# Optimization with strict inequality

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.)

Optimization problems are not often posed with strict inequalities. The reason is that the introduction of strict inequalities calls into question whether an optimal solution exists and is attainable. For example, consider $$\begin{array}{rl} \max&1/x\\ \text{s.t.}&x>0 \end{array}$$ This problem doesn't have an optimal solution, since $$1/x\to\infty$$ as $$x\to0$$. Similarly, consider the problem $$\begin{array}{rl} \min&x\\ \text{s.t.}&x>0 \end{array}$$ Again, this problem doesn't have an optimal solution, since, given any $$x>0$$, $$x/2$$ produces a solution with a strictly better objective value.
These examples are manifestations of the extreme value theorem--in $$\mathbb{R}^n$$ we need our problem to be posed on a compact domain to ensure that our (continuous) objective function obtains its extremum.
So, without more context for your problem (e.g. what is $$U(\cdot,\cdot)$$ etc.), you probably have to do something like what you suggest (solve first and check later, or something like that). In general, these cases can be tricky.