I want to optimize a function $T(x_1,x_2,\ldots,x_n)$ with the constraint that $g(x_1,x_2,\ldots,x_n,y)>0\quad \forall y \in [a,b], a,b \in R$. Both $T$ and $g$ are real-valued functions, defined on $R^n$ and $R^{n+1}$, respectively.
If my constraint was of the form $g(x_1,x_2,\ldots,x_n)=c$ or $g(x_1,x_2,\ldots,x_n)\geq c$, I could use the method of Lagrangian multipliers. However, this is not the case here. Any ideas?