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e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | import Mathlib
import OAI.Combinatorics.MatchingPSD.AffineModel
import OAI.Combinatorics.MatchingPSD.FiniteFactorization
namespace OAI
namespace PerfectMatchingPSD
open MatchingSmoothing Finset Matrix
open scoped BigOperators
noncomputable def slackAffine {n : ℕ} (i : Row n) : (Edge n → ℝ) →ᵃ[ℝ] ℝ := by
classical
cases i with
| inl e => exact (LinearMap.proj e).toAffineMap
| inr U =>
exact (∑ e : Edge n, if Crosses U.1 e then LinearMap.proj e else 0).toAffineMap -
AffineMap.const ℝ (Edge n → ℝ) 1
lemma slackAffine_vertex {n : ℕ} (i : Row n) (M : PerfectMatching n) :
slackAffine i (matchingPoint M) = slack i M := by
classical
cases i with
| inl e => rfl
| inr U =>
simp only [slackAffine, AffineMap.coe_sub, Pi.sub_apply, AffineMap.const_apply,
LinearMap.coe_toAffineMap, slack]
congr 1
rw [LinearMap.sum_apply]
calc
_ = ∑ e : Edge n, if Crosses U.1 e ∧ e ∈ M.1 then (1 : ℝ) else 0 := by
apply Finset.sum_congr rfl
intro e _
by_cases h : Crosses U.1 e <;> simp [h, matchingPoint]
_ = _ := by
simp only [Finset.sum_boole, crossingCount]
apply congrArg (fun s : Finset (Edge n) => (s.card : ℝ))
ext e
simp [and_comm]
lemma slackAffine_nonneg {n : ℕ} (i : Row n) {x : Edge n → ℝ}
(hx : x ∈ matchingPolytope n) : 0 ≤ slackAffine i x := by
have hc : Convex ℝ {x : Edge n → ℝ | 0 ≤ slackAffine i x} := by
exact (convex_Ici (0 : ℝ)).affine_preimage (slackAffine i)
apply convexHull_min _ hc hx
rintro _ ⟨M, rfl⟩
rw [Set.mem_ofPred_eq, slackAffine_vertex]
exact slack_nonneg i M
end PerfectMatchingPSD
end OAI
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