Proper cones

We define a proper cone as a closed, pointed cone. Proper cones are used in defining conic programs which generalize linear programs. A linear program is a conic program for the positive cone. We then prove Farkas' lemma for conic programs following the proof in the reference below. Farkas' lemma is equivalent to strong duality. So, once we have the definitions of conic and linear programs, the results from this file can be used to prove duality theorems.

TODO

The next steps are:

• Add convex_cone_class that extends set_like and replace the below instance
• Define primal and dual cone programs and prove weak duality.
• Prove regular and strong duality for cone programs using Farkas' lemma (see reference).
• Define linear programs and prove LP duality as a special case of cone duality.
• Find a better reference (textbook instead of lecture notes).

References

• [B. Gartner and J. Matousek, Cone Programming][gartnerMatousek]

structure ProperCone (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] extends Submodule : Type u_2

A proper cone is a pointed cone K that is closed. Proper cones have the nice property that they are equal to their double dual, see ProperCone.dual_dual. This makes them useful for defining cone programs and proving duality theorems.

• carrier : Set E
• add_mem' : ∀ {a b : E}, a ∈ self.carrier → b ∈ self.carrier → a + b ∈ self.carrier
• zero_mem' : 0 ∈ self.carrier
• smul_mem' : ∀ (c : { c : 𝕜 // 0 ≤ c }) {x : E}, x ∈ self.carrier → c • x ∈ self.carrier
• isClosed' : IsClosed self.carrier

theorem ProperCone.isClosed' {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (self : ProperCone 𝕜 E) : IsClosed self.carrier

@[reducible, inline]

abbrev ProperCone.toPointedCone {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (C : ProperCone 𝕜 E) : Submodule { c : 𝕜 // 0 ≤ c } E

A PointedCone is defined as an alias of submodule. We replicate the abbreviation here and define toPointedCone as an alias of toSubmodule.

Equations

• ↑C = C.toSubmodule

instance ProperCone.instCoePointedCone {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] : Coe (ProperCone 𝕜 E) (PointedCone 𝕜 E)

Equations

• ProperCone.instCoePointedCone = { coe := ProperCone.toPointedCone }

theorem ProperCone.toPointedCone_injective {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] : Function.Injective ProperCone.toPointedCone

instance ProperCone.instSetLike {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] : SetLike (ProperCone 𝕜 E) E

Equations

• ProperCone.instSetLike = { coe := fun (K : ProperCone 𝕜 E) => K.carrier, coe_injective' := ⋯ }

theorem ProperCone.ext_iff {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {S : ProperCone 𝕜 E} {T : ProperCone 𝕜 E} : S = T ↔ ∀ (x : E), x ∈ S ↔ x ∈ T

theorem ProperCone.ext {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {S : ProperCone 𝕜 E} {T : ProperCone 𝕜 E} (h : ∀ (x : E), x ∈ S ↔ x ∈ T) : S = T

@[simp]

theorem ProperCone.mem_coe {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {x : E} {K : ProperCone 𝕜 E} : x ∈ ↑K ↔ x ∈ K

instance ProperCone.instZero {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (K : ProperCone 𝕜 E) : Zero ↥K

Equations

• K.instZero = PointedCone.instZero K.toSubmodule

theorem ProperCone.nonempty {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (K : ProperCone 𝕜 E) : (↑K).Nonempty

theorem ProperCone.isClosed {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (K : ProperCone 𝕜 E) : IsClosed ↑K

def ProperCone.positive (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E] [TopologicalSpace E] [OrderClosedTopology E] : ProperCone 𝕜 E

The positive cone is the proper cone formed by the set of nonnegative elements in an ordered module.

Equations

• ProperCone.positive 𝕜 E = { toSubmodule := PointedCone.positive 𝕜 E, isClosed' := ⋯ }

@[simp]

theorem ProperCone.mem_positive (𝕜 : Type u_2) (E : Type u_1) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E] [TopologicalSpace E] [OrderClosedTopology E] {x : E} : x ∈ ProperCone.positive 𝕜 E ↔ 0 ≤ x

@[simp]

theorem ProperCone.coe_positive (𝕜 : Type u_2) (E : Type u_1) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E] [TopologicalSpace E] [OrderClosedTopology E] : ↑↑(ProperCone.positive 𝕜 E) = ConvexCone.positive 𝕜 E

instance ProperCone.instZero_1 {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [T1Space E] [Module 𝕜 E] : Zero (ProperCone 𝕜 E)

Equations

• ProperCone.instZero_1 = { zero := { toSubmodule := 0, isClosed' := ⋯ } }

instance ProperCone.instInhabited {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [T1Space E] [Module 𝕜 E] : Inhabited (ProperCone 𝕜 E)

Equations

• ProperCone.instInhabited = { default := 0 }

@[simp]

theorem ProperCone.mem_zero {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [T1Space E] [Module 𝕜 E] (x : E) : x ∈ 0 ↔ x = 0

@[simp]

theorem ProperCone.coe_zero {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [T1Space E] [Module 𝕜 E] : ↑↑0 = 0

theorem ProperCone.pointed_zero {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [T1Space E] [Module 𝕜 E] : (↑↑0).Pointed

theorem ProperCone.pointed {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (K : ProperCone ℝ E) : (↑↑K).Pointed

noncomputable def ProperCone.map {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (f : E →L[ℝ] F) (K : ProperCone ℝ E) : ProperCone ℝ F

The closure of image of a proper cone under a continuous ℝ-linear map is a proper cone. We use continuous maps here so that the comap of f is also a map between proper cones.

Equations

• ProperCone.map f K = { toSubmodule := (PointedCone.map ↑f ↑K).closure, isClosed' := ⋯ }

@[simp]

theorem ProperCone.coe_map {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (f : E →L[ℝ] F) (K : ProperCone ℝ E) : ↑(ProperCone.map f K) = (PointedCone.map ↑f ↑K).closure

@[simp]

theorem ProperCone.mem_map {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {f : E →L[ℝ] F} {K : ProperCone ℝ E} {y : F} : y ∈ ProperCone.map f K ↔ y ∈ (PointedCone.map ↑f ↑K).closure

@[simp]

theorem ProperCone.map_id {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (K : ProperCone ℝ E) : ProperCone.map (ContinuousLinearMap.id ℝ E) K = K

def ProperCone.dual {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (K : ProperCone ℝ E) : ProperCone ℝ E

The inner dual cone of a proper cone is a proper cone.

Equations

• K.dual = { toSubmodule := PointedCone.dual ↑K, isClosed' := ⋯ }

@[simp]

theorem ProperCone.coe_dual {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (K : ProperCone ℝ E) : ↑↑K.dual = (↑K).innerDualCone

@[simp]

theorem ProperCone.mem_dual {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {K : ProperCone ℝ E} {y : E} : y ∈ K.dual ↔ ∀ ⦃x : E⦄, x ∈ K → 0 ≤ inner x y

noncomputable def ProperCone.comap {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (f : E →L[ℝ] F) (S : ProperCone ℝ F) : ProperCone ℝ E

The preimage of a proper cone under a continuous ℝ-linear map is a proper cone.

Equations

• ProperCone.comap f S = { toSubmodule := PointedCone.comap ↑f ↑S, isClosed' := ⋯ }

@[simp]

theorem ProperCone.coe_comap {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (f : E →L[ℝ] F) (S : ProperCone ℝ F) : ↑(ProperCone.comap f S) = ⇑f ⁻¹' ↑S

@[simp]

theorem ProperCone.comap_id {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (S : ConvexCone ℝ E) : ConvexCone.comap LinearMap.id S = S

theorem ProperCone.comap_comap {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {G : Type u_3} [NormedAddCommGroup G] [InnerProductSpace ℝ G] (g : F →L[ℝ] G) (f : E →L[ℝ] F) (S : ProperCone ℝ G) : ProperCone.comap f (ProperCone.comap g S) = ProperCone.comap (g.comp f) S

@[simp]

theorem ProperCone.mem_comap {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {f : E →L[ℝ] F} {S : ProperCone ℝ F} {x : E} : x ∈ ProperCone.comap f S ↔ f x ∈ S

@[simp]

theorem ProperCone.dual_dual {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] (K : ProperCone ℝ E) : K.dual.dual = K

The dual of the dual of a proper cone is itself.

theorem ProperCone.hyperplane_separation {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [CompleteSpace F] (K : ProperCone ℝ E) {f : E →L[ℝ] F} {b : F} : b ∈ ProperCone.map f K ↔ ∀ (y : F), (ContinuousLinearMap.adjoint f) y ∈ K.dual → 0 ≤ inner y b

This is a relative version of ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem, which we recover by setting f to be the identity map. This is also a geometric interpretation of the Farkas' lemma stated using proper cones.

theorem ProperCone.hyperplane_separation_of_nmem {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [CompleteSpace F] (K : ProperCone ℝ E) {f : E →L[ℝ] F} {b : F} (disj : b ∉ ProperCone.map f K) : ∃ (y : F), (ContinuousLinearMap.adjoint f) y ∈ K.dual ∧ inner y b < 0