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Setting: I have a function $f$ of 4 real variables $(x,y,w,z)$ in $\mathbb R^4$. The function $f$ satisfies $f(0,y,w,z)=1$ and $-I \preceq \mathsf{H}f(x,y,w,z) \preceq I$, $\forall x,y,w,z \in [0,1]$, where $\mathsf{H}$ denotes the Hessian operator. Otherwise, the function $f$ does not have any special structure in any of the variables (i.e. no monoticity, convexity, etc). (The analytical expression of $f$ is quite involved and very hard to study analytically.)

Goal: I want to show that $f(\mathbf{v})\leq 1$ for all $\mathbf{v}\triangleq(x,y,w,z)$ in $[0,1]^4$, by showing that $\partial f /\partial x(\mathbf{v})\leq0$, for all $\mathbf{v}$ in $[0,1]^4$.

Since $f(0,y,w,z)=1$, for all $y,w,z\in [0,1]^3$, showing that $\partial f /\partial x(\mathbf{v})\leq0$, for all $\mathbf{v}$ in $[0,1]^4$, will certainly get me what I want. However, the expression of $\partial f/\partial x$ is very involved making it very hard to show any inequality analytically.

Desired Strategy: Numerical optimization suggests that $\partial f /\partial x(\mathbf{v})\leq -0.1$. Given that I also know that $-I \preceq \mathsf{H}f \preceq I$, my strategy was to evalutate $\partial f /\partial x(\mathbf{v})$ on a fine enough grid in $[0,1]^4$ and then invoke Taylor's Theorem to argue that $\partial f /\partial x (\mathbf v)<0$, for all $\mathbf{v} \in[0,1]^4$. Would this be an acceptable proof?

Side Note: The result that I am chasing will ultimately be part of a Theorem in a research article (my field is maths/computer science). And so I want to know if my strategy above for proving the result is still ``acceptable'' even though it relies on the numerical evaluating of a function (in this case $\partial f/\partial x$) on a (large) set of grid points.

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Conterexample: $f(x,y,z,w):=1+\frac{1}{2}x^2.$ Then $f(0,y,z,w)=1$ and $\partial^2 f /\partial x^2=1.$

But $f(1,0,0,0,)=\frac{3}{2}>1.$

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  • $\begingroup$ Your example does not satisfy ∂𝑓/∂𝑥(𝐯)<0. As I explain in the description of my question I can verify (numerically) that ∂𝑓/∂𝑥(𝐯) <-0.1. $\endgroup$ – user6952886 Sep 11 at 11:23

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