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I am following a proof that applies Taylor's theorem on this document (http://www.gautampendse.com/software/lasso/webpage/pendseLassoShooting.pdf)

I am not understanding one of the terms explained on the formula (5.1):

Proposition 5.3 (positive semidefinite Hessian implies Convexity). Suppose x is a p × 1 vector and f(x) is a scalar function of p variables with continuous second order derivatives defined on a convex domain D. If the Hessian ∇2f(x) is positive semidefinite for all x ∈ D then f is convex.

Proof. By Taylor’s theorem for all x, x + h ∈ D we can write:

$$ f(x + h) = f(x) + ∇f(x)^Th + \frac{1}{2}h^T∇^2f(x + \theta h)h $$ for some θ ∈ (0, 1).

I don't understand why the term corresponding to the second derivative is: $$ \frac{1}{2}h^T∇^2f(x + \theta h)h $$

And not:

$$ \frac{1}{2}h^T∇^2f(x )h $$

I understand that by θ ∈ (0, 1), if θ = 0 then the term becomes the latter, but I don't understand what happens when θ > 0.

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This formula uses the Cauchy form of the remainder. $x + \theta h $ is a number between $x$ and $x+h$. http://mathworld.wolfram.com/CauchyRemainder.html

It comes from the mean value theorem.

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  • $\begingroup$ Many thanks! It was driving me crazy! $\endgroup$ – user1064285 Jan 28 '15 at 10:39

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