 , X〉 · · · 〈A m , X〉 ] y
〈S , X〉 ≥ 0
(607)
The converse also follows because
X ≽ 0, S ≽ 0 ⇒ 〈S,X〉 ≥ 0 (1386)
Optimal value of the dual objective thus represents the greatest lower bound
on the primal. This fact is known as the weak duality theorem for semidefinite
programming, [342,1.3.8] and can be used to detect convergence in any
primal/dual numerical method of solution.
4.2.3 Optimality conditions
When any primal feasible point exists relatively interior to A ∩ S n + in S n , or
when any dual feasible point exists relatively interior to C ∗ in S n × R m , then
by Slater’s sufficient condition (p.256) these two problems (584P) (584D)
become strong duals. In other words, the primal optimal objective value
becomes equivalent to the dual optimal objective value: there is no duality
gap (Figure 51); id est, if ∃X ∈ A ∩ int S n + or ∃S,y ∈ rel int C ∗ then
〈 ∑
i
〈C , X ⋆ 〉 = 〈b, y ⋆ 〉
y ⋆ i A i + S ⋆ , X ⋆ 〉
= [ 〈A 1 , X ⋆ 〉 · · · 〈A m , X ⋆ 〉 ] y ⋆
〈S ⋆ , X ⋆ 〉 = 0
(608)
where S ⋆ , y ⋆ denote a dual optimal solution. 4.10 We summarize this:
4.10 Optimality condition 〈S ⋆ , X ⋆ 〉=0 is called a complementary slackness condition, in
keeping with the tradition of linear programming, [82] that forbids dual inequalities in
(584) to simultaneously hold strictly. [265,4]