The reverse mathematics of Ekeland's variational principle

(Joint with David Fernández-Duque, Henry Towsner, and Keita Yokoyama.)
Let $X$ be a complete metric space, and let $V$ be a lower semi-continuous function from $X$ to the non-negative reals. Ekeland's variational principle states that $V$ has a ‘critical point’, which is a point $x^*$ such that $d(x^*, y)> V(x^*) - V(y)$ whenever y is not $x^*$. This theorem has a variety of applications in analysis. For example, it implies that certain optimization problems have approximate solutions, and it implies a number of interesting fixed point theorems, including Caristi's fixed point theorem.
We analyze the proof-theoretic strength of Ekeland's variational principle in the context of second-order arithmetic. We show that the full theorem is equivalent to $\Pi^1_1-\text{CA}_0$. We also show that a few natural special cases, such as when $V$ is assumed to be continuous and/or X is assumed to be compact, are equivalent to the much weaker systems $\text{ACA}_0$ and $\text{WKL}_0$.