For instance, I saw the expression in the Wikipedia article on Lagrange multipliers:
maximize $f(x, y)$
subject to $g(x, y) = c$
What does "subject to" mean?
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It is a way to specify constraints. To put it very simply, the problem "do 'X' subject to 'Y'" means that, you have to do "X" (whatever X is), but you have to do it such that "Y" is also satisfied in the process.
As an example, in 1-D
"minimize $x^2$" would just give the answer $0$; but
"minimize $x^2$ subject to $x \geq 10$ would yield the answer $100$, since you cannot consider $x < 10$ in your problem.