In Constrained Optimization, Restrict Domain to Open Set $A\subset\mathbb{R^N}$? In constrained optimization and context of economics (e.g. utility function with quantity of goods as arguments subject to wealth), why do textbooks always restrict domain of the objective function and constraints to be open set $A\subset\mathbb{R^N}$?
Suppose we are trying to find an extremum for a function $f:\mathbb{R^N}\rightarrow\mathbb{R}$ with $M$ equality or inequality constraints. So this considers constrained maximization problem.
Does this have to do with boundaries? Can someone provide some insight what is the practical and real reason textbooks restrict both obj and constraint functions' domain to an open set?
Thanks.
 A: It is not the case for all kinds of constrained optimisation. Combinatorial, linear, or SDP optimisation books usually deal with closed sets. I am guessing that you are currently reading a chapter on continuous optimisation that will soon speak about $\mathbb{R}$-differentiable functions and local optimas.
One (among other) convenient things about an open set is that a differentiable function defined on it has its local optimas where its derivatives are null. It is about the borders, since it saves lots of paper by avoiding to write many times "if the optimum is on the border ..."
(Following a comment) As an easy example, you can think about a continuous differentiable function $f$ on a closed interval of $\mathbb{R}, [a,b]$ (topologically speaking, a compact subset). $f$ can reach its optimas (minimum or maximum) either in points where $f'=0$, or in $a$, or in $b$. On $]a,b[$ (or on any countable union of open sets, a.k.a an open subset of $\mathbb{R}$ ), it can only reach its optimas in points where $ f'=0 $, which makes less cases to investigate. The same property goes for higher dimensions (but with a generalized definition of differentiability).
I think it is also due to the fact that $\mathbb{R}^n$ is an open subset of $\mathbb{R}^n$. It could also be about talking about functions defined on rather "clean" spaces, since open sets are rather easy to manipulate. 
I suggest that with your knowledge on the subject, you put yourself in the shoes of the writer. What would be both easy to write and general enough to apply to many readers? Here, they went with the open subset.
