Optimization with Dependent Constraints

In the general theory of constrained optimization, is it possible to pose a minimization problem subject to some constraints that are dependent?

For example, suppose we aim to minimize an objective function $\mathscr F(x,y,z)$, for the design parameters $x$, $y$, and $z$, such that $z=x^2+y^2$. Is it correct to have such constraint, theoretically?

EDIT
$z=x^2+y^2$

1 Answer

As Cade Reinberger said in a comment, you can do this by expressing your constraint as $z=z(x,y)$ (might work with any of the variables, if at all) and then minimising the resulting function $F(x,y)=\mathscr F(x,y,z(x,y))$ by solving $\nabla F(x,y)=\mathbf 0$.

Alternatively, you can express your constraint as $g(x,y,z)=0$ and solve the system $$ \left\{ \begin{align} \nabla\mathscr F(x,y,z)&=\lambda\nabla g(x,y,z) \\ g(x,y,z)&=0 \end{align} \right. .$$ If you'd like some explanation of this method, it's called the method of Lagrange multipliers, and it comes from the way the contours of the functions behave at critical points.