2.2Existence of solutions

Let $f_0: \mathbb{R}^n\longrightarrow \mathbb{R}$ and consider the following optimization problem

where $\Omega \subseteq \mathbb{R}^n$ is the feasible set. Under which conditions can we guarantee existence of a solution to the problem (1)? Recall that we say that $x$ is a solution of (1) if the optimal value $p^\star$ is finite, and attained at $x$, i.e., $f_0(x) = p^\star$.

In this course we work in finite dimensions only. The theorem below provides two sufficient conditions, which depend on the nature of the set $\Omega$.

Existence does not mean the solution to a given optimization problem is unique. However, in the important case of convex optimization problems, it is possible to guarantee uniqueness of solutions, as explained in the next section.