relation between positive function and its gradient when its Hessian matrix is bounded


Let f:RnR a positive function, twice differentiableMathworldPlanetmathPlanetmath everywhere. Furthermore, let 𝐇f(𝐱)2M,M>0 𝐱Rn, where 𝐇f(𝐱) is the Hessian matrix of f(𝐱). Then, for any 𝐱Rn,

f(𝐱)22Mf(𝐱)
Proof.

Let 𝐱, 𝐱0Rn be arbitrary points. By positivity of f(𝐱), writing Taylor expansionMathworldPlanetmath of f(𝐱) with Lagrange error formula around 𝐱0, a point 𝐜Rn exists such that:

0 f(𝐱)
= f(𝐱0)+f(𝐱0)(𝐱-𝐱0)+12(𝐱-𝐱0)T𝐇f(𝐜)(𝐱-𝐱0)
= |f(𝐱0)+f(𝐱0)(𝐱-𝐱0)+12(𝐱-𝐱0)T𝐇f(𝐜)(𝐱-𝐱0)|
f(𝐱0)+|f(𝐱0)(𝐱-𝐱0)|+12|(𝐱-𝐱0)T𝐇f(𝐜)(𝐱-𝐱0)|
f(𝐱0)+f(𝐱0)2𝐱-𝐱02+12𝐇f(𝐜)2𝐱-𝐱022 (by Cauchy-Schwartz inequality)
f(𝐱0)+f(𝐱0)2𝐱-𝐱02+12M𝐱-𝐱022

The rightest side is a second degree polynomial in variable 𝐱-𝐱02; for it to be positive for any choice of 𝐱-𝐱02 (that is, for any choice of 𝐱), the discriminant

f(𝐱0)22-412Mf(𝐱0)

must be negative, whence the thesis. ∎

Note: The condition on the boundedness of the Hessian matrix is actually needed. In fact, in the Lagrange form remainder, the constant 𝐜 depends upon the point 𝐱. Thus, if we couldn’t rely on the condition 𝐇f(𝐱)2M, we could only state f(𝐱0)+f(𝐱0)2𝐱-𝐱02+12𝐇f(𝐜(𝐱))2𝐱-𝐱0220 which, not being a second degree polynomial, wouldn’t imply any particular further condition. Moreover, in the case n=1, the lemma assumes the simpler form: Let f: a positive function, twice differentiable everywhere. Furthermore, let f′′(x)M,M>0 x. Then, for any x, |f(x)|2Mf(x).

Title relation between positive function and its gradient when its Hessian matrix is bounded
Canonical name RelationBetweenPositiveFunctionAndItsGradientWhenItsHessianMatrixIsBounded
Date of creation 2013-03-22 15:53:10
Last modified on 2013-03-22 15:53:10
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 16
Author Andrea Ambrosio (7332)
Entry type Theorem
Classification msc 26D10