Convex hull of graphs of polynomial functions over a polytope

5.3.1 Preliminary definitions

Let𝑚, 𝑛∈Nand𝑋 ⊂R^𝑛be a polytope defined by the intersection of finitely many halfspaces, i.e.,𝑋 := {x = (𝑥₁, . . . , 𝑥_𝑛)^𝑇 ∈ R^𝑛 | 𝑎^𝑇_𝑗x ≤ 𝑏_𝑗, 𝑗 = 1, . . . , 𝑚}, where𝑎_𝑗 ∈ R^𝑛, 𝑏_𝑗 ∈ R for𝑗 = 1, . . . , 𝑚. Then, for a given polynomial function𝑓 : 𝑋 → Rwith 𝑓 ∈ R[x], the image of the function𝐹 :𝑋→R^𝑛+1,x↦→(x, 𝑓(x))defines the graph of𝑓. Note that some of definitions and properties are also suitable for general differentiable functions. But in this chapter we focus on polynomial functions.

In this section, we study the convex hull of the set

𝒮:={(x, 𝑧)|𝑧=𝑓(x), x∈𝑋} ⊂R^𝑛+1.

Define projection functions𝜋x:R^𝑛×R→R^𝑛,(x, 𝑧)↦→xand𝜋𝑧 :R^𝑛×R→R,(x, 𝑧)↦→𝑧. For every point(x, 𝑓(x))∈ 𝒮,x=𝜋x((x, 𝑓(x)))is the corresponding domain point.

Theorem 5.1

Let𝑆⊂R^𝑛+1be a compact set. Then the convex hull of𝑆is a compact set which is the intersection of all closed halfspaces containing𝑆.

Proof. This can be directly derived from Corollary 5.33 and 5.83 of the book [AB06]. 2 Without loss of generality, we assume𝑋be to full-dimensional, otherwise we can reduce the dimension by eliminating variables until the new equivalent𝑋is full-dimensional inR^𝑑for 𝑑≤𝑛. Furthermore, with suitable preprocessing, every hyperplane𝑎^𝑇_𝑗x=𝑏_𝑗 in𝑋corresponds to a facet of𝑋.

Let𝜕𝑋 :={x∈ 𝑋 | ∃𝑗, 𝑎^𝑇_𝑗x= 𝑏_𝑗}denote the boundary of𝑋and letint𝑋 := 𝑋∖𝜕𝑋 denote the interior of𝑋. For𝑟∈R+, denote the closed ball of radius𝑟centered at the pointx⁰ by𝐵𝑟(x⁰) ={x∈R^𝑛| ‖x−x⁰‖₂ ≤𝑟}. A hyperplane inR^𝑛+1through point(x⁰, 𝑧⁰)with 𝑧⁰=𝑓(x⁰)and normal vectorn⁰∈R^𝑛,n⁰̸= 0can be defined as

𝐻(x⁰,n⁰) ={(x, 𝑧)|𝑧=𝑧⁰+n⁰·(x−x⁰)}.

The tangent plane to𝑓 at (x⁰, 𝑓(x⁰))is𝑇(x⁰) := 𝐻(x⁰,∇𝑓(x⁰)). The downward closed halfspace associated with hyperplane𝐻(x⁰,n⁰)is then

𝐻(xˇ ⁰,n⁰) ={(x, 𝑧)|𝑧≥𝑧⁰+n⁰·(x−x⁰)}.

The upward closed halfspace𝐻(x^ ⁰,n⁰)is defined similarly.

Hyperplanes𝐻(x⁰,n⁰) are callednonvertical (to space R^𝑛), since𝜋_x(𝐻(x⁰,n⁰)) = R^𝑛. Analogously, averticalhyperplane inR^𝑛+1through point(x⁰, 𝑧⁰)with𝑧⁰ =𝑓(x⁰)and normal vectorn⁰ ∈R^𝑛can be defined as

𝐻^⊥(x⁰,n⁰) ={(x, 𝑧)|n⁰·(x−x⁰) = 0}.

Then the left closed halfspace associated to𝐻^⊥(x⁰,n⁰)is defined by 𝐻´^⊥(x⁰,n⁰) ={(x, 𝑧)|n⁰·(x−x⁰)≤0}

and the right closed halfspace𝐻`^⊥(x⁰,n⁰)is defined similarly. For any halfspace𝐻inR^𝑛+1, let𝑔^𝐻 denote a map which maps a halfspace to its corresponding hyperplane.

Lemma 5.2

We discuss first halfspaces in𝑆^′with corresponding hyperplanes which are nonvertical. For every𝐻ˇ¹ there exists an affine function𝑔1 :R^𝑛 →Rsuch that𝐻ˇ¹ ={(x, 𝑧) |𝑧≥ 𝑔1(x)}.

This is also true for every upward closed halfspaces.

Now we discuss halfspaces in𝑆^′with corresponding hyperplanes which are vertical. Note that for every𝑗 ∈ {1, . . . , 𝑚},{(x, 𝑧)|𝑎^𝑇_𝑗x≤𝑏𝑗}has the form of𝐻´^⊥with𝐻´^⊥⊃ 𝒮. For any

5.3 Convex hull of graphs of polynomial functions over a polytope

Together with the result for all halfspaces in𝑆^′ with corresponding hyperplanes which are

nonvertical, it implies that𝑆^′=𝑆^′′. 2

In this section, we want to find a set of halfspaces that all contain𝒮and the intersection of them is the convex hull of𝒮. Lemma 5.2 shows that we can restrict attention to nonvertical halfspaces whose corresponding hyperplane intersects the graph.

Remark 5.3

Obviously,𝒮 is bounded because of Theorem 5.1. Due to symmetry, we only need to consider the downward closed part of the convex hull since the upward closed part is equivalent to the downward closed part of function−𝑓.

5.3.2 Locally and globally convex points Definition 5.4 (Locally convex points)

A point(x⁰, 𝑓(x⁰))∈ 𝒮 is alocally convex pointif there exists𝜀 >0,n⁰ ∈R^𝑛such that {(x, 𝑧)|x∈𝑋∩𝐵_𝜀(x⁰), 𝑧=𝑓(x)}

⏟ ⏞

local graph

⊂𝐻(xˇ ⁰,n⁰). (5.1) In this case, the domain pointx⁰is the corresponding locally convex domain point.

Let𝒮ˇ^𝑙 ∈ 𝒮 denote the set of all locally convex points and𝑋ˇ^𝑙=𝜋_x( ˇ𝒮^𝑙)the corresponding domain points.

Lemma 5.5

Letx⁰ ∈int𝑋. Ifx⁰is a locally convex domain point, then the gradient vector∇𝑓(x⁰)is the unique normal vectorn⁰tox⁰fulfilling (5.1).

Proof. Consider the function𝑔_n⁰(x) = 𝑓(x)−(n⁰ ·(x−x⁰) +𝑓(x⁰))forn⁰ ∈ R^𝑛and defined on𝑋. The point(x⁰, 𝑓(x⁰))is locally convex if and only if there exists𝜀 >0such that 𝑔_n0(x)≥0over𝑋∩𝐵_𝜀(x⁰). Using “Taylor’s Formula in Several Variables” for𝑔_n0(x)atx⁰ (see the book [Edw94]),𝑔_n⁰(x)may attain a local minimum0atx⁰if and only ifn⁰ =∇𝑓(x⁰).

This implies the result. 2

Definition 5.6 (Globally convex points and valid halfspaces)

A point(x⁰, 𝑓(x⁰))∈ 𝒮 is aglobally convex pointif there exist ann⁰ ∈R^𝑛such that 𝒮 ={(x, 𝑧)|x∈𝑋, 𝑧=𝑓(x)}

⏟ ⏞

total graph

⊂𝐻(xˇ ⁰,n⁰). (5.2) Furthermore, we call𝐻(xˇ ⁰,n⁰)avalidhalfspace,𝐻(x⁰,n⁰)andn⁰the corresponding valid hyperplane and valid normal vector, respectively. The pointx⁰is the corresponding globally convex domain point.

Further we define𝒮ˇ^𝑔 as the set of all globally convex points and𝑋ˇ^𝑔 =𝜋x( ˇ𝒮^𝑔)the set of corresponding domain points.

Remark 5.7

The definition of globally convex points is actually equivalent to the definition of generating sets, which can be found in Chapter4of the book [LS13]. Due to the different point of view, we keep on using the name defined above in this thesis.

Theorem 5.8

Letx⁰∈int𝑋. Thenx⁰is a globally convex domain point if and only if the tangent hyperplane 𝑇(x⁰)is valid.

Proof. It is clear that a globally convex domain pointx⁰must also be a locally convex domain point. The result is then followed from Lemma 5.5, sincen⁰ is the unique candidate normal

vector to satisfy (5.2). 2

For anyx⁰ ∈ 𝑋 we seek a way to determine if it is a globally convex domain point. For x⁰ ∈int𝑋, we need only to check if𝑇(x⁰)is valid due to Theorem 5.8. On the other hand, for a boundary domain pointx⁰ ∈ 𝜕𝑋, the situation is more complex. We require several additional concepts in this case.

5.3.3 Globally convex boundary points Remark 5.9

For polytope𝑋, every face𝐹 possesses a maximal subset𝐼𝐹 ⊂ {1, . . . , 𝑚}such that 𝐹 =^{︁x∈𝑋 |𝑎^𝑇_𝑗x=𝑏_𝑗, 𝑗∈𝐼_𝐹^}︁.

In this section, we assume that the vertices of𝑋are nondegenerate which means there is no pointx∈R^𝑛which satisfies𝑛+ 1of the given𝑚inequalities with equality. This implies two properties:

• For any face𝐹 it holds thatdim𝐹 =𝑛− |𝐼_𝐹|

• For any face𝐹 with|𝐼_𝐹|= 𝑑 ≥ 2and𝐼𝐹 = {1, . . . , 𝑑}without loss of generality, it implies that

𝐹−𝑖 =^{︁x∈𝑋|𝑎^𝑇_𝑗x=𝑏𝑗, 𝑗 ∈𝐼𝐹 ∖ {𝑖}^}︁

is also a face of𝑋with𝐹 (𝐹−𝑖anddim𝐹 = dim𝐹−𝑖−1for any𝑖∈𝐼_𝐹.

Note that the assumption that𝑋is vertex-nondegenerate is not a necessary condition for this section. However, this assumption makes the notation and description easier. For a vertex-degenerate𝑋, the dimension of any face can be determined by calculating the rank of an auxiliary matrix. Since there are finitely many faces, for any given face𝐹 withdim𝐹 ≤𝑛−2, all faces that contain𝐹 of dimension(dim𝐹+ 1)can be determined as well.

Based on this remark, we have the following lemma. Example 5.11 following after its proof illustrates the lemma in a graphical way.

5.3 Convex hull of graphs of polynomial functions over a polytope

Lemma 5.10 (Complete Projection based on the smallest face containingx⁰) Letx⁰ ∈𝜕𝑋and define the index set

𝐼_x0 :=^{︁𝑗∈ {1, . . . , 𝑚} |𝑎^𝑇_𝑗x⁰ =𝑏_𝑗^}︁

which is nonempty. Let𝑑^′ =|𝐼_x0|be the cardinality with1≤𝑑^′ ≤𝑛. Then, the set 𝑃_x⁰ :={x∈R^𝑛|𝑎^𝑇_𝑗x=𝑏_𝑗for all𝑗 ∈𝐼_x⁰}

defines an affine set inR^𝑛 of dimension𝑑 := 𝑛−𝑑^′. Further, the set𝐹_x⁰ := 𝑃_x⁰ ∩𝑋is the smallest face of𝑋which containsx⁰and has dimension𝑑.

By permuting variables inxif necessary, there exists a bijective linear map 𝑔_𝑑:𝑃_x⁰ →R^𝑑,x↦→x_𝑑

wherex_𝑑= (𝑥1, . . . , 𝑥𝑑)^𝑇 such that

• The set

𝒮_𝑑:={(x_𝑑, 𝑧)|x_𝑑=𝑔_𝑑(x), 𝑧=𝑓(x),x∈𝐹_x⁰} ⊂R^𝑑+1 is the graph of some polynomial function over a polytope.

• Either(𝑔𝑑(x⁰), 𝑓(x⁰))is an interior point of𝒮_𝑑or𝒮_𝑑={(𝑔_𝑑(x⁰), 𝑓(x⁰))}.

Proof. Without loss of generality, we assume𝐼_x⁰ ={1, . . . , 𝑑^′}. It is clear that𝐹_x⁰ is a face of𝑋withx⁰ ∈𝐹_x⁰. Assume that there exists another face𝐹 of𝑋withx⁰ ∈𝐹 (𝐹_x⁰. Then there exists𝑗 ∈ {𝑑^′+ 1, . . . , 𝑚}such that𝑎^𝑇_𝑗x⁰ =𝑏_𝑗. This is a contradiction to the definition of𝐼_x⁰. Hence𝐹_x⁰ is the smallest face of𝑋containingx⁰. Similar to the preprocessing of half spaces which form𝑋, which shows that𝐹_x⁰ has dimension𝑑.

Furthermore, we define

𝐴_x⁰ =

⎛

⎜

⎝ 𝑎^𝑇₁

...

𝑎^𝑇_𝑑′

⎞

⎟

⎠∈R^𝑑

′×𝑛,

and𝑏_x⁰ = (𝑏₁, 𝑏₂, . . . , 𝑏_𝑑^′)^𝑇. Then we have

𝑃_x⁰ ={x∈R^𝑛|𝐴_x⁰x=𝑏_x⁰}

and𝑑^′= rank(𝐴_x⁰) =𝑛−𝑑. Via permuting variables, we may assume that the last𝑑^′columns of𝐴_x⁰ are linearly independent and form an invertible matrix𝐴_𝐵. At the same time the first𝑑 columns of𝐴_x0 form the matrix𝐴_𝑑. By settingx_𝐵= (𝑥_𝑑+1, . . . , 𝑥_𝑛)^𝑇, we have

𝐴_x0x= (𝐴_𝑑𝐴_𝐵)(x^𝑇_𝑑 x^𝑇_𝐵)^𝑇 =𝐴_𝑑x_𝑑+𝐴_𝐵x_𝐵 =𝑏_x0,

and so

x_𝐵 =𝐴⁻¹_𝐵 (𝑏_x⁰−𝐴𝑑x_𝑑) =𝐴⁻¹_𝐵 𝑏_x⁰ −𝐴⁻¹_𝐵 𝐴𝑑x_𝑑=:𝑏^′−𝐴^′x_𝑑 for allx∈𝑃_x⁰ with

𝑥_𝑑+𝑖 =𝑏^′_𝑖−𝐴^′_𝑖x_𝑑=𝑏^′_𝑖+

𝑑

∑︁

𝑗=1

𝑐_{𝑑+𝑖,𝑗}𝑥_𝑗 (5.3)

for all𝑖= 1, . . . , 𝑛−𝑑where𝑏^′ =𝐴⁻¹_𝐵 𝑏_x⁰,𝐴^′ =𝐴⁻¹_𝐵 𝐴_𝑑and𝐴^′_𝑖 is the𝑖-th row of𝐴^′,𝑐_{𝑑+𝑗,𝑗} are constants for𝑖= 1, . . . , 𝑛−𝑑and𝑗 = 1, . . . , 𝑑. Note that the𝑑-dimensional affine set𝑃_x0

andR^𝑑are isomorphic. Consider the linear map

𝑔𝑑:𝑃_x⁰ →R^𝑑,x↦→x_𝑑, which is bijective since the inverse exists:

𝑔_𝑑⁻¹:R^𝑑→𝑃_x⁰,x_𝑑↦→

(︃ x_𝑑 𝑏^′−𝐴^′x_𝑑

)︃

.

In the following we investigate the graph of function𝑓 under the restrictionx∈𝐹_x0. The set𝐹_x⁰, function𝑓 and pointx⁰can becompletely projected to𝐹𝑑 =𝑔𝑑(𝐹_x⁰),x⁰_𝑑 = 𝑔𝑑(x⁰) and𝑓_𝑑 = 𝑓(x_𝑑, 𝑏^′₁ −𝐴^′₁x_𝑑, . . . , 𝑏^′_𝑥−𝑑 −𝐴^′_𝑥−𝑑x_𝑑) respectively. It is clear that 𝑓_𝑑 ∈ R[x_𝑑] is a polynomial function and𝐹_𝑑 is a polytope of dimension𝑑. By comparing coefficients, for everyx ∈ 𝐹_x⁰ it implies that𝑓(x) = 𝑓𝑑(𝑔𝑑(x))and for everyx_𝑑 ∈ 𝐹𝑑it implies that 𝑓_𝑑(x_𝑑) =𝑓(𝑔⁻¹_𝑑 (x_𝑑)). The graph of𝑓_𝑑is thus

{(x_𝑑, 𝑧)|𝑧=𝑓_𝑑(x_𝑑), x_𝑑∈𝐹_𝑑}={(x_𝑑, 𝑧)|x_𝑑=𝑔_𝑑(x), 𝑧=𝑓(x),x∈𝐹_x⁰}=𝒮_𝑑, which is the graph of the polynomial function𝑓_𝑑over polytope𝐹_𝑑.

The domain pointx⁰ is an extreme point of𝑋 if𝑑^′ = |𝐼_x0|= 𝑛. In this case𝑑 = 0and 𝐹_𝑑 ={x⁰_𝑑}which contains a single point. It implies thatx⁰_𝑑is also an extreme point of 𝐹_𝑑. Otherwise, we are in case of𝑑^′ < 𝑛. Note that every𝐹𝑑is isomorphic to a face of𝑋. Assume x⁰_𝑑is not an interior point of𝐹_𝑑. Then there exists a face of𝐹_𝑑which containsx⁰_𝑑. It implies there exists a face of𝑋which containsx⁰. This is a contradiction to the definition of𝐼_x0. Thus

x⁰_𝑑is an interior point in𝐹𝑑. 2

Example 5.11

We look at an example with𝑛= 2, i.e.,x= (𝑥, 𝑦). Consider the polynomial function 𝑓(𝑥, 𝑦) =−𝑥⁴+ 3𝑥³+ 10𝑥²𝑦−5𝑥²+ 3𝑥−2𝑦⁴−4𝑦² : [−4,4]×[−4,4]→R. For boundary pointx⁰ = (1,4)we get𝐹_x⁰ ={(𝑥, 𝑦) | −4≤𝑥≤4, 𝑦 = 4},𝑔_𝑑 : (𝑥, 𝑦)↦→𝑥 and𝑔_𝑑⁻¹:𝑥↦→(𝑥,4)and𝑓_𝑑(𝑥) =−𝑥⁴+ 3𝑥³+ 35𝑥²+ 3𝑥−576as defined in Lemma 5.10.

Point(x⁰, 𝑓(x⁰)), set𝒮and a valid hyperplane{(𝑥, 𝑦, 𝑧)|𝑧= 78𝑥+134𝑦−1150}are shown as the red point, the blue surface and the yellow plane in Figure 5.4a and the corresponding (x⁰_𝑑, 𝑓_𝑑(x⁰_𝑑))and𝒮_𝑑are shown as the red point and the blue curve in Figure 5.4b, respectively.♢

5.3 Convex hull of graphs of polynomial functions over a polytope

(a) In spaceR^𝑛+1 (b) In spaceR^𝑑+1

Figure 5.4:Graph and hyperplane restricted on a face in Example 5.11 and the complete projections

Theorem 5.12 (Globally convex boundary points )

Letx⁰ ∈𝜕𝑋. Thenx⁰is a globally convex domain point of𝒮 if and only ifx⁰_𝑑=𝑔_𝑑(x⁰)is also a globally convex domain point of𝒮_𝑑with

𝒮_𝑑={(x_𝑑, 𝑧)|x_𝑑=𝑔_𝑑(x), 𝑧=𝑓(x),x∈𝐹_x0} ⊂R^𝑑+1.

We go back to Example 5.11. Pointx_𝑑⁰= 1is a globally convex domain point of𝒮_𝑑since the tangent plane{(𝑥, 𝑧)|𝑧=𝑓_𝑑(𝑥) = 78𝑥−614}(shown as green line in Figure 5.4b) can be verified to be valid. Using the theorem above it implies that(1,4)is also globally convex. This is verified by the valid hyperplane{(𝑥, 𝑦, 𝑧)|𝑧= 78𝑥+ 134𝑦−1150}.

We prove first the direction “⇒” and postpone the direction “⇐” after introducing a few further definitions and auxiliary lemmas.

Proof (of “⇒”). Consider the case thatx⁰_𝑑andx⁰are both an extreme point in 𝐹_𝑑and𝑋, respectively. Then the single pointx⁰_𝑑∈𝐹_𝑑is surely a globally convex domain point since the single point is a valid hyperplane in that space.

Otherwisex⁰_𝑑is an interior point in𝐹_𝑑. Letv= (𝑣₁, . . . , 𝑣_𝑛)^𝑇 ∈R^𝑛be an arbitrary vector.

Define a function𝑔_𝑐:R^𝑛→R^𝑑,

where all𝑐𝑖𝑗 are defined as in (5.3). We first give and prove the following claim.

Claim Letn∈R^𝑛. The following two sets are affinely isomorphic

{(x, 𝑧)|𝑧=𝑧⁰+n·(x−x⁰),x∈𝑃_x0}∼={(x𝑑, 𝑧)|𝑧=𝑧⁰+𝑔_𝑐(n)·(x_𝑑−x⁰_𝑑),x𝑑∈R^𝑑}. (5.4)

This implies

{(x, 𝑧)|𝑧≥𝑧⁰+n·(x−x⁰),x∈𝑃x⁰}∼={(x𝑑, 𝑧)|𝑧≥𝑧⁰+𝑔𝑐(n)·(x𝑑−x⁰_𝑑),x𝑑∈R^𝑑}. (5.5)

Moreover, the tangent plane onx_𝑑⁰defined as

𝑇_𝑑(x⁰_𝑑) ={(x_𝑑, 𝑧)|𝑧=∇𝑓_𝑑(x⁰_𝑑)·(x_𝑑−x⁰_𝑑) +𝑧⁰,x_𝑑∈R^𝑑}

satisfies

𝑇_𝑑(x_𝑑⁰)∼=𝑇_|𝑃

x0(x⁰) ={(x, 𝑧)∈𝑇(x⁰)|x∈𝑃_x0}. (5.6)

5.3 Convex hull of graphs of polynomial functions over a polytope

To prove (5.6), it suffices to show

𝑔𝑐(∇𝑓(x⁰)) =∇𝑓_𝑑(x⁰_𝑑),

Equation (5.4) says that for a given hyperplane𝐻(x⁰,n), the graph with restrictionx∈𝑃_x⁰ inR^𝑛+1which is{(x, 𝑧)|𝑧=n(x−x⁰)−𝑧⁰,x∈𝑃_x0}is affinely isomorphic to the hyperplane

𝐻(x⁰_𝑑, 𝑔_𝑐(n))inR^𝑑+1. Equation (5.5) translates (5.4) to halfspaces, and (5.6) similarly for tangent hyperplanes.

Now we are ready to prove the remaining part of “⇒”. Let𝐻 ={(x, 𝑧)|𝑧=𝑧⁰+n⁰·(x− x⁰)}be the hypothesized hyperplane for showing globally convexity. Similar to the proof of the claim above, it implies that

Proof. We use all definitions and results from the proof of “⇒” of Theorem 5.12.

Ifx⁰_𝑑andx⁰are both an extreme point in𝐹_𝑑and𝑋respectively, then we have 𝑇_𝐼

x0(x⁰) ={(x⁰, 𝑓(x⁰))} ⊂𝐻.

Otherwisex⁰_𝑑is an interior point in𝐹_𝑑. Similar to the proof of Equation (5.4) in the proof of

“⇒” of Theorem 5.12, it implies for𝐻 =𝐻(x⁰,n)that

where(*)is followed from Theorem 5.8. 2

5.3 Convex hull of graphs of polynomial functions over a polytope

Recall Example 5.11 again. We have𝑇_𝐼

x0(x⁰) ={(𝑥, 𝑦, 𝑧)|𝑧= 78𝑥−614, 𝑦 = 4}for the globally convex pointx⁰ = (1,4), see the green line in Figure 5.4. Corollary 5.13 shows that every valid hyperplane through(x⁰, 𝑓(x⁰))contains𝑇_𝐼

x0(x⁰). Note that𝑇_𝐼

x0(x⁰)is an affine set contained in the tangent plane𝑇(x⁰), we call it a subtangent plane. We give a general definition of subtangent planes that are contained in a valid hyperplane.

Definition 5.14 (Subtangent planes)

Letx⁰ ∈𝜕𝑋 with corresponding𝐼_x0. For every𝐼 ⊂ 𝐼_x0, a correspondingsubtangent plane 𝑇𝐼(x⁰)is defined as

𝑇_𝐼(x⁰) =𝑇(x⁰)∩ {(x, 𝑧)|𝑎^𝑇_𝑖 x=𝑏_𝑖, 𝑖∈𝐼}.

A subtangent plane𝑇𝐼(x⁰)is valid if there exists a valid hyperplane𝐻 with𝑇𝐼(x⁰) ⊂ 𝐻. Note that𝑇(x⁰)may not be valid and yet𝑇_𝐼(x⁰)is valid by choosing𝐻 ̸= 𝑇(x⁰). A valid subtangent plane through(x⁰, 𝑓(x⁰))is said to bemaximally valid if there does not exist any other valid subtangent plane𝑇𝐼^′(x⁰)such that𝑇𝐼(x⁰)(𝑇𝐼^′(x⁰).

We also extend the definition tox⁰ ∈int𝑋since𝐼_x0 =∅and𝑇_∅(x⁰) =𝑇(x⁰).

Note that since|𝐼_x0|=𝑑^′there are2^𝑑′ such subtangent planes and all of them are affine sets.

For every𝐼₁, 𝐼₂with𝐼₁, 𝐼₂ ⊂𝐼_x⁰ it implies that

aff{𝑇_𝐼1(x⁰), 𝑇_𝐼2(x⁰)}=𝑇_𝐼1∩𝐼₂(x⁰) and it is clear that

𝑇_𝐼1(x⁰)⊃𝑇_𝐼2(x⁰) for𝐼1 ⊂𝐼2 ⊂𝐼_x⁰.

Corollary 5.13 shows that every valid hyperplane𝐻through a boundary point(x⁰, 𝑓(x⁰)) satisfies

𝐻 ⊃𝑇_𝐼

x0(x⁰) =^(︁𝑇(x⁰)∩ {(x, 𝑧)|x∈𝑃_x⁰}^)︁⊃^(︁𝑇(x⁰)∩ {(x, 𝑧)|x∈𝐹_x⁰}^)︁. Face𝐹_x0 of dimension less than𝑛−1is contained in a face of𝑋of higher dimension. Exam-ple 5.15 below shows that there could exist another face𝐹_x¹0 of𝑋 such that𝐹_x¹0 )𝐹_x⁰ and there exists also a valid hyperplane𝐻¹with𝐻¹ ⊃(𝑇(x⁰)∩ {(x, 𝑧)|x∈𝐹_x¹0}).

Example 5.15

Consider the following example inR³. Let𝑓(𝑥, 𝑦) =𝑥²−5𝑥𝑦+𝑦²be the polynomial function over domain𝑋 = {(𝑥, 𝑦) ∈ [−3,10]×[−3,10]}, shown in Figure 5.6. The inequalities for the halfspaces of 𝑋with 𝑥 ≥ −3,𝑦 ≥ −3,𝑥 ≤ 10and𝑦 ≤ 10 have corresponding index 1,2,3and4, respectively. The graph of𝑓 inR³is illustrated in Figure 5.5a. The domain point x⁰= (−3,−3)∈𝑋is a boundary point with𝐼_x⁰ ={1,2}. Let𝐹1 ={(𝑥, 𝑦)|𝑥=−3,−3≤ 𝑦 ≤10}and𝐹₂ ={(𝑥, 𝑦) | −3 ≤𝑥 ≤ 10, 𝑦 = −3}be two faces of𝑋 containingx⁰. The

(a) The graph and graph on face(s) (b)𝑇{1}(x⁰)and𝐻¹

(c)𝑇{2}(x⁰)and𝐻² (d)𝑇{1,2}(x⁰)and𝐻³

Figure 5.5:Example 5.15 for valid subtangent planes and corresponding valid hyperplanes graph of𝑓 with restrictionx ∈ 𝐹₁ is the red curve in Figure 5.5a and the graph of 𝑓 with restrictionx∈𝐹₂is the blue curve in Figure 5.5a. In addition, the2² = 4subtangent planes are

𝑇_∅(x⁰) ={(𝑥, 𝑦, 𝑧)|𝑧= 9𝑥+ 9𝑦+ 27}=𝑇(x⁰),

𝑇_{1}(x⁰) =𝑇(x⁰)∩ {(𝑥, 𝑦, 𝑧)|𝑥=−3}={(𝑥, 𝑦, 𝑧)|𝑧= 9𝑦, 𝑥=−3}, 𝑇{2}(x⁰) =𝑇(x⁰)∩ {(𝑥, 𝑦, 𝑧)|𝑦=−3}={(𝑥, 𝑦, 𝑧)|𝑧= 9𝑥, 𝑦=−3}, 𝑇_{1,2}(x⁰) =𝑇(x⁰)∩ {(𝑥, 𝑦, 𝑧)|𝑥=−3, 𝑦=−3}={(−3,−3,−27)}.

Consider the pointx¹ = (10,10)with𝑓(x¹) =−300. In the following, we investigate whether 𝑇∅(x⁰)is valid. It is clear that the point(10,10,9·10 + 9·10 + 27) = (10,10,207)∈𝑇∅(x⁰). Note that the point𝑃¹= (x¹, 𝑓(x¹)) = (10,10,−300)is below𝑇_∅(x⁰)because𝑃¹is below the point(10,10,207). Hence𝑇_∅(x⁰)is not valid. The subtangent plane𝑇_{1}(x⁰)(red line

5.3 Convex hull of graphs of polynomial functions over a polytope

−4 −2 0 2 4 6 8 10

0 5 10

Figure 5.6:Domain𝑋 ={(𝑥, 𝑦)∈[−3,10]×[−3,10]}

in Figure 5.5b and in Figure 5.5d) is valid since the hyperplane𝐻¹(green plane in Figure 5.5b) with

𝐻¹ = aff{𝑇_{1},{𝑃¹}}={(𝑥, 𝑦, 𝑧)|𝑧=−30𝑥+ 9𝑦−90}

can be verified to be valid and𝑇{1}(x⁰)⊂𝐻¹. Similarly,𝑇{2}(x⁰)(blue line in Figure 5.5c and in Figure 5.5d) is valid since the hyperplane𝐻²(green plane in Figure 5.5c) with

𝐻²= aff{𝑇_{2},{𝑃¹}}={(𝑥, 𝑦, 𝑧)|𝑧= 9𝑥−30𝑦−90}

can be verified to be valid and𝑇_{2}(x⁰)⊂𝐻². Let

𝐻³ ={(𝑥, 𝑦, 𝑧)|𝑧=−10.5𝑥−10.5𝑦−90}

with{(−3,−3,−27)} ⊂𝐻³and{𝑃¹} ⊂𝐻³. The affine set𝐻³(green plane in Figure 5.5d) can also be verified to be valid. The subtangent plane𝑇{1,2}(x⁰)(black point in Figure 5.5d) is valid since𝑇_{1,2}(x⁰)⊂𝐻³.

Note that𝑇_{1}(x⁰)and𝑇_{2}(x⁰), both through(x⁰, 𝑓(x⁰)), are maximally valid subtangent planes. Hence a maximally valid subtangent plane does not have to be unique. ♢ Definition 5.16

Let𝑆₁, 𝑆₂⊂R^𝑛+1be two sets. For a given domain𝐷⊂R^𝑛,𝑆₁is said to be(strictly) below𝑆₂ over𝐷if𝜋x(𝑆1)∩𝜋x(𝑆2)∩𝐷̸=∅and for everyx⁰∈𝜋x(𝑆1)∩𝜋x(𝑆2)∩𝐷and𝑧¹, 𝑧² ∈R with(x⁰, 𝑧¹)∈𝑆1and(x⁰, 𝑧²)∈𝑆2, we have

𝑧¹≤𝑧²(𝑧¹ < 𝑧²).

In this case we also call𝑆₂is(strictly) above𝑆₁over𝐷. We omit “over𝐷” if 𝐷⊃(𝜋_x(𝑆₁)∩𝜋_x(𝑆₂)).

Figure 5.7:Example of half hyperplanes for Lemma 5.18 and Lemma 5.19

Remark 5.17

In the definition above, the comparison below or above allows the compared sets to have intersection points. Note that for the special case𝑆₁ =𝑆₂, we also have𝑆₁ is above𝑆₂as well as𝑆1is below𝑆2.

Note that a hyperplane𝐻 is valid if and only if it is below𝒮. In the example shown in Figure 5.11a,𝐻^𝑖is below𝐻^𝑗over[𝑙, 𝑢]for all𝑖, 𝑗with1≤𝑖 < 𝑗 ≤4and𝐻^𝑖is below𝒮for all 𝑖∈ {1,2,3,4}. Note that for any𝑆1, 𝑆2, 𝑆3 ⊂R^𝑛+1 with𝑆1 is below𝑆2 over𝐷¹ ⊂R^𝑛and 𝑆₂is below𝑆₃over𝐷² ⊂R^𝑛,𝑆₁is below𝑆₃over𝐷¹∩𝐷².

Lemma 5.18 (Half hyperplanes) Let

𝐻¹=𝐻(x⁰,n¹) ={(x, 𝑧)|𝑧=𝑧⁰+n¹·(x−x⁰)}

with the normal vectorn¹ ∈R^𝑛and

𝐻²=𝐻(x⁰,n²) ={(x, 𝑧)|𝑧=𝑧⁰+n²·(x−x⁰)}

with the normal vectorn² ∈R^𝑛be two nonvertical and nonparallel hyperplanes with an intersection point(x⁰, 𝑧⁰)∈𝐻¹∩𝐻². Let

𝜋_x(𝐻¹∩𝐻²) ={x∈R^𝑛|𝑧⁰+n¹·(x−x⁰) =𝑧⁰+n²·(x−x⁰)}={x∈R^𝑛|(n¹−n²)·(x−x⁰) = 0}.

5.3 Convex hull of graphs of polynomial functions over a polytope

Then𝐻¹is below𝐻²over𝐿:={x∈R^𝑛|(n¹−n²)·(x−x⁰)≤0}.

In addition, for any set𝑆with𝑆⊂𝐿or𝑆⊂𝑅:={x∈R^𝑛|(n¹−n²)·(x−x⁰)≥0}, either 𝐻¹is below𝐻²over𝑆or𝐻²is below𝐻¹over𝑆.

Proof. For anyx∈𝐿we compare(x, 𝑧¹)∈𝐻¹and(x, 𝑧²)∈𝐻²with

𝑧¹−𝑧²= (𝑧⁰+n¹·(x−x⁰))−(𝑧⁰+n²·(x−x⁰)) = (n¹−n²)·(x−x⁰)≤0, which means𝐻¹is below𝐻² over𝐿.

After that, the second part is trivial to prove. 2

Note that if we write𝜋x(𝐻¹∩𝐻²) ={x∈R^𝑛|(n²−n¹)·(x−x⁰) = 0}, then𝐻¹is above 𝐻²over{x∈R^𝑛|(n²−n¹)·(x−x⁰)≤0}={x∈R^𝑛|(n¹−n²)·(x−x⁰)≥0}=R. An example is shown in Figure 5.7.

Lemma 5.19

Consider the set 𝑃^𝑛−1 = {(x, 𝑧) | 𝑧 = 𝑧⁰ +n¹ ·(x−x⁰),(n¹ −n²) ·(x −x⁰) = 0}

forn¹,n² ∈ R^𝑛,n¹ ̸= n² and x⁰ ∈ 𝑋, 𝑧⁰ ∈ R. Then𝑃^𝑛−1 is an affine set of dimension (𝑛−1). For any compact set𝑆 ∈ R^𝑛+1 with𝜋_x(𝑆) ⊂ {x | (n¹−n²)·(x−x⁰) < 0}or 𝜋x(𝑆)⊂ {x|(n¹−n²)·(x−x⁰)>0}, there exists a hyperplane𝐻⊃𝑃^𝑛−1which is below𝑆.

Proof. The set𝑃^𝑛−1 can be equivalently written as

𝑃^𝑛−1 ={(x, 𝑧)|𝑧=𝑧⁰+n¹·(x−x⁰)} ∩ {(x, 𝑧)|𝑧=𝑧⁰+n²·(x−x⁰)}, which is an intersection of two nonparallel hyperplanes. Hence it is an affine set of dimension (𝑛−1).

Note that for any(x, 𝑧) ∈𝑆, we havex ̸∈ 𝜋_x(𝑃^𝑛−1). The affine setaff{𝑃^𝑛−1,{(x, 𝑧)}}

is then a nonvertical hyperplane. We take an arbitrary fixed pointx^𝑟 ∈𝜋x(𝑆)as areference point, with an affine set{(x^𝑟, 𝑧)|𝑧∈R}. For any(x, 𝑧)∈𝑆,aff{𝑃^𝑛−1,{(x, 𝑧)}}has exactly one intersecting point with the affine set{(x^𝑟, 𝑧)|𝑧 ∈R}, see examples in Figure 5.7 with (x, 𝑧) = (x¹, 𝑧¹)or(x, 𝑧) = (x², 𝑧²). Denoting the intersection point by(x^𝑟, 𝑓^x𝑟(x))with

{(x^𝑟, 𝑓^x𝑟(x))}= aff{𝑃^𝑛−1,{(x, 𝑧)∈𝑆}} ∩ {(x^𝑟, 𝑧)|𝑧∈R}

and𝑓^x𝑟 :𝜋_x(𝑆)→R. Function𝑓^x𝑟 is continuous since the affine function above is continuous.

Let(x¹, 𝑧¹) ∈ 𝑆 and(x², 𝑧²) ∈ 𝑆. Then the hyperplaneaff{𝑃^𝑛−1,{(x¹, 𝑧¹)}} is below aff{𝑃^𝑛−1,{(x², 𝑧²)}}over𝑆if and only if𝑓^x𝑟(x¹)≤𝑓^x𝑟(x²), see an example in Figure 5.7.

Due to the Weierstrass Theorem, we can find(x^*, 𝑧^*)∈𝑆such that𝑓^x𝑟 attains a minimum atx^*. Then we haveaff{𝑃^𝑛−1,{(x^*, 𝑧^*)}}is belowaff{𝑃^𝑛−1,{(x^𝑖, 𝑧^𝑖)}}for any(x^𝑖, 𝑧^𝑖)∈𝑆.

This implies thataff{𝑃^𝑛−1,{(x^*, 𝑧^*)}} is below any(x^𝑖, 𝑧^𝑖) ∈ 𝑆. Hence the hyperplane

𝐻 := aff{𝑃^𝑛−1,{(x^*, 𝑧^*)}}is below𝑆. 2

Figure 5.8:A point sequence in the neighborhood ofx⁰

With the following lemma we prove that a hyperplane𝐻is valid if and only if𝐻is below𝒮 over𝑋ˇ^𝑔.

Lemma 5.20

The hyperplane𝐻(x⁰,n⁰)is below𝒮over𝑋if and only if𝐻(x⁰,n⁰)is below𝒮over𝑋ˇ^𝑔, which means for everyx∈𝑋ˇ^𝑔it satisfies

𝑓(x)≥𝑓(x⁰) +n⁰·(x−x⁰). (5.8) Proof. Direction ”⇒“ simply follows since𝑋⊃𝑋ˇ^𝑔.

For the proof of ”⇐“, first consider the optimization problem

x∈𝑋min 𝑓(x)−𝑓(x⁰)−n⁰·(x−x⁰), (5.9) which has a continuous objective function over a compact feasible region. Since𝑋is compact, an optimum always exists due to the Weierstrass Theorem. Due tox⁰∈𝑋, we have

minx∈𝑋 𝑓(x)−𝑓(x⁰)−n⁰·(x−x⁰)≤𝑓(x⁰)−𝑓(x⁰)−n⁰·(x⁰−x⁰) = 0.

The hyperplane𝐻(x⁰,n⁰)is valid if and only if the optimization problem (5.9) has optimal value0. Note that by assumption, everyx∈𝑋ˇ^𝑔satisfies (5.8) . The hyperplane𝐻(x⁰,n⁰)is valid if the inequality holds also for everyx∈𝑋∖𝑋ˇ^𝑔. We assume there exists anx⁻ ∈𝑋∖𝑋ˇ^𝑔 with

𝑓(x⁻)< 𝑓(x⁰) +n⁰·(x⁻−x⁰).

Then the optimization problem (5.9) possesses an optimal solution(x^*, 𝑧^*)with value<0, i.e., 𝑓(x^*)< 𝑓(x⁰) +n⁰·(x^*−x⁰).

Note thatx^* is a globally convex domain point since𝐻(x^*,n⁰)is valid. This is a contradiction to (5.8) for everyx∈𝑋ˇ^𝑔. It implies thenx⁻does not exist and (5.8) is also satisfied for every x∈𝑋∖𝑋ˇ^𝑔. Hence𝐻(x⁰,n⁰)is a valid hyperplane. 2

5.3 Convex hull of graphs of polynomial functions over a polytope

Proof idea of direction “⇐” of Theorem 5.12. Recall Example 5.11. As shown in Figure 5.4b, for𝒮_𝑑,(x⁰_𝑑, 𝑓𝑑(x⁰_𝑑)) = (1,−536)is globally convex. To show Theorem 5.12, we need to prove that there exists a valid hyperplane through(x⁰, 𝑓(x⁰)) = (1,4,−536), which is the red point in Figure 5.4a. According to Corollary 5.13, every such hyperplane should contain𝑇_𝐼

x0(x⁰) which is the green line in Figure 5.4a. Note that in this example𝑇_𝐼

x0(x⁰)is an affine set of dimension1. According to Lemma 5.19, by setting𝑃^𝑛−1 =𝑇_𝐼

x0(x⁰), for any subset𝑋^′ ⊂𝑋 with𝑋^′∩𝐹_x⁰ =∅, there exists a hyperplane𝐻 ⊃𝑇_𝐼

x0(x⁰)such that𝐻is below𝒮over𝑋^′. The proof idea is as follows. After finding a set𝑃^𝑛−1as described in Lemma 5.19, we split𝑋 into two compact sets𝑋^𝑀 and𝑋^Δsuch that𝜋_x(𝑃^𝑛−1)⊂𝑋^𝑀 and𝑋^Δ∩𝜋_x(𝑃^𝑛−1) =∅as

The following lemma helps us to find𝑋^𝑀 and𝐻^*as mentioned in the proof idea.

Lemma 5.21

For every x⁰ ∈ 𝑋ˇ^𝑔, there exists an 𝜀 > 0 such that 𝑇(x⁰) is below 𝒮 over 𝑋ˇ^𝑔 ∩ 𝐵_𝜀(x⁰). Furthermore, every valid hyperplane𝐻through(x⁰, 𝑓(x⁰))is below𝑇(x⁰)over𝑋.

Proof. Note that for anyx⁰∈int𝑋this lemma is clear due to Lemma 5.5 and Theorem 5.8.

We need only to consider the casex⁰ ∈ 𝜕𝑋. For a given globally convex domain point x⁰∈𝜕𝑋∩𝑋ˇ^𝑔, we want to capture the globally convex domain points in its neighborhood.

Let𝐴, 𝐵⊂R^𝑛+1be two subsets, then

which follows directly from the definition of subtangent planes.

For any𝑇𝐼(x⁰)which is below𝒮,𝑇(x⁰)is then below𝒮 over𝑋ˇ^𝑔∩𝐵𝜀(x⁰)∩𝜋x(𝑇𝐼(x⁰)) for any𝜀 >0. It implies that𝑇(x⁰)is below𝒮over

⋃︁

𝐼⊂𝐼_x0 𝑇𝐼(x⁰)is below𝒮

𝑋ˇ^𝑔∩𝐵_𝜀(x⁰)∩𝜋_x^(︁𝑇_𝐼(x⁰)^)︁.

Note that

𝑋ˇ^𝑔∩𝐵𝜀(x⁰) = ^⋃︁

𝐼⊂𝐼_x0

𝑋ˇ^𝑔∩𝐵𝜀(x⁰)∩𝜋x

(︁𝑇_𝐼(x⁰)^)︁.

In the following we need only to prove that for a sufficiently small𝜀 >0it holds

⋃︁

𝐼⊂𝐼_x0 𝑇𝐼(x⁰)is below𝒮

𝑋ˇ^𝑔∩𝐵_𝜀(x⁰)∩𝜋_x^(︁𝑇_𝐼(x⁰)^)︁= ^⋃︁

𝐼⊂𝐼_x0

𝑋ˇ^𝑔∩𝐵_𝜀(x⁰)∩𝜋_x^(︁𝑇_𝐼(x⁰)^)︁ (5.11)

which implies the first consequence that𝑇(x⁰)is below𝒮over𝑋ˇ^𝑔∩𝐵𝜀(x⁰).

Before proving (5.11) we first prove that for everyx⁰∈𝑋ˇ^𝑔there exists an𝜀_𝐽 >0such that any subtangent plane𝑇_𝐽(x⁰), 𝐽 ⊂ 𝐼_x0 thatdoes not fulfillthe condition𝑇_𝐽(x⁰)is below𝒮 satisfies

𝑋ˇ^𝑔∩𝐵_𝜀𝐽(x⁰)∩relint^(︁𝐹_𝐽(x⁰)^)︁=∅. (5.12) Assume that there does not exist an𝜀𝐽 for which (5.12) holds. Then there exists𝐽 ⊂𝐼_x⁰ such that𝑇_𝐽(x⁰)does not fulfill𝑇_𝐽(x⁰)is below𝒮and for every𝜀_𝐽 >0

𝑋ˇ^𝑔∩𝐵𝜀𝐽(x⁰)∩relint^(︁𝐹𝐽(x⁰)^)︁̸=∅.

Recall that𝐹_𝐽(x⁰)is a compact set. There exists then a point sequence(x^𝑖)_𝑖∈N ⊂𝑋ˇ^𝑔, as shown in Figure 5.8, such that

𝑖→∞lim x^𝑖 =x⁰and{((x^𝑖)_𝑖∈N)} ⊂^(︁𝑋ˇ^𝑔∩𝐵𝜀𝐽(x⁰)∩relint^(︁𝐹𝐽(x⁰)^)︁)︁

for𝜀_𝐽 >0. The pointx^𝑖for any𝑖∈Nis a globally convex domain point and a relative interior point in𝐹𝐽(x⁰). Denotingx^𝑖byy, the function

𝐺_y(x) =𝑓(x)−(∇𝑓(y)·(x−y) +𝑓(y))

evaluates the validity of the tangent plane𝑇(y) through(y, 𝑓(y)): Tangent plane 𝑇(y) is valid if and only if𝐺_y(x)≥0for allx∈𝑋; subtangent plane𝑇_𝐽(y)is below𝒮if and only if 𝐺y(x)≥0for allx∈𝐹𝐽(y). Consider the function

ℎ:𝐹_𝐽(x⁰)→R,y↦→ min

x∈𝐹_𝐽(y)𝐺_y(x).

5.3 Convex hull of graphs of polynomial functions over a polytope

With “⇒” of Theorem 5.12,y ∈ 𝑋ˇ^𝑔 implies that𝑇_𝐽(y) is below𝒮 which is equivalent to 𝑇(y)is below𝒮 over𝐹𝐽(y). Thus,ℎ(x^𝑖) ≥0for every𝑖∈N. For anyyfunction𝐺y(x)is a continuous function, it impliesℎis also continuous since it maps from a compact set to the minimum of𝐺_y(x). Together withlim_𝑖→∞x^𝑖 =x⁰ it implies thatℎ(x⁰)≥ 0. This implies that𝑇𝐽(x⁰)is below𝒮. This is a contradiction to our assumption. Hence there exists an𝜀𝐽 >0 such that

𝑋ˇ^𝑔∩𝐵𝜀𝐽(x⁰)∩relint^(︁𝐹𝐽(x⁰)^)︁=∅.

Since we have finitely many subtangent planes through (x⁰, 𝑓(x⁰)), there exists an𝜀 > 0 which is the minimum of all𝜀𝐽 such that (5.12) holds for every such subtangent plane through (x⁰, 𝑓(x⁰)).

Using (5.14) iteratively and together with (5.12), it implies that 𝑋ˇ^𝑔∩𝑆_𝐽^𝜀(x⁰) = ^⋃︁

𝐽⊂𝐼⊂𝐼_x0

𝑇𝐼(x⁰)is below𝒮

𝑋ˇ^𝑔∩𝑆_𝐼^𝜀(x⁰)

for every𝐽 ⊂𝐼_x⁰. This implies𝑇(x⁰)is below𝒮over

⋃︁

𝐼⊂𝐼_x0

𝑇𝐼(x⁰)is below𝒮

𝑋ˇ^𝑔∩𝑆_𝐼^𝜀(x⁰) = ^⋃︁

𝐼⊂𝐼_x0

𝑋ˇ^𝑔∩𝑆_𝐼^𝜀(x⁰) = ˇ𝑋^𝑔∩𝐵𝜀(x⁰).

Now it remains only to prove that every valid hyperplane𝐻through(x⁰, 𝑓(x⁰))is below 𝑇(x⁰)over𝑋. Assume that𝐻 ̸= 𝑇(x⁰), otherwise we are done. An example for𝑛 = 1is shown in Figure 5.9. Let𝑋 = [𝑙, 𝑢]and𝑙,𝑥^𝑚and𝑢be globally convex points. Tangent plane 𝑇(𝑙), shown as the green line, is not valid. However, the example in Figure 5.9 shows that every valid hyperplane through(𝑙, 𝑓(𝑙))is below𝑇(𝑙)over𝑋; it also shows that every hyperplane through(𝑙, 𝑓(𝑙))which is above𝑇(𝑙)over𝑋cannot be valid. For interior domain point𝑥^𝑚, there does not exist another hyperplane through(𝑥^𝑚, 𝑓(𝑥^𝑚))which is below𝑇(𝑥^𝑚), thus there exists at most one valid hyperplane through(𝑥^𝑚, 𝑓(𝑥^𝑚))which is also implied by Theorem 5.8.

For domain point𝑢,𝑇(𝑢)is valid. In addition, every hyperplane through(𝑢, 𝑓(𝑢))which is below𝑇(𝑢)over𝑋is also valid; every hyperplane through(𝑢, 𝑓(𝑢))which is not𝑇(𝑢)and above𝑇(𝑢)over𝑋is not valid.

We go back to the proof for general𝑛. Under the assumption𝐻 ̸=𝑇(x⁰), hyperplanes𝐻 and𝑇(x⁰)are not parallel due to the intersection point(x⁰, 𝑓(x⁰)). Similar to the notation in Lemma 5.18, denote

𝜋_x(𝐻∩𝑇(x⁰)) ={x∈R^𝑛|n⁰·(x−x⁰) = 0}, and

𝐿:={x∈R^𝑛|n⁰·(x−x⁰)≤0}, 𝑅:={x∈R^𝑛|n⁰·(x−x⁰)≥0}

withn⁰ ∈R^𝑛. Note that sincex⁰is a boundary point of𝑋, we have either𝜋x(𝐻∩𝑇(x⁰))∩ int𝑋 ̸= ∅ or𝜋x(𝐻∩𝑇(x⁰))∩int𝑋 = ∅. If𝜋x(𝐻 ∩𝑇(x⁰))∩int𝑋 ̸= ∅, then we have 𝐿∩int𝑋 ̸=∅and𝑅∩int𝑋̸=∅. Lemma 5.18 implies𝐻is above𝑇(x⁰)over𝐿or𝑅. Without loss of generality,𝐻is above𝑇(x⁰)over𝐿. Consider the graph of𝑓over𝐿∩𝐵_𝜀(x⁰)∩int𝑋̸=∅ for any𝜀 >0. There always exists a pointx¹ ∈𝐿∩𝐵_𝜀(x⁰)∩int𝑋such that(x¹, 𝑓(x¹))is below𝐻. Thus𝐻cannot be valid if𝜋_x(𝐻∩𝑇(x⁰))∩int𝑋 ̸=∅.

Now we consider the case𝜋x(𝐻∩𝑇(x⁰))∩int𝑋=∅. There are two cases, either𝑋 ⊂𝐿or 𝑋⊂𝑅. Without loss of generality, let𝑋⊂𝐿. Lemma 5.18 implies that𝐻is either below𝑇(x⁰)

Now we consider the case𝜋x(𝐻∩𝑇(x⁰))∩int𝑋=∅. There are two cases, either𝑋 ⊂𝐿or 𝑋⊂𝑅. Without loss of generality, let𝑋⊂𝐿. Lemma 5.18 implies that𝐻is either below𝑇(x⁰)

Im Dokument Optimal Operation of Water Supply Networks by Mixed Integer Nonlinear Programming and Algebraic Methods (Seite 75-115)