| import Mathlib | |
| import OAI.Combinatorics.MatchingPSD.OrthogonalCone | |
| import OAI.Combinatorics.MatchingPSD.DeterminantNormalization | |
| namespace OAI | |
| namespace AffinePSD | |
| open Matrix Finset | |
| open scoped BigOperators | |
| variable {I R : Type*} [Fintype I] [Fintype R] [DecidableEq R] | |
| omit [Fintype I] in | |
| lemma evaluationCone_convex (B : I → Matrix R R ℝ) : Convex ℝ (evaluationCone B) := | |
| PerfectMatchingPSD.convex_posSemidef.linear_image (evaluation B) | |
| omit [Fintype I] in | |
| lemma evaluationCone_zero (B : I → Matrix R R ℝ) : 0 ∈ evaluationCone B := | |
| ⟨0, Matrix.PosSemidef.zero, map_zero _⟩ | |
| omit [Fintype I] in | |
| lemma evaluationCone_smul (B : I → Matrix R R ℝ) {y : I → ℝ} | |
| (hy : y ∈ evaluationCone B) {t : ℝ} (ht : 0 ≤ t) : | |
| t • y ∈ evaluationCone B := by | |
| obtain ⟨A, hA, rfl⟩ := hy | |
| exact ⟨t • A, hA.smul ht, map_smul _ _ _⟩ | |
| lemma trace_evaluation_duality (B : I → Matrix R R ℝ) (c : I → ℝ) | |
| (A : Matrix R R ℝ) : | |
| (A * ∑ i, c i • B i).trace = ∑ i, c i * evaluation B A i := by | |
| simp [evaluation, Matrix.mul_sum, Matrix.trace_sum, Matrix.trace_smul] | |
| omit [Fintype I] [DecidableEq R] in | |
| lemma psd_of_trace_nonneg {D : Matrix R R ℝ} (hD : D.IsHermitian) | |
| (h : ∀ A : Matrix R R ℝ, A.PosSemidef → 0 ≤ (A * D).trace) : D.PosSemidef := by | |
| apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg hD | |
| intro x | |
| have hx : (Matrix.vecMulVec x x).PosSemidef := by | |
| simpa using Matrix.posSemidef_vecMulVec_self_star x | |
| have ht := h _ hx | |
| rw [Matrix.trace_mul_comm, Matrix.mul_vecMulVec, Matrix.trace_vecMulVec, | |
| dotProduct_comm] at ht | |
| simpa only [star_trivial] using ht | |
| lemma exists_psd_evaluation (B : I → Matrix R R ℝ) (hB : ∀ i, (B i).PosSemidef) | |
| (y : I → ℝ) | |
| (hy : ∀ c : I → ℝ, (∑ i, c i • B i).PosSemidef → 0 ≤ ∑ i, c i * y i) : | |
| ∃ A : Matrix R R ℝ, A.PosSemidef ∧ evaluation B A = y := by | |
| classical | |
| change y ∈ evaluationCone B | |
| by_contra hn | |
| obtain ⟨f, u, hfy, hu⟩ := geometric_hahn_banach_point_closed | |
| (evaluationCone_convex B) (evaluationCone_isClosed B hB) hn | |
| have hu0 : u < 0 := by simpa using hu 0 (evaluationCone_zero B) | |
| have hfn (z : I → ℝ) (hz : z ∈ evaluationCone B) : 0 ≤ f z := by | |
| by_contra hneg | |
| have hnz : f z < 0 := lt_of_not_ge hneg | |
| have hratio : 0 < u / f z := div_pos_of_neg_of_neg hu0 hnz | |
| have ht := hu ((u / f z + 1) • z) | |
| (evaluationCone_smul B hz (by linarith : 0 ≤ u / f z + 1)) | |
| rw [map_smul, smul_eq_mul] at ht | |
| have he : (u / f z + 1) * f z = u + f z := by | |
| field_simp [ne_of_lt hnz] | |
| rw [he] at ht | |
| linarith | |
| let c : I → ℝ := fun i => f (fun j => if i = j then 1 else 0) | |
| have hf (z : I → ℝ) : f z = ∑ i, c i * z i := by | |
| simpa only [ContinuousLinearMap.coe_coe, c, smul_eq_mul, mul_comm] using f.toLinearMap.pi_apply_eq_sum_univ z | |
| have hD : (∑ i, c i • B i).PosSemidef := by | |
| apply psd_of_trace_nonneg | |
| · exact isSelfAdjoint_sum univ (fun i _ => (hB i).isHermitian.smul (by simp [IsSelfAdjoint])) | |
| · intro A hA | |
| rw [trace_evaluation_duality, ← hf] | |
| exact hfn _ ⟨A, hA, rfl⟩ | |
| have h := hy c hD | |
| rw [← hf] at h | |
| linarith | |
| end AffinePSD | |
| end OAI | |