| 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 |
|
|
