Optimizationtheoryandmethods Sunyuan 2006 -- Chapter 13 -- Section 13325

noncomputable sectionopen scoped Matrix Matrix.Norms.L2Operatorsectionvariable {m n : ℕ}

The Section 13.3 branch hypothesis (Invalid projection: Index `25` is invalid for this structure; the only valid index is 1 Note: The expression 13.3 has type `Float` which has only 1 field13.3.25) for the stage data (sorry, sorry, sorry, sorry, sorry) : ?m.1 × ?m.3 × ?m.5 × ?m.7 × ?m.8(Unknown identifier `c_k`c_k, Unknown identifier `A_k`A_k, Unknown identifier `ξ_k`ξ_k, Unknown identifier `Δ_k`Δ_k, Unknown identifier `b₂`b₂), written directly on the canonical Moore-Penrose pseudoinverse overloaded, errors Unknown identifier `A_k` Unknown identifier `A_k`sorry = sorry : PropA_k⁺ = overloaded, errors 1:10 Unknown identifier `A_k` 1:10 Unknown identifier `A_k`(A_k)⁺.

def powellYuanSection13325
    (ck : EuclideanSpace ℝ (Fin m))
    (Ak : Matrix (Fin n) (Fin m) ℝ)
    (trustRegionRadius ξk b2 : ℝ) : Prop :=
  (b2 * trustRegionRadius ≥ ‖(Ak⁺).transpose.toEuclideanLin ck‖ → ξk = 0) ∧
    (b2 * trustRegionRadius < ‖(Ak⁺).transpose.toEuclideanLin ck‖ →
      ξk ≤ ‖ck - Ak.transpose.toEuclideanLin
          (((b2 * trustRegionRadius) / ‖(Ak⁺).transpose.toEuclideanLin ck‖) •
            (Ak⁺).transpose.toEuclideanLin ck)‖)

Unfolding powellYuanSection13325 sorry sorry sorry sorry sorry : ProppowellYuanSection13325 Unknown identifier `ck`ck Unknown identifier `Ak`Ak Unknown identifier `trustRegionRadius`trustRegionRadius Unknown identifier `ξk`ξk Unknown identifier `b2`b2 recovers the two source branches from (Invalid projection: Index `25` is invalid for this structure; the only valid index is 1 Note: The expression 13.3 has type `Float` which has only 1 field13.3.25).

theorem powellYuanSection13325_iff
    (ck : EuclideanSpace ℝ (Fin m))
    (Ak : Matrix (Fin n) (Fin m) ℝ)
    (trustRegionRadius ξk b2 : ℝ) :
    powellYuanSection13325 ck Ak trustRegionRadius ξk b2 ↔
      (b2 * trustRegionRadius ≥ ‖(Ak⁺).transpose.toEuclideanLin ck‖ → ξk = 0) ∧
        (b2 * trustRegionRadius < ‖(Ak⁺).transpose.toEuclideanLin ck‖ →
          ξk ≤ ‖ck - Ak.transpose.toEuclideanLin
              (((b2 * trustRegionRadius) / ‖(Ak⁺).transpose.toEuclideanLin ck‖) •
                (Ak⁺).transpose.toEuclideanLin ck)‖) :=
  Iff.rfl

'powellYuanSection13325' depends on axioms: [propext, Classical.choice, Quot.sound]

'powellYuanSection13325_iff' depends on axioms: [propext, Classical.choice, Quot.sound]

end