exists_isMinOn_and_gradient_eq_zero

Plain-English statement

- A differentiable, positively strongly convex function on a finite-dimensional real inner-product space has a global minimizer, and its gradient vanishes there. The proof does not use a totalized derivative as a regularity witness: `Differentiable` supplies the genuine gradient and continuity.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem exists_isMinOn_and_gradient_eq_zero
    {V : E → ℝ} {m : ℝ} (hm : 0 < m)
    (hV : Differentiable ℝ V)
    (hsc : StrongConvexOn (Set.univ : Set E) m V) :
    ∃ p : E, IsMinOn V Set.univ p ∧ gradient V p = 0 := by
  let g : E := gradient V 0
  let R : ℝ := 2 * ‖g‖ / m
  have hR : 0 ≤ R := by
    dsimp only [R]
    positivity
  have hfirst (x : E) :
      V 0 + inner ℝ g x + m / 2 * ‖x‖ ^ 2 ≤ V x := by
    simpa only [g, sub_zero] using
      StrongConvexFirstOrder.firstOrder_lower_bound_of_strongConvexOn hsc
        (fun z _ ↦ (hV z).hasGradientAt) (x := 0) (y := x)
        (Set.mem_univ _) (Set.mem_univ _)
  have houtside {x : E} (hx : R < ‖x‖) : V 0 ≤ V x := by
    have hxpos : 0 < ‖x‖ := lt_of_le_of_lt hR hx
    have hscale : 2 * ‖g‖ < m * ‖x‖ := by
      dsimp only [R] at hx
      have hs := (div_lt_iff₀ hm).mp hx
      simpa [mul_comm] using hs
    have hinner : -(‖g‖ * ‖x‖) ≤ inner ℝ g x :=
      (abs_le.mp (abs_real_inner_le_norm g x)).1
    have hquad : ‖g‖ * ‖x‖ ≤ m / 2 * ‖x‖ ^ 2 := by
      nlinarith [mul_pos hm hxpos]
    linarith [hfirst x]
  have hzero : (0 : E) ∈ closedBall 0 R := by
    simp [hR]
  obtain ⟨p, hpBall, hpMinBall⟩ :=
    (isCompact_closedBall (0 : E) R).exists_isMinOn
      ⟨0, hzero⟩ hV.continuous.continuousOn
  have hpMin : IsMinOn V Set.univ p := by
    intro x _
    by_cases hxBall : x ∈ closedBall (0 : E) R
    · exact hpMinBall hxBall
    · have hx : R < ‖x‖ := by
        simpa [mem_closedBall, dist_eq_norm] using hxBall
      exact (hpMinBall hzero).trans (houtside hx)
  refine ⟨p, hpMin, ?_⟩
  simp [gradient, (hpMin.isLocalMin Filter.univ_mem).fderiv_eq_zero]

end AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexMinimizer

Open AutoSamplingTheory/TechnicalLemmas/Analysis/StrongConvexMinimizer.lean:35published source at ed709a7f320d

Proof architecture

SPHMC Lemma 4.16 hidden minimizer-selection prerequisite: construct the minimizer used by the initialization argument

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Hidden assumptions and non-claims

Totalized `fderiv` values must not be read as a differentiability theorem.
Genuine differentiability is localized to the hypotheses shown in the Lean statement.
A Gibbs expression is not automatically a probability law or an invariant law.