I'm very familiar with using Lagrangeans to solve optimization problems with weak inequalities, but I just realized that I don't know how to solve simple optimization problems with strict inequalities.
I have the following problem:
Choose p and x to maximize U(-px, w(p)v(x)) subject to w(p)v(x)>0
If I treat the constraint as a weak inequality, one of the first order conditions will be w(p)v(x)=0, which is a violation. I could ignore the condition and say "check to see if the solution meets the condition after solving", but then what happens if it doesn't satisfy the condition?
(A simpler version of this problem would just be choose x and y to maximize U(x,y) subject to y>0.)