Lecture 18 Subgradients

Lecture 18
Optimality Conditions: Unconstrained Case
Unconstrained optimization
Assumption
minimize f(x)
• The function f is convex (non-differentiable) and proper
[f proper means f(x) > −∞ for all x and dom f ≠ ∅]
Theorem Under this assumption, a vector x ∗ minimizes f over R n if and
only if
0 ∈ ∂f(x ∗ )
• The result is a generalization of ∇f(x ∗ ) = 0
• Proof x ∗ is optimal if and only if f(x) ≥ f(x ∗ ) for all x, or equivalently
f(x) ≥ f(x ∗ ) + 0 T (x − x ∗ ) for all x ∈ R n
Thus, x ∗ is optimal if and only if 0 ∈ ∂f(x ∗ )
Convex Optimization 15