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

Let f:Rn→R a positive function, twice differentiableMathworldPlanetmathPlanetmath everywhere. Furthermore, let ∥𝐇f⁢(𝐱)∥2≤M,M>0 ∀𝐱∈Rn, where 𝐇f⁢(𝐱) is the Hessian matrix of f⁢(𝐱). Then, for any 𝐱∈Rn,


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

0 ≤ f⁢(𝐱)
= f⁢(𝐱0)+∇⁡f⁢(𝐱0)⋅(𝐱-𝐱0)+12⁢(𝐱-𝐱0)T⋅𝐇f⁢(𝐜)⋅(𝐱-𝐱0)
= |f⁢(𝐱0)+∇⁡f⁢(𝐱0)⋅(𝐱-𝐱0)+12⁢(𝐱-𝐱0)T⋅𝐇f⁢(𝐜)⋅(𝐱-𝐱0)|
≤ f⁢(𝐱0)+|∇⁡f⁢(𝐱0)⋅(𝐱-𝐱0)|+12⁢|(𝐱-𝐱0)T⋅𝐇f⁢(𝐜)⋅(𝐱-𝐱0)|
≤ f⁢(𝐱0)+∥∇⁡f⁢(𝐱0)∥2⁢∥𝐱-𝐱0∥2+12⁢∥𝐇f⁢(𝐜)∥2⁢∥𝐱-𝐱0∥22⁢ (by Cauchy-Schwartz inequality)
≤ f⁢(𝐱0)+∥∇⁡f⁢(𝐱0)∥2⁢∥𝐱-𝐱0∥2+12⁢M⁢∥𝐱-𝐱0∥22

The rightest side is a second degree polynomial in variable ∥𝐱-𝐱0∥2; for it to be positive for any choice of ∥𝐱-𝐱0∥2 (that is, for any choice of 𝐱), the discriminant


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)∥2≤M, we could only state f⁢(𝐱0)+∥∇⁡f⁢(𝐱0)∥2⁢∥𝐱-𝐱0∥2+12⁢∥𝐇f⁢(𝐜⁢(𝐱))∥2⁢∥𝐱-𝐱0∥22≥0 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:ℝ→ℝ a positive function, twice differentiable everywhere. Furthermore, let f′′⁢(x)≤M,M>0 ∀x∈ℝ. Then, for any x∈ℝ, |f′⁢(x)|≤2⁢M⁢f⁢(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