Ekeland's variational principle

Remarks

The proof builds a sequence $x_0, x_1, x_2, \ldots$. Each $x_{n+1}$ is chosen in the set of points $y$ with $f(y) \le f(x_n) - (\varepsilon / \lambda) d(x_n, y)$, with $f(x_{n+1})$ close to the infimum of $f$ on that set. These sets are closed and nested, their diameters tend to $0$, and completeness gives a limit point $v$, which has the three properties. Taking $\lambda = 1$ on a Banach space with $f$ Gateaux differentiable gives a point $v$ with $f(v) \le \inf f + \varepsilon$ and $\norm{f'(v)} \le \varepsilon$: an almost minimizer with small derivative. With this, a $C^1$ function that is bounded below and satisfies the Palais–Smale condition attains its minimum. Completeness is needed: on $X = (0, 1]$ with $f(x) = x$, $\varepsilon = 1/2$ and $\lambda = 1$, every $v$ violates clause 3 for any $w < v$, because $f(w) = v - (v - w) < v - \tfrac12 (v - w)$.

Going deeper

For a student

The principle is a basic tool of nonsmooth analysis and of critical point theory. It produces Palais–Smale sequences for functionals bounded below, and it gives short proofs of the Caristi fixed point theorem and of the Bishop–Phelps theorem. Conversely, a metric space in which the conclusion holds for all such functions is complete, so the principle characterizes completeness.