Consider the following natural exponential function:

$f\left( {\bf{x}} \right) = {e^{{{\bf{a}}^T}{\bf{x}} + \frac{1}{2}{{\bf{x}}^T}{\bf{Cx}}}}$

where $\bf C$ is a positive definite matrix. Is there any tight lower bound for this function?


1 Answer 1



Notice that the exponential function is an increasing function, hence the lower bound is the smallest when $g(x)=a^Tx+\frac12x^TCx$ attains the smallest value.

You can differentiate $g(x)$ to solve for the minimal value.

  • $\begingroup$ Thank you very much. But, is there any proper lower bound for this function? $\endgroup$
    – user51780
    May 16, 2018 at 10:04
  • $\begingroup$ what do you mean by "proper" lower bound? $\endgroup$ May 16, 2018 at 16:55

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