statement stringlengths 1 615 | proof stringlengths 0 8.26k | type stringclasses 11
values | symbolic_name stringlengths 1 64 | library stringclasses 29
values | filename stringclasses 121
values | imports listlengths 0 15 | deps listlengths 0 20 | docstring stringlengths 0 1.5k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
rewrite_objFun_6 (hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → f x = g x) :
⟨f, [[c1, c2, c3, c4, c5, c6]]⟩ ≡ ⟨g, [[c1, c2, c3, c4, c5, c6]]⟩ | rewrite_objFun (fun x _ => by apply hrw x <;> tauto) | def | Minimization.Equivalence.rewrite_objFun_6 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_objFun_7 (hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → f x = g x) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7]]⟩ ≡ ⟨g, [[c1, c2, c3, c4, c5, c6, c7]]⟩ | rewrite_objFun (fun x _ => by apply hrw x <;> tauto) | def | Minimization.Equivalence.rewrite_objFun_7 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_objFun_8 (hrw :
∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → c8 x → f x = g x) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8]]⟩ ≡ ⟨g, [[c1, c2, c3, c4, c5, c6, c7, c8]]⟩ | rewrite_objFun (fun x _ => by apply hrw x <;> tauto) | def | Minimization.Equivalence.rewrite_objFun_8 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_objFun_9
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → c8 x → c9 x → f x = g x) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9]]⟩ ≡
⟨g, [[c1, c2, c3, c4, c5, c6, c7, c8, c9]]⟩ | rewrite_objFun (fun x _ => by apply hrw x <;> tauto) | def | Minimization.Equivalence.rewrite_objFun_9 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_objFun_10
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → c8 x → c9 x → c10 x → f x = g x) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9, c10]]⟩ ≡
⟨g, [[c1, c2, c3, c4, c5, c6, c7, c8, c9, c10]]⟩ | rewrite_objFun (fun x _ => by apply hrw x <;> tauto) | def | Minimization.Equivalence.rewrite_objFun_10 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
"prove_phi_feasibility_of_rw_constr" hrw:ident : tactic =>
`(tactic| (simp [feasible]; intros; rw [← $hrw] <;> tauto)) | macro | prove_phi_feasibility_of_rw_constr | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | ||
"prove_psi_feasibility_of_rw_constr" hrw:ident : tactic =>
`(tactic| (simp [feasible]; intros; rw [($hrw)] <;> tauto)) | macro | prove_psi_feasibility_of_rw_constr | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | ||
"equivalence_of_rw_constr" hrw:ident : term =>
`(Equivalence.ofStrongEquivalence <|
{ phi := id,
psi := id,
phi_feasibility := by prove_phi_feasibility_of_rw_constr $hrw
psi_feasibility := by prove_psi_feasibility_of_rw_constr $hrw
phi_optimality := fun {x} _ => le_refl _
psi_optimal... | macro | equivalence_of_rw_constr | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | ||
rewrite_constraints (hrw : ∀ x, cs x ↔ cs' x) : ⟨f, [[cs]]⟩ ≡ ⟨f, [[cs']]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraints | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_1 (hrw : ∀ x, cs x → (c1 x ↔ c1' x)) : ⟨f, [[c1, cs]]⟩ ≡ ⟨f, [[c1', cs]]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_1 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_1_last (hrw : ∀ x, c1 x ↔ c1' x) : ⟨f, [[c1]]⟩ ≡ ⟨f, [[c1']]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_1_last | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_2 (hrw : ∀ x, c1 x → cs x → (c2 x ↔ c2' x)) :
⟨f, [[c1, c2, cs]]⟩ ≡ ⟨f, [[c1, c2', cs]]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_2 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_2_last (hrw : ∀ x, c1 x → (c2 x ↔ c2' x)) :
⟨f, [[c1, c2]]⟩ ≡ ⟨f, [[c1, c2']]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_2_last | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_3 (hrw : ∀ x, c1 x → c2 x → cs x → (c3 x ↔ c3' x)) :
⟨f, [[c1, c2, c3, cs]]⟩ ≡ ⟨f, [[c1, c2, c3', cs]]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_3 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_3_last (hrw : ∀ x, c1 x → c2 x → (c3 x ↔ c3' x)) :
⟨f, [[c1, c2, c3]]⟩ ≡ ⟨f, [[c1, c2, c3']]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_3_last | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_4 (hrw : ∀ x, c1 x → c2 x → c3 x → cs x → (c4 x ↔ c4' x)) :
⟨f, [[c1, c2, c3, c4, cs]]⟩ ≡ ⟨f, [[c1, c2, c3, c4', cs]]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_4 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_4_last (hrw : ∀ x, c1 x → c2 x → c3 x → (c4 x ↔ c4' x)) :
⟨f, [[c1, c2, c3, c4]]⟩ ≡ ⟨f, [[c1, c2, c3, c4']]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_4_last | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_5 (hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → cs x → (c5 x ↔ c5' x)) :
⟨f, [[c1, c2, c3, c4, c5, cs]]⟩ ≡ ⟨f, [[c1, c2, c3, c4, c5', cs]]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_5 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_5_last (hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → (c5 x ↔ c5' x)) :
⟨f, [[c1, c2, c3, c4, c5]]⟩ ≡ ⟨f, [[c1, c2, c3, c4, c5']]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_5_last | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_6 (hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → cs x → (c6 x ↔ c6' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, cs]]⟩ ≡ ⟨f, [[c1, c2, c3, c4, c5, c6', cs]]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_6 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_6_last (hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → (c6 x ↔ c6' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6]]⟩ ≡ ⟨f, [[c1, c2, c3, c4, c5, c6']]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_6_last | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_7
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → cs x → (c7 x ↔ c7' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, cs]]⟩ ≡ ⟨f, [[c1, c2, c3, c4, c5, c6, c7', cs]]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_7 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_7_last
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → (c7 x ↔ c7' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7]]⟩ ≡ ⟨f, [[c1, c2, c3, c4, c5, c6, c7']]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_7_last | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_8
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → cs x → (c8 x ↔ c8' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, cs]]⟩ ≡ ⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8', cs]]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_8 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_8_last
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → (c8 x ↔ c8' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8]]⟩ ≡ ⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8']]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_8_last | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_9
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → c8 x → cs x → (c9 x ↔ c9' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9, cs]]⟩ ≡
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9', cs]]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_9 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_9_last
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → c8 x → (c9 x ↔ c9' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9]]⟩ ≡
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9']]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_9_last | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_10
(hrw :
∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → c8 x → c9 x → cs x → (c10 x ↔ c10' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, cs]]⟩ ≡
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9, c10', cs]]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_10 | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_10_last
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → c8 x → c9 x → (c10 x ↔ c10' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9, c10]]⟩ ≡
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9, c10']]⟩ | equivalence_of_rw_constr hrw | def | Minimization.Equivalence.rewrite_constraint_10_last | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
add_constraint {cs' : D → Prop} (h : ∀ x, cs x → cs' x) : ⟨f, cs⟩ ≡ ⟨f, [[cs', cs]]⟩ | Equivalence.ofStrongEquivalence <|
{ phi := id,
psi := id,
phi_feasibility := fun _ hcs => ⟨h _ hcs, hcs⟩,
psi_feasibility := fun _ ⟨_, hcs⟩ => hcs,
phi_optimality := fun _ _ => le_refl _,
psi_optimality := fun _ _ => le_refl _ } | def | Minimization.Equivalence.add_constraint | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | We can always add a redundant constraint. This might be useful to help the reduction algorithm
infer some constraints that cannot be easily infered by `arith`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
map_le_constraint_standard_form [Zero R] {cs' : D → Prop} {fi : D → R} {g : R → R}
(hcs : ∀ x, cs x ↔ fi x ≤ 0 ∧ cs' x) (hg : ∀ x, g x ≤ 0 ↔ x ≤ 0) :
⟨f, cs⟩ ≡ ⟨f, fun x => g (fi x) ≤ 0 ∧ cs' x⟩ | by
apply rewrite_constraints; intros x; rw [hcs x, hg (fi x)] | def | Minimization.Equivalence.map_le_constraint_standard_form | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | See [BV04,p.131] where `g` is `ψᵢ`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
map_eq_constraint_standard_form [Zero R] {cs' : D → Prop} {hi : D → R} {g : R → R}
(hcs : ∀ x, cs x ↔ hi x = 0 ∧ cs' x) (hg : ∀ x, g x = 0 ↔ x = 0) :
⟨f, cs⟩ ≡ ⟨f, fun x => g (hi x) = 0 ∧ cs' x⟩ | by
apply rewrite_constraints; intros x; rw [hcs x, hg (hi x)] | def | Minimization.Equivalence.map_eq_constraint_standard_form | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | See [BV04,p.131] where `g` is `ψₘ₊ᵢ`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
add_slack_variable_standard_form {cs' : D → Prop} {fi : D → ℝ}
(hcs : ∀ x, cs x ↔ fi x ≤ 0 ∧ cs' x) :
⟨f, cs⟩ ≡ ⟨fun (_, x) => f x, fun (si, x) => 0 ≤ (si : ℝ) ∧ fi x + si = 0 ∧ cs' x⟩ | Equivalence.ofStrongEquivalence <|
{ phi := fun x => (-fi x, x),
psi := fun (_, x) => x,
phi_feasibility := fun {x} h_feas_x => by
simp [feasible, hcs x] at h_feas_x ⊢; exact h_feas_x,
psi_feasibility := fun (si, x) ⟨hsi, hfi, hc⟩ => by
simp [feasible, hcs x]; refine ⟨?_, hc⟩; linarith
phi... | def | Minimization.Equivalence.add_slack_variable_standard_form | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | Adding a slack variable [BV04,p.131]. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
eliminate_eq_constraint_standard_form [Inhabited E] {cs' : D → Prop} {hi : D → ℝ}
{g : E → D} (hcs : ∀ x, cs x ↔ hi x = 0 ∧ cs' x) (hg : ∀ x, hi x = 0 ↔ ∃ z, x = g z) :
⟨f, cs⟩ ≡ ⟨fun x => f (g x), fun x => cs' (g x)⟩ | Equivalence.ofStrongEquivalence <|
{ phi := fun x => if h : hi x = 0 then Classical.choose ((hg x).mp h) else default,
psi := g,
phi_feasibility := fun {x} h_feas_x => by
simp [feasible, hcs x] at h_feas_x ⊢
replace ⟨h_hix_eq_0, h_cx⟩ := h_feas_x
simp [h_hix_eq_0]
rwa [Classical.choose... | def | Minimization.Equivalence.eliminate_eq_constraint_standard_form | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | Eliminate equality constraints [BV04,p.132]. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
decompose_constraint (g : D → E) (cs' : D → E → Prop) (hc : ∀ x, cs x ↔ cs' x (g x)) :
⟨f, cs⟩ ≡ ⟨fun (x, _) => f x, fun (x, y) => y = g x ∧ cs' x y⟩ | Equivalence.ofStrongEquivalence <|
{ phi := fun x => (x, g x),
psi := fun (x, _) => x,
phi_feasibility := fun {x} h_feas_x => by simpa [feasible, ← hc, h_feas_x],
psi_feasibility := fun (x, y) ⟨h_x_eq_gy, h_cyx⟩ => by simp [feasible, hc, h_x_eq_gy ▸ h_cyx],
phi_optimality := fun {_} _ => le_refl _,
... | def | Minimization.Equivalence.decompose_constraint | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | Decompose constraint by introducing another equality constraint [BV04,p.132]. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
epigraph_form : ⟨f, cs⟩ ≡ ⟨fun (t, _) => t, fun (t, x) => f x ≤ t ∧ cs x⟩ | Equivalence.ofStrongEquivalence <|
{ phi := fun x => (f x, x),
psi := fun (_, x) => x,
phi_feasibility := fun {x} h_feas_x => by simpa [feasible],
psi_feasibility := fun (t, x) ⟨_, h_csx⟩ => by simpa [feasible],
phi_optimality := fun {_} _ => le_refl _,
psi_optimality := fun {_} ⟨h_fx_le_t, _⟩ => ... | def | Minimization.Equivalence.epigraph_form | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | Epigraph form [BV04,p.134]. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
eq_to_le_left {S} [Preorder S] (e : D ≃ S × E) (g : E → S) (cs' : D → Prop)
(hcs : ∀ {x}, cs x ↔ ((e x).1 = g (e x).2 ∧ cs' x))
(hf : ∀ y r s, f (e.symm (r, y)) = f (e.symm (s, y)))
(h_mono: ∀ y r s, r ≤ s → cs' (e.symm (r, y)) → cs' (e.symm (s, y))) :
⟨f, cs⟩ ≡ ⟨f, fun x => (e x).1 ≤ g (e x).2 ∧ cs' x⟩ | Equivalence.ofStrongEquivalence <|
{ phi := fun x => x,
psi := fun x => e.symm (g (e x).2, (e x).2),
phi_feasibility := fun {x} h_feas_x => ⟨le_of_eq (hcs.1 h_feas_x).1, (hcs.1 h_feas_x).2⟩,
psi_feasibility := fun {x} h_feas_x => by
have hcx : cs' x := h_feas_x.2
have hcegex : cs' (e.symm (g (... | def | Minimization.Equivalence.eq_to_le_left | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | Suppose `D ≃ S × E`. Let problem `p := ⟨f, cs⟩` be defined over `D`. Every `x : D` maps
one-to-one to `(s, y) : S × E`. Assume that `x` is `p`-feasible iff `s = g y` and `cs' x`. We can
think of `s` as a new variable. If changing `s` does not change the objective function and the new
constraints `c` respect monotonicit... | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
eq_to_le_right {S} [Preorder S] (e : Equiv D (S × E)) (g : E → S) (cs' : D → Prop)
(hcs : ∀ {x}, cs x ↔ (g (e x).2 = (e x).1 ∧ cs' x))
(hf : ∀ x r s, f (e.symm ⟨r, x⟩) = f (e.symm ⟨s, x⟩))
(h_mono : ∀ x r s, r ≤ s → cs' (e.symm (s, x)) → cs' (e.symm ⟨r, x⟩)) :
⟨f, cs⟩ ≡ ⟨f, fun x => g (e x).2 ≤ (e x).1... | Equivalence.ofStrongEquivalence <|
{ phi := fun x => x,
psi := fun x => e.symm ⟨g (e x).2, (e x).2⟩,
phi_feasibility := fun {x} h_feas_x => ⟨le_of_eq (hcs.1 h_feas_x).1, (hcs.1 h_feas_x).2⟩,
psi_feasibility := fun {x} h_feas_x => by
have hcx : cs' x := h_feas_x.2
have hcegex : cs' (e.symm ⟨g (... | def | Minimization.Equivalence.eq_to_le_right | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | Similar to `eq_to_le_left` with the monotonicity condition on `c` flipped. In this case we have
that `P` is equivalent to `⟨f, g y ≤ s ∧ cs' x⟩`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
domain_equiv (e : E ≃ D) : ⟨f, cs⟩ ≡ ⟨f ∘ e, cs ∘ e⟩ | Equivalence.ofStrongEquivalence <|
{ phi := e.symm,
psi := e,
phi_feasibility := fun {x} h_feas_x => by simpa [feasible],
psi_feasibility := fun {x} h_feas_x => by simpa [feasible],
phi_optimality := fun {x} _ => by simp,
psi_optimality := fun {x} _ => by simp } | def | Minimization.Equivalence.domain_equiv | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | Changing the domain to an equivalent type yields an equivalent problem. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
linearization_mono {S} [Preorder S] (g : D → S) (c : S → D → Prop) (h : S → D → R)
(hf : ∀ x, f x = h (g x) x)
(hcs : ∀ x, cs x = c (g x) x)
(h_mono_f : ∀ x r s, r ≤ s → h s x ≤ h r x)
(h_mono_cs : ∀ x r s, r ≤ s → c r x → c s x) :
⟨f, cs⟩ ≡ ⟨fun (s, y) => h s y, fun (s, y) => s ≤ g y ∧ c s y⟩ | Equivalence.ofStrongEquivalence <|
{ phi := fun x => (g x, x),
psi := fun (_, y) => y,
phi_feasibility := fun {x} h_feas_x => ⟨le_refl _, hcs x ▸ h_feas_x⟩,
psi_feasibility := fun (s, y) ⟨h_s_le_gy, h_csy⟩ => by
simp only [feasible, hcs]; exact h_mono_cs y _ _ h_s_le_gy h_csy
phi_optimality := f... | def | Minimization.Equivalence.linearization_mono | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | Introduce a new variable `s` that replaces occurrences of (non-linear) `g x` in the original
problem. The resulting problem has an extra constraint `s ≤ g y`. The objective funciton and the
rest of the constraints need to satisfy the appropriate monotonicity conditions [Gra05,4.2.1]. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
linearization_antimono {S} [Preorder S] (g : D → S) (c : S → D → Prop) (h : S → D → R)
(hf : ∀ x, f x = h (g x) x)
(hcs : ∀ x, cs x = c (g x) x)
(h_mono_f : ∀ x r s, r ≤ s → h r x ≤ h s x)
(h_mono_cs : ∀ x r s, r ≤ s → c s x → c r x) :
⟨f, cs⟩ ≡ ⟨fun (s, y) => h s y, fun (s, y) => g y ≤ s ∧ c s y⟩ | Equivalence.ofStrongEquivalence <|
{ phi := fun x => (g x, x),
psi := fun (_, y) => y,
phi_feasibility := fun {x} h_feas_x => ⟨le_refl _, hcs x ▸ h_feas_x⟩,
psi_feasibility := fun (s, y) ⟨h_gy_le_s, h_csy⟩ => by
simp only [feasible, hcs]; exact h_mono_cs y _ _ h_gy_le_s h_csy
phi_optimality := f... | def | Minimization.Equivalence.linearization_antimono | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | Similar to `linearization_mono` with the monotonicity conditions flipped. The resulting problem
adds the exactra constraint `g y ≤ s` in this case [Gra05,4.2.1]. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
graph_expansion_greatest {S} [Preorder S] (g : D → S) (c d : S → D → Prop) (h : S → D → R)
(hg : ∀ x v, c v x → IsGreatest {y | d y x} (g x))
(hf : ∀ x, f x = h (g x) x)
(hcs : ∀ x, cs x = c (g x) x)
(h_mono_f : ∀ x r s, r ≤ s → h s x ≤ h r x)
(h_mono_cs : ∀ x r s, r ≤ s → c r x → c s x) :
⟨f, c... | Equivalence.ofStrongEquivalence <|
{ phi := fun x => (g x, x),
psi := fun (_, y) => y,
phi_feasibility := fun {x} h_feas_x => by
simp only [feasible, hcs] at h_feas_x; exact ⟨(hg x (g x) h_feas_x).1, h_feas_x⟩,
psi_feasibility := fun (s, y) ⟨h_dsy, h_csy⟩ => by
simp only [feasible, hcs]; exact... | def | Minimization.Equivalence.graph_expansion_greatest | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | This can be seen as a generalization of `linearization_mono`, where `d` is the graph
implementation of `g`. This is not used by the DCP procedure. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
graph_expansion_least {S} [Preorder S] (g : D → S) (c d : S → D → Prop) (h : S → D → R)
(hg : ∀ x v, c v x → IsLeast {y | d y x} (g x))
(hf : ∀ x, f x = h (g x) x)
(hcs : ∀ x, cs x = c (g x) x)
(h_mono_f : ∀ x r s, r ≤ s → h r x ≤ h s x)
(h_mono_cs : ∀ x r s, r ≤ s → c s x → c r x) :
⟨f, cs⟩ ≡ ⟨... | Equivalence.ofStrongEquivalence <|
{ phi := fun x => (g x, x),
psi := fun (_, y) => y,
phi_feasibility := fun {x} h_feas_x => by
simp only [feasible, hcs] at h_feas_x; exact ⟨(hg x (g x) h_feas_x).1, h_feas_x⟩,
psi_feasibility := fun (s, y) ⟨h_dsy, h_csy⟩ => by
simp only [feasible, hcs]; exact... | def | Minimization.Equivalence.graph_expansion_least | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | Similar to `graph_expansion_greatest` but in the flipped monotonicity context, c.f.
`linearization_antimono`. This is not used by the DCP procedure. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
graph_expansion_least_forall {S I : Type} [Preorder S] [Inhabited I] (g : D → I → S)
(c d : S → D → Prop) (hg : ∀ x v i, c v x → IsLeast {y | d y x} (g x i))
(hcs : ∀ x, cs x = ∀ i, c (g x i) x) (h_mono_cs : ∀ x r s, r ≤ s → c s x → c r x) :
⟨f, cs⟩ ≡
⟨fun sy : (I → S) × D => f sy.2,
fun sy : (I → ... | graph_expansion_least (f := f) (cs := cs) (g := g) (c := fun y x => ∀ i, c (y i) x)
(d := fun y x => ∀ i, d (y i) x) (h := fun _ x => f x)
(hg := fun x v hc =>
⟨fun i => (hg x (v i) i (hc i)).1, fun _ c i => (hg x (v i) i (hc i)).2 (c i)⟩)
(hf := fun _ => rfl) (hcs := hcs) (h_mono_f := fun _ _ _ _ => ... | def | Minimization.Equivalence.graph_expansion_least_forall | Lib | CvxLean/Lib/Equivalence.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Minimization",
"CvxLean.Meta.Attributes"
] | [] | Version of `graph_expansion_least` that works with vectors. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
Minimization (D R : Type) where
objFun : D → R
constraints : D → Prop | structure | Minimization | Lib | CvxLean/Lib/Minimization.lean | [
"Mathlib.Order.Bounds.Basic"
] | [] | Type of an optimization problem. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
feasible (x : D) : Prop | p.constraints x | def | Minimization.feasible | Lib | CvxLean/Lib/Minimization.lean | [
"Mathlib.Order.Bounds.Basic"
] | [] | A point `x : D` is feasible in `p` if it satisfies the constraints. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
optimal (x : D) : Prop | p.feasible x ∧ ∀ y, p.feasible y → p.objFun x ≤ p.objFun y | def | Minimization.optimal | Lib | CvxLean/Lib/Minimization.lean | [
"Mathlib.Order.Bounds.Basic"
] | [] | A point `x : D` is optimal in `p` if it is feasible and for any feasible point `y : D` the
value of `x` is a lower bound to the value of `y`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
Solution where
point : D
isOptimal : p.optimal point | structure | Minimization.Solution | Lib | CvxLean/Lib/Minimization.lean | [
"Mathlib.Order.Bounds.Basic"
] | [] | A solution is simply an optimal point. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
Reduction where
psi : E → D
psi_optimality : ∀ x, q.optimal x → p.optimal (psi x) | structure | Minimization.Reduction | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [] | We read `Reduction p q` as `p` reduces to `q` [Pap93,10.1]. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
refl : p ≼ p | { psi := id,
psi_optimality := fun _ hy => hy } | def | Minimization.Reduction.refl | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
trans (R₁ : p ≼ q) (R₂ : q ≼ r) : p ≼ r | { psi := R₁.psi ∘ R₂.psi,
psi_optimality := fun x h => R₁.psi_optimality (R₂.psi x) (R₂.psi_optimality x h) } | def | Minimization.Reduction.trans | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
toBwd (R : p ≼ q) : Solution q → Solution p | fun sol =>
{ point := R.psi sol.point,
isOptimal := R.psi_optimality sol.point sol.isOptimal } | def | Minimization.Reduction.toBwd | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
ofEquivalence (E : p ≡ q) : p ≼ q | { psi := E.psi,
psi_optimality := E.psi_optimality } | def | Minimization.Reduction.ofEquivalence | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
map_objFun_of_order_reflecting {D R} [Preorder R] {f : D → R} {g : R → R} {cs : D → Prop}
(h : ∀ {r s}, cs r → cs s → g (f r) ≤ g (f s) → f r ≤ f s) :
⟨f, cs⟩ ≼ ⟨fun x => g (f x), cs⟩ | { psi := id,
psi_optimality := fun x ⟨h_feas_x, h_opt_x⟩ =>
⟨h_feas_x,
fun y h_feas_y =>
have h_gfx_le_gfy : g (f x) ≤ g (f y) := h_opt_x y h_feas_y;
h h_feas_x h_feas_y h_gfx_le_gfy⟩ } | def | Minimization.Reduction.map_objFun_of_order_reflecting | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [] | Weaker version of `Equivalence.map_objFun`. For a reduction we only need the map to be
order-reflecting on the image of the objective function. Note that for an equivalence we also need
it to be order-preserving (order-reflecting + order-preserving = order embedding). | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
map_objFun {D R} [Preorder R] {f : D → R} {g : R → R} {cs : D → Prop}
(h : ∀ {r s}, cs r → cs s → (g (f r) ≤ g (f s) ↔ f r ≤ f s)) :
⟨f, cs⟩ ≼ ⟨fun x => g (f x), cs⟩ | ofEquivalence <| Equivalence.map_objFun h | def | Minimization.Reduction.map_objFun | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [] | See `Equivalence.map_objFun`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
map_objFun_log {f : D → ℝ} (h : ∀ x, cs x → f x > 0) :
⟨f, cs⟩ ≼ ⟨fun x => (Real.log (f x)), cs⟩ | ofEquivalence <| Equivalence.map_objFun_log h | def | Minimization.Reduction.map_objFun_log | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [] | See `Equivalence.map_objFun_log`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
map_objFun_sq {f : D → ℝ} (h : ∀ x, cs x → f x ≥ 0) :
⟨f, cs⟩ ≼ ⟨fun x => (f x) ^ (2 : ℝ), cs⟩ | ofEquivalence <| Equivalence.map_objFun_sq h | def | Minimization.Reduction.map_objFun_sq | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [] | See `Equivalence.map_objFun_sq`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
map_domain {f : D → R} {cs : D → Prop} {fwd : D → E} {bwd : E → D}
(h : ∀ x, cs x → bwd (fwd x) = x) :
⟨f, cs⟩ ≼ ⟨fun x => f (bwd x), (fun x => cs (bwd x))⟩ | ofEquivalence <| Equivalence.map_domain h | def | Minimization.Reduction.map_domain | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [] | See `Equivalence.map_domain`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
rewrite_objFun (hrw : ∀ x, cs x → f x = g x) : ⟨f, cs⟩ ≼ ⟨g, cs⟩ | ofEquivalence <| Equivalence.rewrite_objFun hrw | def | Minimization.Reduction.rewrite_objFun | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraints (hrw : ∀ x, cs x ↔ cs' x) : ⟨f, [[cs]]⟩ ≼ ⟨f, [[cs']]⟩ | ofEquivalence <| Equivalence.rewrite_constraints hrw | def | Minimization.Reduction.rewrite_constraints | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_1 (hrw : ∀ x, cs x → (c1 x ↔ c1' x)) : ⟨f, [[c1, cs]]⟩ ≼ ⟨f, [[c1', cs]]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_1 hrw | def | Minimization.Reduction.rewrite_constraint_1 | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_1_last (hrw : ∀ x, c1 x ↔ c1' x) : ⟨f, [[c1]]⟩ ≼ ⟨f, [[c1']]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_1_last hrw | def | Minimization.Reduction.rewrite_constraint_1_last | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_2 (hrw : ∀ x, c1 x → cs x → (c2 x ↔ c2' x)) :
⟨f, [[c1, c2, cs]]⟩ ≼ ⟨f, [[c1, c2', cs]]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_2 hrw | def | Minimization.Reduction.rewrite_constraint_2 | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_2_last (hrw : ∀ x, c1 x → (c2 x ↔ c2' x)) :
⟨f, [[c1, c2]]⟩ ≼ ⟨f, [[c1, c2']]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_2_last hrw | def | Minimization.Reduction.rewrite_constraint_2_last | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_3 (hrw : ∀ x, c1 x → c2 x → cs x → (c3 x ↔ c3' x)) :
⟨f, [[c1, c2, c3, cs]]⟩ ≼ ⟨f, [[c1, c2, c3', cs]]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_3 hrw | def | Minimization.Reduction.rewrite_constraint_3 | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_3_last (hrw : ∀ x, c1 x → c2 x → (c3 x ↔ c3' x)) :
⟨f, [[c1, c2, c3]]⟩ ≼ ⟨f, [[c1, c2, c3']]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_3_last hrw | def | Minimization.Reduction.rewrite_constraint_3_last | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_4 (hrw : ∀ x, c1 x → c2 x → c3 x → cs x → (c4 x ↔ c4' x)) :
⟨f, [[c1, c2, c3, c4, cs]]⟩ ≼ ⟨f, [[c1, c2, c3, c4', cs]]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_4 hrw | def | Minimization.Reduction.rewrite_constraint_4 | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_4_last (hrw : ∀ x, c1 x → c2 x → c3 x → (c4 x ↔ c4' x)) :
⟨f, [[c1, c2, c3, c4]]⟩ ≼ ⟨f, [[c1, c2, c3, c4']]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_4_last hrw | def | Minimization.Reduction.rewrite_constraint_4_last | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_5 (hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → cs x → (c5 x ↔ c5' x)) :
⟨f, [[c1, c2, c3, c4, c5, cs]]⟩ ≼ ⟨f, [[c1, c2, c3, c4, c5', cs]]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_5 hrw | def | Minimization.Reduction.rewrite_constraint_5 | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_5_last (hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → (c5 x ↔ c5' x)) :
⟨f, [[c1, c2, c3, c4, c5]]⟩ ≼ ⟨f, [[c1, c2, c3, c4, c5']]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_5_last hrw | def | Minimization.Reduction.rewrite_constraint_5_last | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_6 (hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → cs x → (c6 x ↔ c6' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, cs]]⟩ ≼ ⟨f, [[c1, c2, c3, c4, c5, c6', cs]]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_6 hrw | def | Minimization.Reduction.rewrite_constraint_6 | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_6_last (hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → (c6 x ↔ c6' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6]]⟩ ≼ ⟨f, [[c1, c2, c3, c4, c5, c6']]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_6_last hrw | def | Minimization.Reduction.rewrite_constraint_6_last | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_7
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → cs x → (c7 x ↔ c7' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, cs]]⟩ ≼ ⟨f, [[c1, c2, c3, c4, c5, c6, c7', cs]]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_7 hrw | def | Minimization.Reduction.rewrite_constraint_7 | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_7_last
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → (c7 x ↔ c7' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7]]⟩ ≼ ⟨f, [[c1, c2, c3, c4, c5, c6, c7']]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_7_last hrw | def | Minimization.Reduction.rewrite_constraint_7_last | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_8
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → cs x → (c8 x ↔ c8' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, cs]]⟩ ≼ ⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8', cs]]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_8 hrw | def | Minimization.Reduction.rewrite_constraint_8 | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_8_last
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → (c8 x ↔ c8' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8]]⟩ ≼ ⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8']]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_8_last hrw | def | Minimization.Reduction.rewrite_constraint_8_last | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_9
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → c8 x → cs x → (c9 x ↔ c9' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9, cs]]⟩ ≼
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9', cs]]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_9 hrw | def | Minimization.Reduction.rewrite_constraint_9 | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_9_last
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → c8 x → (c9 x ↔ c9' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9]]⟩ ≼
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9']]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_9_last hrw | def | Minimization.Reduction.rewrite_constraint_9_last | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_10
(hrw :
∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → c8 x → c9 x → cs x → (c10 x ↔ c10' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, cs]]⟩ ≼
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9, c10', cs]]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_10 hrw | def | Minimization.Reduction.rewrite_constraint_10 | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
rewrite_constraint_10_last
(hrw : ∀ x, c1 x → c2 x → c3 x → c4 x → c5 x → c6 x → c7 x → c8 x → c9 x → (c10 x ↔ c10' x)) :
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9, c10]]⟩ ≼
⟨f, [[c1, c2, c3, c4, c5, c6, c7, c8, c9, c10']]⟩ | ofEquivalence <| Equivalence.rewrite_constraint_10_last hrw | def | Minimization.Reduction.rewrite_constraint_10_last | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [
"c1",
"c5"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
ofReductions (R₁ : p ≼ q) (R₂ : q ≼ p) : p ≡ q | { phi := R₂.psi,
psi := R₁.psi,
phi_optimality := R₂.psi_optimality,
psi_optimality := R₁.psi_optimality } | def | Minimization.Equivalence.ofReductions | Lib | CvxLean/Lib/Reduction.lean | [
"CvxLean.Lib.Equivalence"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
Relaxation where
phi : D → E
phi_feasibility : ∀ x, p.feasible x → q.feasible (phi x)
phi_optimality : ∀ x, p.feasible x → q.objFun (phi x) ≤ p.objFun x | structure | Minimization.Relaxation | Lib | CvxLean/Lib/Relaxation.lean | [
"CvxLean.Lib.Equivalence"
] | [] | We read `Relaxation p q` as `p` relaxes to `q` or `q` is a relaxation of `p`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
refl : p ≽' p | { phi := id,
phi_feasibility := fun _ h => h,
phi_optimality := fun _ _ => le_refl _ } | def | Minimization.Relaxation.refl | Lib | CvxLean/Lib/Relaxation.lean | [
"CvxLean.Lib.Equivalence"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
trans (Rx₁ : p ≽' q) (Rx₂ : q ≽' r) : p ≽' r | { phi := Rx₂.phi ∘ Rx₁.phi,
phi_feasibility := fun x h => Rx₂.phi_feasibility (Rx₁.phi x) (Rx₁.phi_feasibility x h),
phi_optimality := fun x hx =>
-- `h(φ₂(φ₁(x))) ≤ g(φ₁(x))`
have h₁ := Rx₂.phi_optimality (Rx₁.phi x) (Rx₁.phi_feasibility x hx)
-- `g(φ₁(x)) ≤ f(x)`
have h₂ := Rx₁.phi_opt... | def | Minimization.Relaxation.trans | Lib | CvxLean/Lib/Relaxation.lean | [
"CvxLean.Lib.Equivalence"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
feasible_and_bounded_of_feasible (Rx : p ≽' q) {x : D} (h_feas_x : p.feasible x) :
q.feasible (Rx.phi x) ∧ q.objFun (Rx.phi x) ≤ p.objFun x | ⟨Rx.phi_feasibility x h_feas_x, Rx.phi_optimality x h_feas_x⟩ | lemma | Minimization.Relaxation.feasible_and_bounded_of_feasible | Lib | CvxLean/Lib/Relaxation.lean | [
"CvxLean.Lib.Equivalence"
] | [] | First property in [RelaxationWiki]. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
induces_original_problem_optimality (Rx : p ≽' q) (phi_inv : E → D)
(phi_left_inv : Function.LeftInverse Rx.phi phi_inv)
(h_objFun : ∀ x, p.feasible x → p.objFun x = q.objFun (Rx.phi x)) {y : E}
(h_opt_y : q.optimal y) (h_feas_y : p.feasible (phi_inv y)) : p.optimal (phi_inv y) | by
refine ⟨h_feas_y, ?_⟩
intros x h_feas_x
rw [h_objFun _ h_feas_y, phi_left_inv]
have h_bound := h_opt_y.2 (Rx.phi x) (Rx.phi_feasibility x h_feas_x)
rwa [← h_objFun _ h_feas_x] at h_bound | lemma | Minimization.Relaxation.induces_original_problem_optimality | Lib | CvxLean/Lib/Relaxation.lean | [
"CvxLean.Lib.Equivalence"
] | [] | Second property in [RelaxationWiki]. NOTE: This does not use `Rx.phi_optimality`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
ofStrongEquivalence (E : p ≡' q) : p ≽' q | { phi := E.phi,
phi_feasibility := E.phi_feasibility,
phi_optimality := E.phi_optimality } | def | Minimization.Relaxation.ofStrongEquivalence | Lib | CvxLean/Lib/Relaxation.lean | [
"CvxLean.Lib.Equivalence"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
ofRelaxations (Rx₁ : p ≽' q) (Rx₂ : q ≽' p) : p ≡' q | { phi := Rx₁.phi,
psi := Rx₂.phi,
phi_feasibility := Rx₁.phi_feasibility,
psi_feasibility := Rx₂.phi_feasibility,
phi_optimality := Rx₁.phi_optimality,
psi_optimality := Rx₂.phi_optimality } | def | Minimization.StrongEquivalence.ofRelaxations | Lib | CvxLean/Lib/Relaxation.lean | [
"CvxLean.Lib.Equivalence"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
ofRelaxations (Rx₁ : p ≽' q) (Rx₂ : q ≽' p) : p ≡ q | Equivalence.ofStrongEquivalence (StrongEquivalence.ofRelaxations Rx₁ Rx₂) | def | Minimization.Equivalence.ofRelaxations | Lib | CvxLean/Lib/Relaxation.lean | [
"CvxLean.Lib.Equivalence"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
remove_constraint {c cs' : D → Prop} (hcs : ∀ x, cs x ↔ c x ∧ cs' x) : ⟨f, cs⟩ ≽' ⟨f, cs'⟩ | { phi := id,
phi_feasibility := fun x h_feas_x => ((hcs x).mp h_feas_x).2,
phi_optimality := fun _ _ => le_refl _ } | def | Minimization.Relaxation.remove_constraint | Lib | CvxLean/Lib/Relaxation.lean | [
"CvxLean.Lib.Equivalence"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
weaken_constraints (cs' : D → Prop) (hcs : ∀ x, cs x → cs' x) : ⟨f, cs⟩ ≽' ⟨f, cs'⟩ | { phi := id,
phi_feasibility := fun x h_feas_x => hcs x h_feas_x,
phi_optimality := fun _ _ => le_refl _ } | def | Minimization.Relaxation.weaken_constraints | Lib | CvxLean/Lib/Relaxation.lean | [
"CvxLean.Lib.Equivalence"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
expCone (x y z : ℝ) : Prop | (0 < y ∧ y * exp (x / y) ≤ z) ∨ (y = 0 ∧ 0 ≤ z ∧ x ≤ 0) | def | Real.expCone | Lib.Cones | CvxLean/Lib/Cones/ExpCone.lean | [
"Mathlib.Data.Complex.Exponential"
] | [] | The exponential cone:
`𝒦ₑ := S₁ ∪ S₂ ⊆ ℝ³`, where
`S₁ := { (x, y, z) | 0 < y ∧ exp(x / y) ≤ z }`, and
`S₂ := { (x, 0, z) | 0 ≤ z ∧ x ≤ 0 }`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
Vec.expCone {n} [Fintype n] (x y z : n → ℝ) : Prop | ∀ i, Real.expCone (x i) (y i) (z i) | def | Real.Vec.expCone | Lib.Cones | CvxLean/Lib/Cones/ExpCone.lean | [
"Mathlib.Data.Complex.Exponential"
] | [
"Real.expCone"
] | The `n`-dimensional exponential cone `𝒦ₑⁿ`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
exp_iff_expCone (t x : ℝ) : exp x ≤ t ↔ expCone x 1 t | by
unfold expCone
rw [iff_def]
split_ands
{ intro hexp
apply Or.intro_left
split_ands
{ apply Real.zero_lt_one }
{ rwa [div_one, one_mul] } }
{ intro h
cases h with
| inl h =>
have h : 1 * exp (x / 1) ≤ t := h.2
rwa [div_one, one_mul] at h
| inr h =>
exfalso... | theorem | Real.exp_iff_expCone | Lib.Cones | CvxLean/Lib/Cones/ExpCone.lean | [
"Mathlib.Data.Complex.Exponential"
] | [] | We have `exp(x) ≤ t ↔ (x, 1, t) ∈ 𝒦ₑ`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
nonnegOrthCone (x : ℝ) : Prop | 0 ≤ x | def | Real.nonnegOrthCone | Lib.Cones | CvxLean/Lib/Cones/NonnegOrthCone.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Matrix"
] | [] | The nonnegative orthant
`ℝ₊ := { x | 0 ≤ x } ⊆ ℝ`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
Vec.nonnegOrthCone {n} [Fintype n] (x : n → ℝ) : Prop | ∀ i, Real.nonnegOrthCone (x i) | def | Real.Vec.nonnegOrthCone | Lib.Cones | CvxLean/Lib/Cones/NonnegOrthCone.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Matrix"
] | [
"Real.nonnegOrthCone"
] | The `n`-dimensional nonnegative orthant `ℝ₊ⁿ`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
Matrix.nonnegOrthCone {n m} [Fintype n] [Fintype m] (M : Matrix n m ℝ) : Prop | ∀ i j, Real.nonnegOrthCone (M i j) | def | Real.Matrix.nonnegOrthCone | Lib.Cones | CvxLean/Lib/Cones/NonnegOrthCone.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Matrix"
] | [
"Real.nonnegOrthCone"
] | The `n×m`-dimensional nonnegative orthant `ℝ₊ⁿˣᵐ`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
Matrix.PSDCone {n} [Fintype n] (A : Matrix n n ℝ) : Prop | Matrix.PosSemidef A | def | Real.Matrix.PSDCone | Lib.Cones | CvxLean/Lib/Cones/PSDCone.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.LinearAlgebra.Matrix.PosDef"
] | [] | The cone of `n×n` positive semidefinite matrices
`𝒮₊ⁿ := { A | A is symmetric ∧ 0 ≼ A } ⊆ ℝⁿˣⁿ`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
soCone (t : ℝ) (x : n → ℝ) : Prop | sqrt (∑ i, x i ^ 2) ≤ t | def | Real.soCone | Lib.Cones | CvxLean/Lib/Cones/SOCone.lean | [
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [] | The `n`-dimensional second-order cone
`𝒬ⁿ⁺¹ := { (t, x) | ‖x‖₂ = sqrt(x₁² + ⋯ + xₙ²) ≤ t } ⊆ ℝ × ℝⁿ`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
rotatedSoCone (v w : ℝ) (x : n → ℝ) : Prop | (∑ i, x i ^ 2) ≤ (v * w) * 2 ∧ 0 ≤ v ∧ 0 ≤ w | def | Real.rotatedSoCone | Lib.Cones | CvxLean/Lib/Cones/SOCone.lean | [
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [] | The `n`-dimensional rotated second-order cone
`𝒬ᵣⁿ⁺² := { (v, w, x) | x₁² + ⋯ + xₙ² ≤ 2vw ∧ 0 ≤ v, w } ⊆ ℝ × ℝ × ℝⁿ`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
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