What if LICQ does not hold?

Question

Suppose given a nonlinear optimization programming:

$min_{x,y} f(x,y)$ st

$g_1(x,y)\ge 0$ , $g_2(x,y)\ge 0$ , $g_3(x,y)\ge 0$

and suppose that at the solution $(x*, y*)$ the three constraints are active. This means that the constraints gradients will not be linearly dependent and the LICQ fails to hold. How could we handle this case since the LICQ is a necessary condition for optimality.

Answer 1

LICQ is not a necessary condition for optimality. It is a prerequisite that the KKT conditions are necessary for optimality.

You might check other constraint qualifications (MFCQ, linearity, convexity + Slater point, etc).

LICQ fails trivially if $g_1=g_2=g_3$. Nevertheless, Lagrange multipliers might exist as other constraint qualifications might hold.