Can we construct / associate object and constraint functions $u(x,y),v(x,y) $ with a function of a complex variable?
For object/constraint function solution we have Lagrange Multiplier
$$\dfrac{u_x}{u_y}=\dfrac{v_x}{v_y} = \lambda \tag1 $$
For a complex number $ z= u+iv $ by Cauchy-Riemann relations
$$u_x= v_y ; \, u_y=-v_x;$$
Dividing
$$\dfrac{u_x}{u_y}=-\dfrac{v_y}{v_x} \tag2$$
The right hand side 2) is negative reciprocal of 1). In other words negative reciprocal Lagrange multiplier. ( Could orthogonal curve slopes be manipulated here?)
Agreed it makes no direct sense this way but it could provide a complex algebra tool towards a two-variable optimization problems. Just a thought.
Please indicate earlier references for linking the two topics this way or another.