Formulation of "significantly" close/far I am writing a tech document and I would like to formulate "significantly" close/far. For example a certain suboptimal solution $x^{\dagger}$ is "significantly close" to the optimal solution $x^{*}$. I was tempted to write something like $x^{*} - x^{\dagger} \simeq 0 $ however this is not correct because the "closeness" is relative to the problem and its parameters. In other words the difference might be 0.01, 0.1, 1, 5, 10 and still it would be considered to be "significantly close". Thank you

How about $\frac{x^{*} - x^{\dagger}}{x^{*}} \simeq 0$? The denominator is a stab in the dark but often the notion of 'relative' closeness is conveyed as a ratio of the distance to the solution normalized by some function of the parameters.

You could also try something like $f(x^{*}) - f(x^{\dagger}) \simeq 0$ if there is a function $f$ that you can define for your problem to convey the notion of a 'value' of a solution.