Conjugacy in Convex Analysis
In classical real analysis, the gradient of a differentiable function f : ℝn → ℝ. plays a key role - to say the least. Considering this gradient as a mapping x ↦ s(x) = ∇f(x) from (some subset X of) ℝn to (some subset S of) ℝn, an interesting object is then its inverse: to a given s ∈ S, associate the x ∈ X such that s = ∇f(x). This question may be meaningless: not all mappings are invertible! but could for example be considered locally, taking for X x S a neighborhood of some (x 0, s 0 = ∇f(x 0)), with ∇2 f continuous and invertible at x 0 (use the local inverse theorem).
KeywordsSupport Function Convex Analysis Conjugate Function Strict Convexity Closed Convex Cone
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