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3Maximum over a Collection of Convex Functions is Convex

# 3Maximum over a Collection of Convex Functions is Convex -...

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The Maximum over a Collection of Convex Functions is Convex Consider V ( t ) = max x C f ( x,t ) where t lies in a convex subset T of R m and C is a subset of R n , n,m 1. Let a solution exist for every t T . Theorem If f ( x, · ) is convex for every x C , then V ( · ) is convex. Proof : Exercise. (Hint: Let t 0 , t 00 T , λ [0 , 1], and let x λ solve the problem at t = λt 00 +(1 - λ ) t 0 . Then V ( λt 00 + (1 - λ ) t 0 ) = f ( x λ ,λt 00 + (1 - λ ) t 0 ) ...). Example : π ( p,α ) = max q 0 { αpq - c ( q ) } , where c ( · ) is strictly increasing and strictly convex and α,p > 0. Since the objective function is aﬃne in α , π ( p, · ) is convex in α . If the solution is unique, then, by

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Unformatted text preview: the Envelope Theorem, ∂π/∂α = pq * ( p,α ). But since π is convex in α , ∂π/∂α is nondecreasing in α . And since p > 0, q * ( p,α ) cannot decrease in α . (If π is C 2 , we have ≤ ∂ 2 π/∂α 2 = p∂q * ( p,α ) /∂α so that ∂q * ( p,α ) /∂α ≥ 0.) f ( x 1 , t ) f ( x 2 , t ) V ( t ) t V , f The Maximum of a Collection of Convex Functions is Convex V ( t ) = max x ∈ { x 1 , x 2 } f ( x , t )...
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3Maximum over a Collection of Convex Functions is Convex -...

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