EnvelopeTheorem

EnvelopeTheorem - The Envelope Theorem L A T E X file...

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Unformatted text preview: The Envelope Theorem L A T E X file: Envelope-nb-all — Daniel A. Graham <[email protected]>, June 22, 2005 An example without constraints Here we consider a simple maximization problem without constraints, with a single choice variable and a single parameter. The object is to maximize f = ( a- (x- a) ∧ 2 ) / 2; In: xbest = Solve [D [f,x] ==0 ,x] In: {{ x → a }} Out: Substituting this solution, x(a)=a, into the objective function gives F = (f /. xbest )[[ 1 ]] In: a 2 Out: as the optimized value of the objective function. Now the envelope theorem states that the derivative of F with respect to the parameter a D [F,a] In: 1 2 Out: and the partial derivate of f with respect to a, evaluated at the optimal x D [f,a] /. xbest In: 1 2 Out: π p In: π p Out: are exactly the same. To see why this is the case lets plot F(a) and f(x,a) for several fixed choices of x: Plot F,f /. x->3 ,f /. x->5 ,f /. x->7 , { a, , 10 } , PlotRange-> { , 5 } , PlotStyle-> {{ Thickness [ . 004 ] } , { Thickness...
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EnvelopeTheorem - The Envelope Theorem L A T E X file...

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