File size: 2,373 Bytes
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 53 54 55 | 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
|