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4 (Fenchel-Young inequality) Any points φ in E and x in the domain of a function h : E → (−∞, +∞] satisfy the inequality h(x) + h∗ (φ) ≥ φ, x . Equality holds if and only if φ ∈ ∂h(x). 2 we analyzed the standard inequality-constrained convex program by studying its optimal value under perturbations. A similar approach works for another model for convex programming, particularly suited to problems with linear constraints. An interesting byproduct is a convex analogue of the chain rule for diﬀerentiable functions, ∇(f + g ◦ A)(x) = ∇f (x) + A∗ ∇g(Ax) (for a linear map A).

B) Compute dφ when φ(t) = t2 /2 and when φ is the function p deﬁned in Exercise 27. (c) Suppose φ is three times diﬀerentiable. Prove dφ is convex if and only if −1/φ is convex on int (dom φ). (d) Extend the results above to the function Dφ : (dom φ)n × (int (dom φ))n → R deﬁned by Dφ (x, y) = 17. i dφ (xi , yi ). ) (b) ⎧ ⎪ ⎨ x21 /x2 (x2 > 0) 0 (x = 0) ⎪ ⎩ +∞ (otherwise). 50 18. Fenchel duality ∗ Prove the function f (x) = −(x1 x2 . . xn )1/n (x ∈ Rn+ ) +∞ (otherwise) is convex. 19. (Domain of subdiﬀerential) If the function f : R2 → (−∞, +∞] is deﬁned by f (x1 , x2 ) = √ max{1 − x1 , |x2 |} (x1 ≥ 0) +∞ (otherwise), prove that f is convex but that dom ∂f is not convex.

M, x ∈ E. Geometrically, Gordan’s theorem says that 0 does not lie in the convex hull of the set {a0 , a1 , . . , am } if and only if there is an open halfspace {y | y, x < 0} containing {a0 , a1 , . . , am } (and hence its convex hull). This is another illustration of the idea of separation (in this case we separate 0 and the convex hull). Theorems of the alternative like Gordan’s theorem may be proved in a variety of ways, including separation and algorithmic approaches. We employ a less standard technique, using our earlier analytic ideas, and leading to a rather uniﬁed treatment.