Ekeland's variational principle #
Let φ be a lower semicontinuous real function, bounded below, on a complete metric space, and
let ε > 0. Ekeland's variational principle produces, from any starting point x, a point
y with φ y + ε * dist y x ≤ φ x which is the strict global minimum of the perturbed function
z ↦ φ z + ε * dist z y.
In Banach space theory this is the device behind the Bishop–Phelps theorem: applied to minus the real part of a functional on a closed set, it produces a point admitting a supporting cone.
Main results #
LowerSemicontinuous.exists_add_mul_dist_le_and_lt_add_mul_dist: Ekeland's variational principle.
References #
- I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353.
- R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd ed., Lecture Notes in Mathematics 1364, Springer, 1993, Chapter 3.
theorem
LowerSemicontinuous.exists_add_mul_dist_le_and_lt_add_mul_dist
{X : Type u_1}
[MetricSpace X]
[CompleteSpace X]
{φ : X → ℝ}
(hφ : LowerSemicontinuous φ)
(hbdd : BddBelow (Set.range φ))
{ε : ℝ}
(hε : 0 < ε)
(x : X)
:
Ekeland's variational principle. Let φ be a lower semicontinuous function, bounded below,
on a complete metric space, and let ε > 0. For every x there is a point y such that
φ y + ε * dist y x ≤ φ x and y is the strict global minimum of z ↦ φ z + ε * dist z y.