How should substitution be noted?

Question

In my thesis, I want to describe replacing equality constraints with inequalities in a constrained optimization problem. But I'm not sure how it should be mathematically noted. This is my current notation:

$$ h_i(x)=0 \rightarrow |h_i(x)|<\varepsilon_i, i=1,2, ... ,p \\ \therefore |h_i(x)|-\varepsilon_i<0 \rightarrow h'_i(x)<0 $$

I used "$\rightarrow$" twice with two different meanings. In first line I am assuming that two sides of arrow are equivalent and in second line I defined a new function ($h'(x)$). I know that the right-arrow is used for implication, which is totally irrelevant here, but what is the correct notation?

Answer 1

$A\rightarrow B$ is conventional for generative grammars and rewrite systems, $[A/B]$ for logical calculi. But please use $\text{replace }A\text{ with }B$ if your audience isn't an expert in such systems and $\Rightarrow$ for implication to avoid overloading $\rightarrow$.

Edit: Making some educated guesses on what you mean, I would probably write what you wrote as something like

Consider the relaxation of Equation [ref. number of eq. block for program] such that, for each $i$ in $\{1,2,\ldots,p\}$, the equality constraint $$h_i(x)=0$$ is replaced by an inequality $$\lvert h_i(x)\rvert<\varepsilon_i$$ for some positive real $\varepsilon_i$. Then by defining auxiliary functions $$\begin{align}k_i&\colon X\to\mathbb{R}& k_i(x)&\stackrel{\text{def}}{=} \lvert h_i(x)\rvert-\varepsilon_i& (i&\in\{1,2,\ldots,p\})\end{align}$$ the new constraints may be simplified to the standard form $$\begin{align}k_i(x)&<0&(i&\in\{1,2,\ldots,p\})\text{.}\end{align}$$

In particular, I'd avoid using a prime to denote a new function (too easily confused with a derivative), a "$\rightarrow$" between two propositions to define a function with restricted domain and range (it's almost certain to be read as implication), and shorthand proof symbols like "$\therefore$" (useful when drafting proofs, but potentially disruptive in the final prose).