# What does “subject to” mean in math?

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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While your question is legitimate, you could look up the phrase in an English dictionary (e.g. the online versions of Cambridge or Oxford dictionaries) first before you posted your question here. – user1551 Nov 29 '11 at 4:54
It comes immediately after subject one. – Will Jagy Nov 29 '11 at 4:55

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.

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It means that the solution to the optimization problem should satisfy the constraint $g(x,y) = c$.

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