# Optimization problems and strict vs non-strict inqualities?

If there are strict inequalities can we always replace them with non-strict ones? I'm inclined to say yes, but I'm struggling to think of an example where we could do this.

• One has to guess a little about your context. When the domain of a function to optimize is discrete, then the difference between a strict equality and a non-strict one can be crucial. For instance, the minima of the function $f(x)=x^2$ on the sets $\{n\in\mathbb{Z}\mid 0 \lt n\}$ and $\{n\in\mathbb{Z}\mid 0 \le n\}$ differ substantially. – whuber Jun 14 '18 at 13:27
• How about in the case where the domain of the function to optimize is assumed to be continuous? – Fruh Jun 14 '18 at 20:48
• Then it depends on whether $f$ is continuous. If it's continuous, then there's only an infinitesimal difference between its value on the boundary and its values near the boundary. – whuber Jun 14 '18 at 21:03

$$\text{Maximise }f(\mathbf{x}) \quad \text{subject to} \quad \mathbf{x} \in \mathcal{G},$$
where $\mathcal{G}$ is the constraint set in the problem. The constraint set can consist of equality constraints or strict/non-strict inequality constraints.
There is a famous mathematical result called the Weierstrauss extreme-value theorem, which gives sufficient conditions for the existence of a maximising value. If $f$ is continuous and $\mathcal{G}$ is a compact set (bounded and closed) then there will exist one or more maximising values in the problem, and so the supremum can be obtained exactly as a maximum.