Lecture 18 Subgradients

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Lecture 18 Subdifferential Set Properties Theorem 1 A subdifferential set ∂f(ˆx) is convex and closed Proof H7. Theorem 2 (Existence) Let f be convex with a nonempty dom f. Then: (a) For x ∈ relint(dom f), we have ∂f(x) ≠ ∅. (b) ∂f(x) ≠ ∅ is nonempty and bounded if and only if x ∈ int(dom f). Implications • The subdifferential ∂f(ˆx) is nonempty compact convex set for every ˆx in the interior of dom f. • When dom f = R n , ∂f(x) is nonempty compact convex set for all x Convex Optimization 5
Lecture 18 Partial Proof: If ˆx ∈ int(dom f), then ∂f(ˆx) is nonempty and bounded. • ∂f(ˆx) Nonempty. Let ˆx be in the interior of dom f. The vector (ˆx, f(ˆx)) does not belong to the interior of epi f. The epigraph epi f is convex and by the Supporting Hyperlane Theorem, there is a vector (d, β) ∈ R n+1 , (d, β) ≠ 0 such that d T ˆx + βf(ˆx) ≤ d T x + βw for all (x, w) ∈ epi f We have epi f = {(x, w) | f(x) ≤ w, x ∈ dom f}. Hence, d T ˆx + βf(ˆx) ≤ d T x + βw for all x ∈ dom f, f(x) ≤ w We must have β ≥ 0. We cannot have β = 0 (it would imply d = 0). Dividing by β, we see that −d/β is a subgradient of f at ˆx Convex Optimization 6
Page 1 and 2: Lecture 18 Subgradients November 3,
Page 3 and 4: Convex-Constrained Non-differentiab
Page 5: Lecture 18 Subgradients and Epigrap
Page 9 and 10: Lecture 18 Boundedness of the Subdi
Page 11 and 12: Lecture 18 Continuity of the Subdif
Page 13 and 14: Lecture 18 Proof When x ∈ int(dom
Page 15 and 16: Lecture 18 Letting z = x, w ↓ f(z
Page 17 and 18: Lecture 18 Examples • The functio
Page 19: Lecture 18 Optimality Conditions: C

Lecture 18 Subgradients

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