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For $f : \mathbb{R}^{n \times 1} \to \mathbb{R}$ a double differentiable function with bounded hessian, not necessarily convex, is any known polynomial algorithm, in the general case, which can assert if for a given $C \in \mathbb{R}^{n \times 1}$ with $\|C\| \geq k > 0$ exists $X \in \mathcal{B}(0_{n \times 1}, R)$ such that $$ f(X) + C^T \cdot X = 0$$

Consider $k$ and $R$ as some positive constants. Is any work in this direction published?

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I doubt you can do so in polynomial time without any additional condition.

Let $g$ be an arbitrary polynomial function, and let $f$ be defined by $f(X) = g(X) - C^{T}X$ over $B(0, R)$, and an appropriate extension so that it's twice differentiable with bounded Hessian over $\mathbb{R}^{n}$. Then, $f(X) + C^{T}X = 0$ reduces to $g(X) = 0$, i.e., your setting is equivalent to finding a root of $g$ in $B(0, R)$.

(I believe that deciding whether an arbitrary polynomial has a root of at most norm $R$ is already NP-hard, but I'm not 100% sure).

Now, since 3SAT can be written as finding a root of a polynomial (make a polynomial for each clause and multiply all of them), finding a root of an arbitrary polynomial is NP-hard.

Thus, unless P = NP, there is little chance that you can solve your problem for general $f$ in polynomial time. (note that the value of $R, k$ are insignificant, since we can always re-scale everything).

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  • $\begingroup$ Assuming that an algorithm is found, but which requires for instance something like $ \|C\| \geq \| \frac{\partial f}{\partial x} \| } + R $. Can this be still used to assert the zeros of $f(x)$ ? Can you expand a little on the insignificancy of $R$ and $k$ ? $\endgroup$
    – C Marius
    May 30 at 16:01
  • $\begingroup$ For the example I give, you can assume without loss of generality that $R=1$ by replacing $X$ with $X / R$. This does not affect the assumptions on $f$. You can also assume w.l.o.g that $k=1$ by multiplying $f$ by $1/k$. $\endgroup$
    – mtanneau
    May 31 at 20:23

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