| import Mathlib |
| import OAI.Combinatorics.MatchingPSD.PSDCompression |
| import OAI.Combinatorics.MatchingPSD.PSDTopology |
| import OAI.Combinatorics.MatchingPSD.MatrixBounds |
|
|
| namespace OAI |
|
|
| namespace AffinePSD |
| open Matrix Finset |
| open scoped BigOperators |
| variable {I R : Type*} [Fintype I] [Fintype R] [DecidableEq R] |
|
|
| lemma evaluationCone_isClosed_of_diagonal (B : I → Matrix R R ℝ) |
| (hB : ∀ i, (B i).PosSemidef) (d : R → ℝ) |
| (hs : ∑ i, B i = diagonal d) : IsClosed (evaluationCone B) := by |
| classical |
| apply IsSeqClosed.isClosed |
| intro u y hu hy |
| choose A hA heA using hu |
| have hc (k : ℕ) := diagonal_compression B hB d hs (A k) (hA k) |
| choose C hC heC hzC using hc |
| have heval (k : ℕ) : evaluation B (C k) = u k := (heC k).trans (heA k) |
| have hsum : Filter.Tendsto (fun k => ∑ i, u k i) Filter.atTop (nhds (∑ i, y i)) := |
| tendsto_finsetSum _ (fun i _ => (tendsto_pi_nhds.mp hy) i) |
| obtain ⟨b, hb⟩ := hsum.bddAbove_range |
| have hbound (k : ℕ) : ∑ i, u k i ≤ b := hb (Set.mem_range_self k) |
| let a : ℝ := ∑ j, (d j)⁻¹ |
| have ha : 0 ≤ a := Finset.sum_nonneg (fun j _ => |
| inv_nonneg.mpr (sum_diagonal_nonneg B hB d hs j)) |
| have htr (k : ℕ) : (C k).trace ≤ a * b := by |
| refine (diagonal_trace_bound B hB d hs (C k) (hC k) (hzC k)).trans ?_ |
| rw [heval] |
| exact mul_le_mul_of_nonneg_left (hbound k) ha |
| let lo : R → R → ℝ := fun _ _ => -(a * b) |
| let hi : R → R → ℝ := fun _ _ => a * b |
| have hm (k : ℕ) : C k ∈ Set.Icc lo hi := by |
| constructor <;> intro i j |
| · exact (abs_le.mp ((PerfectMatchingPSD.posSemidef_entry_abs_le_trace (hC k) i j).trans |
| (htr k))).1 |
| · exact (abs_le.mp ((PerfectMatchingPSD.posSemidef_entry_abs_le_trace (hC k) i j).trans |
| (htr k))).2 |
| obtain ⟨L, hL, φ, hφ, ht⟩ := IsCompact.tendsto_subseq (X := R → R → ℝ) (isCompact_Icc : IsCompact (Set.Icc lo hi)) hm |
| have hpsd : Matrix.PosSemidef L := psd_isClosed.mem_of_tendsto ht |
| (Filter.Eventually.of_forall (fun k => hC (φ k))) |
| refine ⟨L, hpsd, ?_⟩ |
| have hlim := (evaluation_continuous B).continuousAt.tendsto.comp ht |
| have hlim' := hy.comp hφ.tendsto_atTop |
| have hEq : evaluation B ∘ (C ∘ φ) = u ∘ φ := by ext k i; exact congrFun (heval (φ k)) i |
| have hlim := hlim.congr (fun k => congrFun hEq k) |
| exact tendsto_nhds_unique hlim hlim' |
|
|
| end AffinePSD |
|
|
| end OAI |
|
|
