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Corollary 2.17: μ-PŁ ⟹ μ-MB (Main Theorem)

Statement

If f : E → ℝ is C² and satisfies μ-PŁ around a local minimizer x₀ ∈ S, then f satisfies μ-MB at x₀. The constant μ is preserved.

Proof outline (algebraic shortcut, bypassing QG)

The proof combines two ingredients:

1. C² + PŁ ⟹ Hessian coercive on ker(Hess)⊥ (via Rayleigh quotient): Direct proof using 1D Taylor + PŁ + eigenvector analysis, without going through quadratic growth (QG) or gradient flow.

2. C² + PŁ ⟹ S is submanifold (Theorem 2.16): Using constant rank of Hessian on S (Cor 2.13), gradient alignment (Lemma 2.14), and the implicit function theorem (Lemma 2.15).

Combining these two gives μ-MB.

References

• Rebjock & Boumal, "Fast convergence to non-isolated minima: four equivalent conditions for C² functions", Corollary 2.17.

theorem PLAcceleratedNesterovLean.muPL_implies_muMB {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] (f : E → ℝ) (μ : ℝ) (x₀ : E) (hμ : 0 < μ) (hf : ContDiffAt ℝ 2 f x₀) (hmin : IsLocalMin f x₀) (hPL : MuPL f μ x₀) : IsMuMB f μ x₀

Corollary 2.17 (μ-PŁ ⟹ μ-MB). If f : E → ℝ is C² and satisfies μ-PŁ around a local minimizer x₀, then f satisfies μ-MB at x₀. The constant μ is preserved.

Proof: Combine two facts:

1. C² + PŁ ⟹ Hessian coercive on ker(Hess)⊥ (via Rayleigh quotient)
2. C² + PŁ ⟹ S is C¹ submanifold with T = ker(Hess) (Thm 2.16)

Note: This uses the algebraic shortcut (Rayleigh quotient + eigenvector analysis) to prove Hessian coercivity directly from PŁ, bypassing the gradient flow argument (PŁ ⟹ QG) and the distance lemma entirely.