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Proof CVaR in Convex Form(a) The minimizer ζ ⋆ of F α (w, ζ) satisfies: 0 ∈ ∂ ζ F α (w, ζ ⋆ ).example we choose the following subgradient:s ζ F α (w, ζ ⋆ ) = 1 − 11 − α∫I (f (w, r) ≥ ζ ⋆ ) p (r) dr= 1 − 11 − α Pr (f (w, r) ≥ ζ⋆ ) = 0Forwhere I (·) is the indicator function. Consequently,Pr (f (w, r) ≥ ζ ⋆ ) = 1 − α =⇒ ζ ⋆ = VaR α (f (w, r)) .Daniel P. Palomar 18