Minimizing by Ekeland’s Variational Principle.

Minimizing by Ekeland’s Variational Principle.

Motivation of Ekeland’s paper “On the variational principle.” JMAA, 47. (1974). If f is a continous function on \mathbb{R}^n or Banach space X or complete Riemannian manifold \overline{M} and f has not minimum (of course f is bounded below) then we can choose a sequence (x_n) such that f(x_n) is “near” to the infimum of f.

We provide following Ekeland’s theorem with X is complete metric space

Theorem (Ekeland’s Variational Principle).  Let X be a complete metric space and F: X \to \mathbb{R} \cup \{+\infty\} a lower semi continous function, \ne +\infty, bounded from below. For every point u \in X satisfying \inf F \le F(u) \le \inf F + \epsilon and every \lambda > 0, there exists some point v \in X such that

F(v) \le F(u),

d(u,v) \le \lambda,

\forall w \ne v, F(w) > F(v) - (\epsilon/\lambda)d(v,w).

In part 5 of the paper, the author states an interesting result on complete Riemannian manifold

Proposition. Let F be a C^1 function on a complete Riemannian manifold \overline{M}. If F bounded from below, then, for every \epsilon \ge 0, there exists some point p_\epsilon \in \overline{M} such that

F(p_\epsilon) \le \inf F + \epsilon^2,

\|gradF(p_\epsilon)\|_{p_\epsilon} \le \epsilon.

Conclusion. p_\epsilon is the sequence of minimizer.