Formulating three functions as Lagrangian multipliers :

Question

so I have two functions f(x, y) and g(u, v) to minimize and we know a third function that maps the variables: h(x,y) = h(u,v). How can I define the Lagrange equation for f and g to make an optimization problem?

Answer 1

Lagrange Multipliers

Given differentiable functions $f(x,y)$ and $g(x,y)$ such that $f(x,y)$ has a local maximum (or minimum) on the constraint curve $g(x,y) = k$ at a point $P = (a,b)$ and $\nabla g_P \neq (0,0),$ there exists a nonzero scalar $\lambda$ such that $\nabla f(a,b) = \lambda \cdot \nabla g(a,b),$ i.e., $f_x(a,b) = \lambda \cdot g_x(a,b)$ and $f_y(a,b) = \lambda \cdot g_y(a,b).$

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Considering the variables $u$ and $v$ as fixed, we have that $h(u, v) = k$ so that the constraint curve is given by $h(x, y) = h(u, v) = k.$ By the theory of Lagrange Multipliers, then, so long as $\nabla h_P \neq (0, 0)$ at the point $P$ on $h(x, y) = h(u, v) = k$ such that $f$ has a local minimum, there exists a nonzero scalar $\lambda$ such that $\nabla f_P = \lambda \cdot \nabla h_P.$ Once we solve for $x$ and $y$ in terms of $u$ and $v$ here, we can view the constraint curve for $g(u, v)$ as $h(u, v) = h(x, y) = \ell,$ and we can again use Lagrange Multipliers (if the hypotheses hold).