what are the generic methods to prove solution existence!?

Question

Suppose we are in $\mathbb R^n$ and consider $$\min f(x) \qquad s.t. \qquad x\in P.$$ Under what conditions on $f$ and $P$, we can guarantee this problem obtains a solution?

The most generic situation that I know of is that when $f$ is lower-semicontinuous with a bounded sublevel (or coercive which implies a bounded sublevel set) and $P$ is closed, then the problem has a solution. Do we have other generic cases?

What other methods are often used to show the existence of a solution?

Answer 1

From my point of view, the proof of existence (almost) always follows the classical steps:

• Define infimal value $j := \inf\{ f(x) \mid x \in P\}$ and take minimizing sequence $(x_k) \subset P$ with $f(x_k) \to j$.
• Somehow, choose a certain subsequence $(x_{k_l})$ which converges in some sense to some limit point $\hat x$.
• Show that $\hat x \in P$.
• Use some kind of lower semicontinuity to get $$ j \le f(\hat x) \le \liminf_{l \to \infty} f(x_{k_l}) = j.$$

Consequently, $\hat x$ is a solution.

From this recipe, one can see, that one needs some way to get boundedness of $(x_k)$ to extract a convergent subsequence, closedness of $P$ to get $\hat x \in P$ and lower semicontinuity to get the final inequality.