TITLE: Well-posed vs Well-conditioned
QUESTION [9 upvotes]: What's the difference between a well-posed (ill-posed) and well-conditioned (ill-conditioned) problem?
According to Wikipedia:

Even if a problem is well-posed, it may still be ill-conditioned, meaning that a small error in the initial data can result in much larger errors in the answers.

REPLY [6 votes]: According to the book Numerical Analysis in Modern Scientific Computing: An Introduction, 2nd edition, by Andreas Hohmann and Peter Deuflhard, second chapter on "Error Analysis":

Definition 2.2
The absolute norm-wise condition number of the problem $(f, x)$ is the smallest number $K_{abs} \geq 0$, such that
$$|f(\tilde{x}) - f(x)| \dot{\leq} K_{abs} \cdot |\tilde{x} - x|, \text{ for $\tilde{x} \mapsto x$}$$
The problem $(f, x)$ is said to be ill-posed whenever such a number ($K_{abs}$) does not exist, which is formally equivalent to $K_{abs} = \infty$.
Analogously, the relative norm-wise condition number of $(f, x)$ is the smallest number $K_{rel} \geq 0$, such that
$$\frac{|f(\tilde{x}) - f(x)|}{|f(x)|} \dot{\leq} K_{rel} \cdot \frac{|\tilde{x} - x|}{|x|}, \text{ for $\tilde{x} \mapsto x$}$$
Thus $K_{abs}$ and $K_{rel}$ describe the increase of the absolute and the relative errors, respectively.
A problem $(f, x)$ is said to be well-conditioned when its condition number is small and ill-conditioned when its large. Naturally, the meaning of "small" and "large" has to be considered separately for each problem.