Bivariate polynomial functions: a case study

Letℋˇ^◁▷be the set of all downward valid and tight hyperplanes andℋ^^◁▷be the set of all upward valid and tight hyperplanes. Letℋˇ^◇ ⊂ℋˇ^◁▷andℋ^^◇ ⊂ℋ^^◁▷be two finite sets.

Corollary 5.30 (Polyhedral relaxation) The set

𝑆( ˇℋ^◇,ℋ^^◇) =

⎛

⎝

⋂︁

𝐻ˇ∈ℋˇ^◇

𝐻ˇ

⎞

⎠∩

⎛

⎝

⋂︁

𝐻^∈ℋˇ^◇

𝐻^

⎞

⎠ is a polyhedral set which fulfills

conv(𝒮)⊂𝑆( ˇℋ^◇,ℋ^^◇).

So far, we gave a description of the convex hull of the graph of polynomial functions. Recalling the definitions and theorems, our work focused mainly on theoretical point of view. Instead of obtaining algorithms to compute valid hyperplanes, we dealt with proof of existence. Indeed, algorithmically, it is very hard to verify if a given hyperplane is valid in a general dimension and for a general degree of polynomial functions.

In the next section, we concentrate on bivariate polynomial functions with a limited degree.

Algorithms are developed to find tight hyperplanes. Computations show that these tight hyperplanes accelerate MINLP solving processes.

5.4 Bivariate polynomial functions: a case study

In this section we design algorithms to find finitely many tight valid hyperplanes for the graph of bivariate polynomial functions with degree up to3. Every given bivariate polynomial function with degree up to3has the form

𝑓(𝑥, 𝑦) = ^∑︁

0≤𝑖,𝑗≤3 0≤𝑖+𝑗≤3

𝑎_𝑖𝑗𝑥^𝑖𝑦^𝑗, (5.17)

where all𝑎𝑖𝑗 ∈Rare constants and𝑋 ⊂R²is the domain which is a polytope. Then𝑋is a convex polygon with𝑚≥3edges and vertices. Every edge is a line segment as well as a facet of𝑋and every vertex is an extreme point of𝑋.

Again, we only consider the downward closed part. Recall that𝑋ˇ^𝑔is the set of all globally convex domain points and𝑋ˇ^𝑙is the set of all locally convex domain points. Theoretically, for anyx⁰ ∈int𝑋we need only to check if𝑇(x⁰)is valid. However, in practice, this is not easy even for𝑓 given as in (5.17). Instead of getting valid hyperplanes starting from interior domain points, we pay more attention to those boundary domain points.

Using the result from Section 5.3, the graph of the bivariate polynomial function𝑓 on a facet of𝑋is isomorphic to the graph of a univariate polynomial function on a corresponding projected domain. We show later that finding𝑋ˇ^𝑔 for univariate polynomial functions with

degree≤ 3is tractable. Thus we can easily find the set𝑋ˇ^𝑔∩𝜕𝑋 for bivariate polynomial functions with degree≤3. In the following we design algorithms which first compute a few hyperplanes that are below𝒮over𝑋ˇ^𝑔∩𝜕𝑋. For each of the hyperplanes which are below the boundary of𝒮, we solve a NLP globally either to verify if the hyperplane is valid or to find a valid hyperplane which is parallel to this hyperplane. These NLPs contain only two variables and can be globally solved by SCIP in less than one second.

Going back to our applications, all of these hyperplanes can be found in an offline way, i.e., before we start to solve the MINLPs. For every instance we need only to calculate these hyperplanes once. Every globally solved NLP above yields a tight valid hyperplane.

Remark 5.31

In this section we discuss hyperplanes and graphs of polynomial functions inR³. As before, we use(𝑥, 𝑦, 𝑧)to denote a point inR³. Similar to Section 5.3, we usex= (𝑥, 𝑦)∈R²to denote domain points and use e.g.,x⁰ = (𝑥₀, 𝑦₀)∈R² to denote a certain domain point.

For a boundary point(𝑥, 𝑦) ∈𝜕𝑋 there exists at least one facet𝐹𝑖 of𝑋with(𝑥, 𝑦) ∈𝐹𝑖. Since𝐹_𝑖is a line segment, it must be contained in a line denoted by

{(𝑥, 𝑦)|𝑎𝑖𝑥+𝑏𝑖𝑦+𝑐𝑖 = 0}=: aff{𝐹_𝑖},

where𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖 ∈ Rare constants and at least one of𝑎_𝑖 and𝑏_𝑖 is nonzero. Without loss of generality we assume𝑏𝑖 ̸= 0(otherwise permute𝑥and𝑦) and set𝑏𝑖 = 1(otherwise scale𝑎𝑖, 𝑏𝑖

and𝑐_𝑖). Facet𝐹_𝑖can be then be represented as

𝐹𝑖 ={(𝑥, 𝑦)|𝑦=−𝑎_𝑖𝑥−𝑐𝑖, 𝑥∈[𝑥^min_𝑖 , 𝑥^max_𝑖 ]},

where𝑥^min_𝑖 , 𝑥^max_𝑖 ∈Rare constants with𝑥^min_𝑖 < 𝑥^max_𝑖 . Recalling the definitions in Section 5.3 and using the same notations, we have the projection map

𝑔_𝑑: aff{𝐹_𝑖} →R,(𝑥, 𝑦)↦→𝑥 and its inverse map

𝑔_𝑑⁻¹ :R→aff{𝐹_𝑖}, 𝑥↦→

(︃ 𝑥

−𝑎_𝑖𝑥−𝑐𝑖

)︃

as well as

𝑓_𝑖(𝑥) =𝑓(𝑥,−𝑎_𝑖𝑥−𝑐_𝑖) = ^∑︁

0≤𝑖,𝑗≤3 0≤𝑖+𝑗≤3

𝑎_𝑖𝑗𝑥^𝑖(−𝑎_𝑖𝑥−𝑐_𝑖)^𝑗 =𝑎𝑥³+𝑏𝑥²+𝑐𝑥+𝑑,

where𝑎, 𝑏, 𝑐, 𝑑are constants depending on𝑎𝑖,𝑐𝑖and all𝑎𝑖𝑗. An example has been shown in Figure 5.4 and discussed in Section 5.3.

5.4 Bivariate polynomial functions: a case study

Corollary 5.32

A boundary domain point(𝑥0, 𝑦0)on facet𝐹𝑖of𝑋is globally convex for𝒮 if and only if𝑥0is globally convex for the graph of𝑓_𝑖(𝑥)over[𝑥^min_𝑖 , 𝑥^max_𝑖 ].

Proof. The result is a special case of Theorem 5.12. 2 Let𝑋ˇ_𝑖^𝑙 ⊂[𝑥^min_𝑖 , 𝑥^max_𝑖 ]denote the set of all locally convex domain points for the graph of 𝑓_𝑖(𝑥)and𝑋ˇ_𝑖^𝑔 ⊂𝑋ˇ_𝑖^𝑙the set of the globally convex domain points. Note that𝑋ˇ^𝑔∩𝐹_𝑖 =𝑔_𝑑⁻¹( ˇ𝑋_𝑖^𝑔). Thus, finding𝑋ˇ_𝑖^𝑔for every𝑖∈ {1, . . . , 𝑚}will find𝑋ˇ^𝑔∩𝜕𝑋.

Lemma 5.33

The set of globally convex domain points𝑋ˇ_𝑖^𝑔 ⊂[𝑥^min_𝑖 , 𝑥^max_𝑖 ]has one of the four following forms 1. {𝑥^min_𝑖 , 𝑥^max_𝑖 },

2. [𝑥^min_𝑖 , 𝑥^max_𝑖 ],

3. [𝑥^min_𝑖 , 𝑥^mid_𝑖 ]∪ {𝑥^max_𝑖 }, 4. {𝑥^min_𝑖 } ∪[𝑥^mid_𝑖 , 𝑥^max_𝑖 ].

In the two latter cases,𝑥^mid_𝑖 is a constant with𝑥^min_𝑖 < 𝑥^mid_𝑖 < 𝑥^max_𝑖 .

Proof. If𝑎= 0, then𝑓𝑖(𝑥)is a convex function (if𝑏≥0) or a concave function (if𝑏≤0). For this reason, we only need to consider the case𝑎̸= 0. We first seek the locally convex points𝑥₀ since every globally convex point is also locally convex. Let𝑓_𝑖^(𝑛)denote the𝑛th derivative of 𝑓_𝑖. We have

𝑓_𝑖⁽¹⁾(𝑥) = 3𝑎𝑥²+ 2𝑏𝑥+𝑐, 𝑓_𝑖⁽²⁾(𝑥) = 6𝑎𝑥+ 2𝑏, 𝑓_𝑖⁽³⁾(𝑥) = 6𝑎̸= 0, 𝑓_𝑖^(𝑛)(𝑥) = 0for all𝑛≥4.

Similar to the proof of Lemma 5.5, using Taylor’s Formula, we can easily prove that for any 𝑥∈(𝑥^min_𝑖 , 𝑥^max_𝑖 ),𝑥is locally convex if𝑓_𝑖⁽²⁾(𝑥) = 6𝑎𝑥+ 2𝑏 >0. Note that the extreme points 𝑥^min_𝑖 and𝑥^max_𝑖 are globally convex and thus locally convex which is implied by Corollary 5.22.

Since𝑓_𝑖⁽²⁾(𝑥) = 6𝑎𝑥+ 2𝑏is a monotonic function and has at most one root, depending on the value of𝑎,𝑏 𝑥^min_𝑖 and𝑥^max_𝑖 , the set of locally convex domain𝑋ˇ_𝑖^𝑙has one of the following four forms:

1. {𝑥^min_𝑖 , 𝑥^max_𝑖 }, 2. [𝑥^min_𝑖 , 𝑥^max_𝑖 ],

3. [𝑥^min_𝑖 ,−𝑏/3𝑎)∪ {𝑥^max_𝑖 },

4. {𝑥^min_𝑖 } ∪(−𝑏/3𝑎, 𝑥^max_𝑖 ].

We now discuss the set𝑋ˇ_𝑖^𝑔with the four cases above.

1. It is clear that𝑋ˇ_𝑖^𝑔 ={𝑥^min_𝑖 , 𝑥^max_𝑖 }if𝑋ˇ_𝑖^𝑙={𝑥^min_𝑖 , 𝑥^max_𝑖 }.

2. If𝑋ˇ_𝑖^𝑙 = [𝑥^min_𝑖 , 𝑥^max_𝑖 ], then𝑓_𝑖⁽²⁾(𝑥)≥0for all𝑥∈[𝑥^min_𝑖 , 𝑥^max_𝑖 ]which implies that𝑓𝑖(𝑥) is a convex function with domain[𝑥^min_𝑖 , 𝑥^max_𝑖 ]. Since𝑓_𝑖(𝑥)is differentiable, the tangent plane{(𝑥, 𝑦)|𝑦= (3𝑎𝑥²₀+ 2𝑏𝑥₀+𝑐)(𝑥−𝑥₀) +𝑓_𝑖(𝑥₀)}at point(𝑥₀, 𝑓_𝑖(𝑥₀))for every 𝑥0∈𝐹𝑖 is below the graph of𝑓𝑖over𝐹𝑖. Hence𝑋ˇ_𝑖^𝑔 = [𝑥^min_𝑖 , 𝑥^max_𝑖 ].

3. Examples of this case can be seen in Figure 5.14. Note that𝑋ˇ_𝑖^𝑔 ⊂[𝑥^min_𝑖 ,−𝑏/3𝑎)∪ {𝑥^max_𝑖 } and{𝑥^min_𝑖 , 𝑥^max_𝑖 } ⊂𝑋ˇ_𝑖^𝑔. Theorem 5.8 implies that𝑥₀∈(𝑥^min_𝑖 ,−𝑏/3𝑎)is globally convex if and only if the corresponding tangent plane

𝑇_𝑑(𝑥₀) ={(𝑥, 𝑦)|𝑦= (3𝑎𝑥²₀+ 2𝑏𝑥₀+𝑐)(𝑥−𝑥₀) +𝑓_𝑖(𝑥₀)}

is valid. Note that 𝑇𝑑(𝑥0) is below the graph of 𝑓𝑖 in [𝑥^min_𝑖 ,−𝑏/3𝑎]. With 𝑋ˇ_𝑖^𝑔 ⊂ [𝑥^min_𝑖 ,−𝑏/3𝑎)∪ {𝑥^max_𝑖 }, Lemma 5.20 implies that𝑥₀ ∈(𝑥^min_𝑖 ,−𝑏/3𝑎)is globally convex if and only if𝑇_𝑑(𝑥₀)is below the point(𝑥^max_𝑖 , 𝑓_𝑖(𝑥^max_𝑖 )). Define

𝑔max(𝑥) = (3𝑎𝑥²+ 2𝑏𝑥+𝑐)(𝑥^max_𝑖 −𝑥) +𝑓𝑖(𝑥)

such that point (𝑥^max_𝑖 , 𝑔_max(𝑥)) ∈ 𝑇_𝑑(𝑥) for any 𝑥 ∈ [𝑥^min_𝑖 ,−𝑏/3𝑎]. The tangent plane 𝑇_𝑑(𝑥0)for𝑥0 ∈ [𝑥^min_𝑖 ,−𝑏/3𝑎]is below the point(𝑥^max_𝑖 , 𝑓𝑖(𝑥^max_𝑖 ))if and only if𝑔_max(𝑥₀) ≤ 𝑓_𝑖(𝑥^max_𝑖 ). Thus we only need to compare𝑔_max(𝑥₀)and𝑓_𝑖(𝑥^max_𝑖 ). Con-sider the first derivative of𝑔_max(𝑥)

𝑔⁽¹⁾_max(𝑥) = (𝑥^max_𝑖 −𝑥)(6𝑎𝑥+ 2𝑏𝑥)−(3𝑎𝑥²+ 2𝑏𝑥+𝑐) +𝑓_𝑖⁽¹⁾(𝑥)

= (𝑥^max_𝑖 −𝑥)(6𝑎𝑥+ 2𝑏𝑥).

Thus,𝑔_maxis strictly increasing on[𝑥^min_𝑖 ,−𝑏/3𝑎)since we have𝑔⁽¹⁾max(𝑥)>0for any𝑥∈ [𝑥^min_𝑖 ,−𝑏/3𝑎); similarly𝑔maxis strictly decreasing on(−𝑏/3𝑎, 𝑥^max_𝑖 )since𝑔max⁽¹⁾ (𝑥)<0 for any𝑥∈[𝑥^min_𝑖 ,−𝑏/3𝑎). It is then clear that

𝑔max(−𝑏/3𝑎)> 𝑔max(𝑥^max_𝑖 ) =𝑓𝑖(𝑥^max_𝑖 ).

Now we compare𝑔_max(𝑥^min_𝑖 )and𝑓_𝑖(𝑥^max_𝑖 ). If𝑔_max(𝑥^min_𝑖 )> 𝑓_𝑖(𝑥^max_𝑖 ), see an example in Figure 5.14a, we have𝑔_max(𝑥)> 𝑓_𝑖(𝑥^max_𝑖 )for all𝑥∈(𝑥^min_𝑖 ,−𝑏/3𝑎). Hence no point in(𝑥^min_𝑖 ,−𝑏/3𝑎)is globally convex, which implies𝑋ˇ_𝑖^𝑔 ={𝑥^min_𝑖 , 𝑥^max_𝑖 }.

Otherwise we have𝑔max(𝑥^min_𝑖 )≤𝑓𝑖(𝑥^max_𝑖 ), see an example in Figure 5.14b. Consider the strictly increasing function𝑔_max(𝑥)−𝑓_𝑖(𝑥^max_𝑖 )over(𝑥^min_𝑖 ,−𝑏/3𝑎), with𝑔_max(𝑥^min_𝑖 )− 𝑓𝑖(𝑥^max_𝑖 )≤0and𝑔max(−𝑏/3𝑎)−𝑓𝑖(𝑥^max_𝑖 )>0. This function has exactly one real root over[𝑥^min_𝑖 ,−𝑏/3𝑎), say𝑥^mid_𝑖 . Then we have𝑋ˇ_𝑖^𝑔 = [𝑥^min_𝑖 , 𝑥^mid_𝑖 ]∪ {𝑥^max_𝑖 }if𝑥^min_𝑖 < 𝑥^mid_𝑖 and𝑋ˇ_𝑖^𝑔 ={𝑥^min_𝑖 , 𝑥^max_𝑖 }if𝑥^min_𝑖 =𝑥^mid_𝑖 .

5.4 Bivariate polynomial functions: a case study

4. Similar to case3, we need only to know whether the polynomial function of𝑥 (3𝑎𝑥²+ 2𝑏𝑥+𝑐)(𝑥^min_𝑖 −𝑥) +𝑓_𝑖(𝑥)

⏟ ⏞

=:𝑔min(𝑥)

−𝑓_𝑖(𝑥^min_𝑖 )

has a real root over 𝑥 ∈ (−𝑏/3𝑎, 𝑥^max_𝑖 ). If the root exists, say 𝑥^mid_𝑖 , then we have 𝑋ˇ_𝑖^𝑔 ={𝑥^min_𝑖 } ∪[𝑥^mid_𝑖 , 𝑥^max_𝑖 ]; otherwise, we have𝑋ˇ_𝑖^𝑔 ={𝑥^min_𝑖 , 𝑥^max_𝑖 }as well. 2

(a) Case1 (b) Case2

Figure 5.14:Examples for globally and locally convex domain points

Considering the four cases, the set of globally convex points𝑋ˇ_𝑖^𝑔 ⊂[𝑥^min_𝑖 , 𝑥^max_𝑖 ]has either two points, or is an interval plus a point, or an interval. Since the projection function𝑔⁻¹_𝑑 is bijective, the set of globally convex points on𝐹𝑖, denoted by𝑋ˇ^𝑔∩𝐹𝑖 =𝑔⁻¹_𝑑 ( ˇ𝑋_𝑖^𝑔), also consists of either two extreme points, or is a line segment inR²plus an extreme point, or a line segment inR². Note that every extreme point of 𝑋is globally convex. We call an extreme point an isolated extreme point if it is not contained in a line segment that consists of globally convex boundary domain points only. We then get the following lemma easily.

Lemma 5.34

The set 𝑋ˇ^𝑔 ∩𝜕𝑋 of globally convex boundary domain points for the graph of 𝑓(𝑥, 𝑦) over the polytope 𝑋 ∈ R² is a union of 𝑚₁ line segments and 𝑚₂ isolated extreme points with 𝑚1, 𝑚2 ∈N0, 𝑚1 ≤𝑚and𝑚2 ≤𝑚.

Let𝐿1, 𝐿2, . . . , 𝐿𝑚1 be the𝑚1line segments andx^𝑒₁,x^𝑒₂, . . . ,x^𝑒_𝑚2 be the𝑚2isolated extreme points. With this notation we have

𝑋ˇ^𝑔∩𝜕𝑋 =𝐿1∪ · · · ∪𝐿𝑚1 ∪ {x^𝑒₁, . . . ,x^𝑒_𝑚

2}.

Furthermore, let𝒮_𝐿𝑖be the graph of𝑓(𝑥, 𝑦)on𝐿_𝑖with

𝒮_𝐿𝑖 ={(𝑥, 𝑦, 𝑧)|𝑧=𝑓(𝑥, 𝑦),(𝑥, 𝑦)∈𝐿_𝑖}

Figure 5.15:Hyperplane that intersects𝒮𝐿_𝑖,𝒮𝐿_𝑗 and below them

for every𝑖∈ {1, . . . , 𝑚}and let

𝒮_𝑋^𝑒 ={(𝑥, 𝑦, 𝑧)|𝑧=𝑓(𝑥, 𝑦),(𝑥, 𝑦)∈𝑋^𝑒}.

In the following, for any(𝑥₀, 𝑦₀)∈𝐿_𝑖, we show that there exists a hyperplane𝐻through (𝑥0, 𝑦0, 𝑓(𝑥0, 𝑦0))such that𝐻is below𝒮_𝐿𝑖over𝑋ˇ^𝑔∩𝜕𝑋. In Lemma 5.36 we have more details included. We show later in Theorem 5.37 that either𝐻 is a tight valid hyperplane or a tight valid hyperplane𝐻^*can be found very easily which is parallel to𝐻.

To find the hyperplane𝐻with the properties described above, we first prove Lemma 5.35, which implies that for any𝐿𝑗 there exists a hyperplane𝐻_𝑖^𝑗 which is below𝒮 over𝐿𝑖∪𝐿𝑗

with 𝑖, 𝑗 ∈ {1, . . . , 𝑚₁}, 𝑖 ̸= 𝑗. Using this result, we show that a hyperplane 𝐻 through (𝑥0, 𝑦0, 𝑓(𝑥0, 𝑦0))exists such that𝐻is below𝒮_𝐿𝑗 for any𝑗∈ {1, . . . , 𝑚₁}, 𝑖̸=𝑗. In addition,

a hyperplane𝐻can be found that it is below(x^𝑒_𝑘, 𝑓(x^𝑒_𝑘))for any𝑘∈ {1, . . . , 𝑚₁}. Lemma 5.35

For any𝐿_𝑖 and𝐿_𝑗 with𝑖, 𝑗 ∈ {1, . . . , 𝑚₁}, 𝑖 ̸= 𝑗 and for any (𝑥₀, 𝑦₀) ∈ 𝐿_𝑖, there exists a hyperplane𝐻through(𝑥0, 𝑦0, 𝑓(𝑥0, 𝑦0))with𝐻∩ 𝒮_𝐿𝑗 ̸=∅and𝐻is below𝒮_𝐿𝑖and𝒮_𝐿𝑗.

Moreover, such a hyperplane𝐻is unique for any(𝑥₀, 𝑦₀)∈𝐿_𝑖∖𝑋^𝑒.

5.4 Bivariate polynomial functions: a case study

Proof. An example is shown in Figure 5.15. The two blue curves are𝒮_𝐿𝑖 and𝒮_𝐿𝑗. We need to find a hyperplane𝐻through(𝑥0, 𝑦0, 𝑓(𝑥0, 𝑦0))that intersects both𝒮_𝐿𝑖 and𝒮_𝐿𝑗 and at the same time𝐻is below them.

For the special case(𝑥₀, 𝑦₀)∈𝐿_𝑗, we can easily check that𝐻 =𝑇(𝑥₀, 𝑦₀), i.e., the tangent plane at(𝑥0, 𝑦0)fulfills all the conditions. In this case𝐻 is not unique.

Assume that(𝑥₀, 𝑦₀)̸∈𝐿_𝑗. We discuss the case(𝑥₀, 𝑦₀)∈𝐿_𝑖∖𝑋^𝑒, e.g.,(𝑥₀, 𝑦₀) = (𝑥₁, 𝑦₁) in Figure 5.15. Corollary 5.13 implies that a hyperplane𝐻through(𝑥₀, 𝑦₀, 𝑓(𝑥₀, 𝑦₀))which is below𝒮_𝐿𝑖 contains the subtangent plane

Ł(𝑥0, 𝑦0) =𝑇(𝑥0, 𝑦0)∩ {(𝑥, 𝑦, 𝑧)|(𝑥, 𝑦)∈aff{𝐿_𝑖}} (5.18) which is the lower left green line in Figure 5.15. Denote𝑃0 = (𝑥0, 𝑦0, 𝑓(𝑥0, 𝑦0)). For every point𝑃𝑗 = (𝑥𝑗, 𝑦𝑗, 𝑧𝑗) ∈ 𝒮_𝐿𝑗, we define𝐻(Ł(𝑥0, 𝑦0), 𝑃𝑗) = aff{Ł(𝑥₀, 𝑦0),{𝑃_𝑗}}which is a hyperplane below𝑃_𝑗. Similar to the proof of Theorem 5.27, there exists a point𝑃^*∈ 𝒮_𝐿𝑗 such that𝐻^* =𝐻(Ł(𝑥0, 𝑦0), 𝑃^*)is below𝒮_𝐿𝑗. Note that𝐻^* is unique since it is associated to the objective value of an optimization problem introduced in Theorem 5.27 which always has an optimum.

Finally, we discuss the case(𝑥0, 𝑦0) ∈ 𝐿𝑖 ∩𝑋^𝑒, e.g., (𝑥0, 𝑦0) = (𝑥2, 𝑦2) in Figure 5.15.

Consider a lineŁ^𝑙(𝑥0, 𝑦0)⊂ {(𝑥, 𝑦, 𝑧)|(𝑥, 𝑦)∈aff{𝐿_𝑖}}through(𝑥0, 𝑦0, 𝑓(𝑥0, 𝑦0))which is belowŁ(𝑥₀, 𝑦₀)defined by (5.18) such that(𝑥, 𝑦)∈𝐿_𝑖. In the example in Figure 5.15,Ł^𝑙(𝑥₀, 𝑦₀) is the red line andŁ(𝑥0, 𝑦0)is the upper right green line. Every hyperplane𝐻which contains Ł^𝑙(𝑥0, 𝑦0)is through(𝑥0, 𝑦0, 𝑓(𝑥0, 𝑦0))and below𝒮_𝐿𝑖. Similar to the discussion above, there exists a point𝑃^* ∈ 𝒮_𝐿𝑗such that𝐻^*= aff{Ł^𝑙(𝑥₀, 𝑦₀),{𝑃_𝑗}}is below𝒮_𝐿𝑗. Note that for every fixed chosenŁ^𝑙(𝑥₀, 𝑦₀)there exists a unique𝐻^*. However, we have infinitely manyŁ^𝑙(𝑥₀, 𝑦₀)

to choose. 2

Now we discuss how to algorithmically find𝐻 which fulfills Lemma 5.35. Note that for (𝑥₀, 𝑦₀)∈𝐿_𝑖∩𝑋^𝑒we may chooseŁ^𝑙(𝑥₀, 𝑦₀) = Ł(𝑥₀, 𝑦₀)which can be computed easily. For any (𝑥0, 𝑦0)∈𝐿𝑖we compute a hyperplane𝐻that fulfills Lemma 5.35 and satisfies𝐻⊃Ł(𝑥0, 𝑦0) which is a line defined in (5.18). This is equivalent to finding a point(𝑥^*, 𝑦^*)∈𝐿𝑗 such that 𝐻 = aff{Ł(𝑥₀, 𝑦₀),{((𝑥^*, 𝑦^*), 𝑓(𝑥^*, 𝑦^*))}}is below𝒮_𝐿𝑗. Consider the two linesaff{𝐿_𝑖}and aff{𝐿_𝑗}. They are either not parallel or parallel. Examples for both cases are in Figure 5.16.

As mentioned before, for the case(𝑥₀, 𝑦₀) ∈ 𝐿𝑗, we set 𝐻 =𝑇(𝑥₀, 𝑦₀) and we are done.

Otherwise, let

𝐿𝑖 ={(𝑥, 𝑦)|𝑦=𝑎𝑖𝑥+𝑏𝑖, 𝑥∈[𝑥^min_𝑖 , 𝑥^max_𝑖 ]}

and

𝐿_𝑗 ={(𝑥, 𝑦)|𝑦=𝑎_𝑗𝑥+𝑏_𝑗, 𝑥∈[𝑥^min_𝑗 , 𝑥^max_𝑗 ]}.

Define 𝑓𝐿𝑖(𝑥) = 𝑓(𝑥, 𝑎𝑖𝑥 +𝑏𝑖) for 𝑥 ∈ [𝑥^min_𝑖 , 𝑥^max_𝑖 ]and define𝑓𝐿𝑗(𝑥) = 𝑓(𝑥, 𝑎𝑗𝑥 +𝑏𝑗) for𝑥 ∈ [𝑥^min_𝑗 , 𝑥^max_𝑗 ]. 𝑓_𝐿𝑖(𝑥)and𝑓_𝐿𝑗(𝑥) are univariate functions with degree up to3. The

(a)aff{𝐿𝑖}andaff{𝐿𝑗}are not parallel (b)aff{𝐿𝑖}andaff{𝐿𝑗}are parallel Figure 5.16:Two linesaff{𝐿_𝑖}andaff{𝐿_𝑗}can be parallel or not parallel

lineŁ_𝑖(𝑥₀, 𝑦₀) = 𝑇(𝑥₀, 𝑦₀)∩ {(𝑥, 𝑦, 𝑧) |(𝑥, 𝑦) ∈aff{𝐿_𝑖}}for𝑥₀ ∈ [𝑥^min_𝑖 , 𝑥^max_𝑖 ]with𝑦₀ = 𝑎_𝑗𝑥₀+𝑏_𝑗 can also be represented as

Ł_𝑖(𝑥₀, 𝑦₀) ={(𝑥, 𝑦, 𝑧)|𝑥∈R, 𝑦=𝑎_𝑖𝑥+𝑏_𝑖, 𝑧 =𝑓_𝐿^′_𝑖(𝑥₀)(𝑥−𝑥₀) +𝑓_𝐿𝑖(𝑥₀)}. (5.19) Analogously, for every𝑥1 ∈[𝑥^min_𝑗 , 𝑥^max_𝑗 ]and𝑦1 =𝑎𝑗𝑥1+𝑏𝑗, we get

Ł_𝑗(𝑥₁, 𝑦₁) ={(𝑥, 𝑦, 𝑧)|𝑥∈R, 𝑦=𝑎_𝑗𝑥+𝑏_𝑗, 𝑧=𝑓_𝐿^′_𝑗(𝑥₁)(𝑥−𝑥₁) +𝑓_𝐿𝑗(𝑥₁)}.

First we discuss the case thataff{𝐿_𝑖}andaff{𝐿_𝑗}are not parallel, see an example in Fig-ure 5.16a. Since𝑎𝑖 ̸=𝑎𝑗, the intersection ofaff{𝐿_𝑖}andaff{𝐿_𝑗}is(𝑥𝑖𝑗, 𝑦𝑖𝑗)with

𝑥𝑖𝑗 = 𝑏𝑗−𝑏𝑖

𝑎_𝑖−𝑎_𝑗 and𝑦𝑖𝑗 = 𝑏𝑗𝑎𝑖−𝑏𝑖𝑎𝑗

𝑎_𝑖−𝑎_𝑗 . Consider the point𝑃_𝑖𝑗 = (𝑥_𝑖𝑗, 𝑦_𝑖𝑗, 𝑧_𝑖𝑗)with𝑧_𝑖𝑗 =𝑓_𝐿^′

𝑖(𝑥₀)(𝑥_𝑖𝑗 −𝑥₀) +𝑓_𝐿𝑖(𝑥₀). We can check that𝑃_𝑖𝑗 ∈Ł_𝑖(𝑥₀, 𝑦₀)which implies that𝑃_𝑖𝑗 ∈𝐻for every𝐻that fulfills Lemma 5.35. Since𝐻 also intersects𝒮_𝐿𝑗, finding𝐻fulfilling Lemma 5.35 is equivalent to finding a point𝑃^* ∈ 𝒮_𝐿𝑗 such thataff{Ł_𝑖(𝑥₀, 𝑦₀),{𝑃^*}}is below𝒮_𝐿𝑗. Consider the function

𝑔𝑗(𝑥) =𝑓_𝐿^′_𝑗(𝑥)(𝑥𝑖𝑗−𝑥) +𝑓𝐿𝑗(𝑥)

for𝑥 ∈ [𝑥^min_𝑗 , 𝑥^max_𝑗 ]. Note that the point (𝑥1, 𝑎𝑗𝑥1 +𝑏𝑗, 𝑔𝑗(𝑥1))lies in lineŁ_𝑗(𝑥1, 𝑦1) for 𝑥1 ∈ [𝑥^min_𝑗 , 𝑥^max_𝑗 ]. As we analyzed before by considering the sign of the first derivative, 𝑔_𝑗(𝑥)is a strictly decreasing function if 𝑥_𝑖𝑗 < 𝑥^min_𝑗 and is a strictly increasing function if

5.4 Bivariate polynomial functions: a case study

(a) For case𝑧𝑖𝑗< 𝑔𝑗(𝑥^min𝑗 )< 𝑔𝑗(𝑥^max𝑗 ) (b) For case𝑔𝑗(𝑥^max𝑗 )< 𝑔𝑗(𝑥^min𝑗 )< 𝑧𝑖𝑗

(c) For case𝑧𝑖𝑗< 𝑔𝑗(𝑥^max_𝑗 )< 𝑔𝑗(𝑥^min_𝑗 ) (d) For case𝑔𝑗(𝑥^min_𝑗 )< 𝑔𝑗(𝑥^max_𝑗 )< 𝑧𝑖𝑗

Figure 5.17:Example for the4cases in the proof of Lemma 5.35

𝑥𝑖𝑗 > 𝑥^max_𝑗 . There are no other cases since𝑥𝑖𝑗 ̸∈[𝑥^min_𝑗 , 𝑥^max_𝑗 ]. In both cases we have𝑔𝑗(𝑥^min_𝑗 )̸=

𝑔𝑗(𝑥^max_𝑗 ). We compare𝑧𝑖𝑗,𝑔𝑗(𝑥^min_𝑗 )and𝑔𝑗(𝑥^max_𝑗 ). If𝑧𝑖𝑗 is between𝑔𝑗(𝑥^min_𝑗 )and𝑔𝑗(𝑥^max_𝑗 ), i.e., 𝑔_𝑗(𝑥^min_𝑗 ) ≤𝑧_𝑖𝑗 ≤𝑔_𝑗(𝑥^max_𝑗 )or𝑔_𝑗(𝑥^max_𝑗 ) ≤𝑧_𝑖𝑗 ≤𝑔_𝑗(𝑥^min_𝑗 ), then the increasing or decreasing function𝑔𝑗(𝑥)−𝑧𝑖𝑗 has a unique root𝑥^* ∈[𝑥^min_𝑗 , 𝑥^max_𝑗 ]with𝑔𝑗(𝑥^*)−𝑧𝑖𝑗 = 0. Note that since 𝑔𝑗(𝑥)is a polynomial function of degree up to3,𝑥^*can be computed very easily. It implies that the lineŁ_𝑗(𝑥^*, 𝑦^*)with𝑦^* =𝑎_𝑗𝑥^*+𝑏_𝑗contains also𝑃_𝑖𝑗 and

𝐻 = aff{Ł_𝑖(𝑥₀, 𝑦₀),Ł_𝑗(𝑥^*, 𝑦^*)}

is a hyperplane which fulfills Lemma 5.35.

Otherwise, if𝑧_𝑖𝑗 is not between𝑔_𝑗(𝑥^min_𝑗 )and𝑔_𝑗(𝑥^max_𝑗 ), we have the following four cases 1. 𝑧𝑖𝑗 < 𝑔𝑗(𝑥^min_𝑗 )< 𝑔𝑗(𝑥^max_𝑗 ),

2. 𝑔_𝑗(𝑥^max_𝑗 )< 𝑔_𝑗(𝑥^min_𝑗 )< 𝑧_𝑖𝑗,

3. 𝑧_𝑖𝑗 < 𝑔_𝑗(𝑥^max_𝑗 )< 𝑔_𝑗(𝑥^min_𝑗 ), 4. 𝑔_𝑗(𝑥^min_𝑗 )< 𝑔_𝑗(𝑥^max_𝑗 )< 𝑧_𝑖𝑗.

Examples for the four cases are shown in Figure 5.17. In the first two cases we set 𝑃^*= (𝑥^min_𝑗 , 𝑎_𝑗𝑥^min_𝑗 +𝑏_𝑗, 𝑓_𝐿𝑗(𝑥^min_𝑗 )).

In the last two cases we set

𝑃^* = (𝑥^max_𝑗 , 𝑎_𝑗𝑥^max_𝑗 +𝑏_𝑗, 𝑓_𝐿𝑗(𝑥^max_𝑗 )).

For all four cases, the lineaff{𝑃^*, 𝑃^𝑖𝑗}, shown as the red line in the corresponding subgraphs, is below𝒮_𝐿𝑗 (the corresponding blue curve). We then have 𝐻 = aff{Ł_𝑖(𝑥₀, 𝑦₀),{𝑃^*}} ⊃ aff{𝑃^*, 𝑃^𝑖𝑗}which fulfills Lemma 5.35 and is the hyperplane we are looking for.

Now it only remains to discuss the case thataff{𝐿_𝑖}andaff{𝐿_𝑗}are parallel, see an example in Figure 5.16b. Note that 𝑎_𝑖 = 𝑎_𝑗 and for any 𝑥₁ ∈ [𝑥^min_𝑗 , 𝑥^max_𝑗 ] with 𝑦₁ = 𝑎_𝑗𝑥₁ +𝑏_𝑗, Ł_𝑗(𝑥1, 𝑦1)is contained in the hyperplane{(𝑥, 𝑦, 𝑧) | 𝑦 = 𝑎𝑗𝑥+𝑏𝑗}. The other hyperplane {(𝑥, 𝑦, 𝑧)|𝑦=𝑎𝑖𝑥+𝑏𝑖}which containsŁ_𝑖(𝑥0, 𝑦0)is parallel to the hyperplane{(𝑥, 𝑦, 𝑧)|𝑦= 𝑎_𝑗𝑥+𝑏_𝑗}. It implies thataff{Ł_𝑖(𝑥₀, 𝑦₀),Ł_𝑗(𝑥₁, 𝑦₁)}is a hyperplane if and only ifŁ_𝑖(𝑥₀, 𝑦₀) andŁ_𝑗(𝑥1, 𝑦1)are parallel. This is equivalent to 𝑓_𝐿^′_𝑗(𝑥1) = 𝑓_𝐿^′_𝑖(𝑥0). Note that𝑓𝐿𝑗(𝑥) is a convex function for𝑥∈[𝑥^min_𝑗 , 𝑥^max_𝑗 ]and is a polynomial function with degree up to3. Then 𝑓_𝐿^′

𝑗(𝑥)is an increasing function with𝑓_𝐿^′

𝑗(𝑥^min_𝑗 ) < 𝑓_𝐿^′

𝑗(𝑥^max_𝑗 ). Compare𝑓_𝐿^′

𝑖(𝑥₀),𝑓_𝐿^′

𝑗(𝑥^min_𝑗 ) and𝑓_𝐿^′_𝑗(𝑥^max_𝑗 ). If𝑓_𝐿^′_𝑗(𝑥^min_𝑗 ) ≤ 𝑓_𝐿^′_𝑖(𝑥0) ≤ 𝑓_𝐿^′_𝑗(𝑥^max_𝑗 ), then the function 𝑓_𝐿^′_𝑗(𝑥) −𝑓_𝐿^′_𝑖(𝑥0) has a unique root 𝑥^* for𝑥 ∈ [𝑥^min_𝑗 , 𝑥^max_𝑗 ]. The function 𝑓_𝐿^′_𝑗(𝑥) is a polynomial function with degree up to2, for which we can easily compute the root𝑥^*. In this case we set𝑃^* = (𝑥^*, 𝑎_𝑗𝑥^* +𝑏_𝑗, 𝑓_𝐿𝑗(𝑥^*)). See the example in Figure 5.16b again. Otherwise if 𝑓_𝐿^′

𝑖(𝑥₀) <

𝑓_𝐿^′_𝑗(𝑥^min_𝑗 ) we set 𝑃^* = (𝑥^min_𝑗 , 𝑎𝑗𝑥^min_𝑗 +𝑏𝑗, 𝑓𝐿𝑗(𝑥^min_𝑗 ))and if𝑓_𝐿^′_𝑖(𝑥0) > 𝑓_𝐿^′_𝑗(𝑥^max_𝑗 )we set 𝑃^* = (𝑥^max_𝑗 , 𝑎𝑗𝑥^max_𝑗 +𝑏𝑗, 𝑓𝐿𝑗(𝑥^max_𝑗 )). Examples for both cases are shown in Figure 5.18. We can easily prove that the lineaff{Ł_𝑖(𝑥₀, 𝑦₀),{𝑃^*}} ∩ {(𝑥, 𝑦, 𝑧)|(𝑥, 𝑦)∈aff{𝐿_𝑗}}, shown as the red line in the corresponding subgraphs, is below𝒮_𝐿𝑗 (the corresponding blue curve). Thus 𝐻= aff{Ł_𝑖(𝑥0, 𝑦0),{𝑃^*}}fulfills the requirements of Lemma 5.35 and is the hyperplane we are looking for.

Consequently, for all cases above, we can always find a unique point𝑃^*∈ 𝒮_𝐿𝑗such that𝐻 = aff{Ł_𝑖(𝑥0, 𝑦0),{𝑃^*}}fulfills Lemma 5.35. Note that𝑃_𝑖^min = (𝑥^min_𝑖 , 𝑎𝑖𝑥^min_𝑖 +𝑏𝑖, 𝑓𝐿𝑖(𝑥^min_𝑖 )) and𝑃_𝑖^max= (𝑥^max_𝑖 , 𝑎_𝑖𝑥^max_𝑖 +𝑏_𝑖, 𝑓_𝐿𝑖(𝑥^max_𝑖 ))are two different points on𝒮_𝐿𝑖. Since𝑃_𝑖^min,𝑃_𝑖^max and𝑃^* do not lie on the same line, we set𝐻 = aff{𝑃_𝑖^min, 𝑃_𝑖^max, 𝑃^*}. We can compute𝐻by solving a system of linear equations. The algorithm is summarized in Algorithm 5.1.

Define𝜕𝒮 = {(𝑥, 𝑦, 𝑧) |(𝑥, 𝑦) ∈ 𝜕𝑋, 𝑧 = 𝑓(𝑥, 𝑦)}and for each𝑖∈ {1, . . . , 𝑚}and the corresponding facet𝐹_𝑖we further define𝒮_𝐹𝑖 ={(𝑥, 𝑦, 𝑧)|(𝑥, 𝑦)∈𝐹_𝑖, 𝑧=𝑓(𝑥, 𝑦)}.

Lemma 5.36

For any𝐿𝑖 ⊂ 𝑋ˇ^𝑔∩𝜕𝑋 with𝑖∈ {1, . . . , 𝑚₁}and for any(𝑥0, 𝑦0) ∈𝐿𝑖 ⊂ 𝐹𝑖, there exists a hyperplane𝐻

5.4 Bivariate polynomial functions: a case study

Algorithm 5.1:Algorithm that computes a hyperplane that intersects𝒮_𝐿𝑖,𝒮_𝐿𝑗and is below them

Input: Line segments𝐿_𝑖and𝐿_𝑗with𝑖, 𝑗∈ {1, . . . , 𝑚₁}, a pointx⁰= (𝑥₀, 𝑦₀)∈𝐿_𝑖 Output: The unique hyperplane𝐻(x⁰, 𝐿𝑖, 𝐿𝑗)with𝐻⊃Ł_𝑖(𝑥0, 𝑦0),𝐻∩ 𝒮_𝐿𝑗 ̸=∅and𝐻

is below𝒮_𝐿𝑖and𝒮_𝐿𝑗

1 ifaff{𝐿_𝑖}andaff{𝐿_𝑗}are not parallelthen

2 ifx⁰ ∈𝐿_𝑗then

3 return𝑇(x₀)

4 end

5 Compute the intersection point(𝑥_𝑖𝑗, 𝑦_𝑖𝑗), value𝑔_𝑗(𝑥^min_𝑗 )and𝑔_𝑗(𝑥^max_𝑗 );

6 Compute𝑧𝑖𝑗 =𝑓_𝐿^′_𝑖(𝑥0)(𝑥𝑖𝑗 −𝑥0) +𝑓𝐿𝑖(𝑥0)and set𝑃𝑖𝑗 = (𝑥𝑖𝑗, 𝑦𝑖𝑗, 𝑧𝑖𝑗);

7 if𝑔𝑗(𝑥^min_𝑗 )≤𝑧𝑖𝑗 ≤𝑔𝑗(𝑥^max_𝑗 )or𝑔𝑗(𝑥^max_𝑗 )≤𝑧𝑖𝑗 ≤𝑔𝑗(𝑥^min_𝑗 )then

8 Compute the unique root𝑥^*of𝑔𝑗(𝑥)−𝑧𝑖𝑗 = 0for𝑥∈[𝑥^min_𝑗 , 𝑥^max_𝑗 ];

9 Set𝑃^* = (𝑥^*, 𝑎_𝑗𝑥^*+𝑏_𝑗, 𝑓_𝐿𝑗(𝑥^*))

10 end

11 if𝑧_𝑖𝑗 < 𝑔_𝑗(𝑥^min_𝑗 )< 𝑔_𝑗(𝑥^max_𝑗 )or𝑔_𝑗(𝑥^max_𝑗 )< 𝑔_𝑗(𝑥^min_𝑗 )< 𝑧_𝑖𝑗 then

12 Set𝑃^* = (𝑥^min_𝑗 , 𝑎_𝑗𝑥^min_𝑗 +𝑏_𝑗, 𝑓_𝐿𝑗(𝑥^min_𝑗 ))

13 end

14 if𝑧𝑖𝑗 < 𝑔𝑗(𝑥^max_𝑗 )< 𝑔𝑗(𝑥^min_𝑗 )or𝑔𝑗(𝑥^min_𝑗 )< 𝑔𝑗(𝑥^max_𝑗 )< 𝑧𝑖𝑗 then

15 Set𝑃^* = (𝑥^max_𝑗 , 𝑎𝑗𝑥^max_𝑗 +𝑏𝑗, 𝑓𝐿𝑗(𝑥^max_𝑗 ))

16 end

17 end

18 else

19 Compute𝑓_𝐿^′_𝑗(𝑥^min_𝑗 ), 𝑓_𝐿^′_𝑗(𝑥^max_𝑗 )and𝑓_𝐿^′_𝑖(𝑥0);

20 if𝑓_𝐿^′_𝑗(𝑥^min_𝑗 )≤𝑓_𝐿^′_𝑖(𝑥0)≤𝑓_𝐿^′_𝑗(𝑥^max_𝑗 )then

21 Compute the unique root𝑥^*of𝑓_𝐿^′

𝑗(𝑥)−𝑓_𝐿^′

𝑖(𝑥₀)for𝑥∈[𝑥^min_𝑗 , 𝑥^max_𝑗 ];

22 Set𝑃^* = (𝑥^*, 𝑎𝑗𝑥^*+𝑏𝑗, 𝑓_𝐿𝑗(𝑥^*)).

23 end

24 if𝑓_𝐿^′

𝑖(𝑥₀)< 𝑓_𝐿^′

𝑗(𝑥^min_𝑗 )then

25 Set𝑃^* = (𝑥^min_𝑗 , 𝑎_𝑗𝑥^min_𝑗 +𝑏_𝑗, 𝑓_𝐿𝑗(𝑥^min_𝑗 ))

26 end

27 if𝑓_𝐿^′

𝑗(𝑥^max_𝑗 )< 𝑓_𝐿^′

𝑖(𝑥₀)then

28 Set𝑃^* = (𝑥^max_𝑗 , 𝑎𝑗𝑥^max_𝑗 +𝑏𝑗, 𝑓𝐿𝑗(𝑥^max_𝑗 ))

29 end

30 end

31 ComputeŁ_𝑖(𝑥₀, 𝑦₀)

32 Compute𝐻((𝑥₀, 𝑦₀), 𝐿_𝑖, 𝐿_𝑗) = aff{Ł_𝑖(𝑥₀, 𝑦₀),{𝑃^*}}and return𝐻((𝑥₀, 𝑦₀), 𝐿_𝑖, 𝐿_𝑗)

(a) For the case𝑓_𝐿^′_𝑖(𝑥0)< 𝑓_𝐿^′_𝑗(𝑥^min_𝑗 ) (b) For the case𝑓_𝐿^′_𝑖(𝑥0)> 𝑓_𝐿^′_𝑗(𝑥^max_𝑗 ) Figure 5.18:Two cases by𝑓_𝐿^′

𝑖(𝑥₀)̸∈[𝑓_𝐿^′

𝑗(𝑥^min_𝑗 ), 𝑓_𝐿^′

𝑗(𝑥^max_𝑗 )] in Algorithm 5.1

1. either with𝐻=𝑇(𝑥₀, 𝑦₀) 2. or with(𝐻∩𝜕𝒮)∖ 𝒮_𝐹𝑖 ̸=∅

such that𝐻 ⊃Ł_𝑖(𝑥0, 𝑦0). In addition,𝐻is below𝒮over(𝑥, 𝑦)∈𝑋ˇ^𝑔∩𝜕𝑋. Proof. We develop an algorithm to find the hyperplane𝐻.

Denote aff{𝐹_𝑖} = {(𝑥, 𝑦) | 𝑎_𝑖𝑥+𝑏_𝑖𝑥 = 𝑐_𝑖} with three constants 𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖 ∈ R. Since 𝐹𝑖 is a facet of𝑋, we have𝑋 ⊂ {(𝑥, 𝑦) | 𝑎𝑖𝑥+𝑏𝑖𝑥 ≥ 𝑐𝑖}or𝑋 ⊂ {(𝑥, 𝑦) | 𝑎𝑖𝑥+𝑏𝑖𝑥 ≤ 𝑐𝑖}. Lemma 5.18 implies that for any two nonvertical 𝐻1, 𝐻2 with 𝐻1 ⊃ Ł_𝑖(𝑥0, 𝑦0) and 𝐻₂ ⊃ Ł_𝑖(𝑥₀, 𝑦₀), either 𝐻₁ is below 𝐻₂ over 𝑋 or𝐻₂ is below 𝐻₁ over 𝑋. Recall that 𝑋ˇ^𝑔∩𝜕𝑋 =𝐿1∪ · · · ∪𝐿𝑚1∪ {x^𝑒₁, . . . ,x^𝑒_𝑚2}. For every𝐿𝑗 with𝑗̸=𝑖and𝑗 ∈ {1, . . . , 𝑚₁}, compute𝐻_𝑗^𝐿 = 𝐻((𝑥0, 𝑦0), 𝐿𝑖, 𝐿𝑗)as output of Algorithm 5.1. For everyx^𝑒_𝑗 with x_𝑗^𝑒 ̸∈ 𝐹𝑖

and𝑗 ∈ {1, . . . , 𝑚₂}, compute𝐻_𝑗^𝑒 = aff{Ł_𝑖(𝑥₀, 𝑦₀),{(x_𝑗^𝑒, 𝑓(x^𝑒_𝑗))}}which is a nonvertical hyperplane since(x^𝑒_𝑗, 𝑓(x^𝑒_𝑗))̸∈Ł_𝑖(𝑥0, 𝑦0)andx^𝑒_𝑗 ̸∈𝐹𝑖. Consider the set

ℋ(𝑥₀, 𝑦₀) ={𝐻_𝑗^𝐿|𝑗̸=𝑖, 𝑗∈ {1, . . . , 𝑚₁}} ∪ {𝐻_𝑗^𝑒|x^𝑒_𝑗 ̸∈𝐹𝑖, 𝑗 ∈ {1, . . . , 𝑚₂}}

of finitely many hyperplanes that all containŁ_𝑖(𝑥0, 𝑦0). There exists𝐻^* ∈ ℋ(𝑥₀, 𝑦0)such that𝐻^*is below𝐻over𝑋for every𝐻 ∈ ℋ(𝑥₀, 𝑦₀). The hyperplane𝐻^*is below𝒮_𝐹𝑖 since Ł_𝑖(𝑥₀, 𝑦₀)⊂𝐻^*which is below𝒮_𝐹𝑖. The hyperplane𝐻^*is below every𝒮_𝐹𝑗 with𝑗 ̸=𝑖since 𝐻^*is below𝐻_𝑗^𝐿over𝐹𝑗 ⊂𝑋and𝐻_𝑗^𝐿is below𝒮_𝐹𝑗. Similarly,𝐻^*is below every(x^𝑒_𝑗, 𝑓(x^𝑒_𝑗)) since𝐻^*is below𝐻_𝑗^𝑒over𝑋 ∋x^𝑒_𝑗. Thus𝐻^*is below𝒮over(𝑥, 𝑦)∈𝑋ˇ^𝑔∩𝜕𝑋.

5.4 Bivariate polynomial functions: a case study

If𝐻^* =𝐻_𝑘^𝐿for some𝑘∈ {1, . . . , 𝑚₁}, Algorithm 5.1 implies that either𝐻^* =𝑇(𝑥₀, 𝑦₀) or there exists a pointx_𝑘∈𝐿𝑘withx_𝑘̸∈𝐹𝑖and(x_𝑘, 𝑓(x_𝑘))∈𝐻^*. Otherwise if𝐻^* =𝐻_𝑘^𝑒, there exists a pointx_𝑘∈𝑋^𝑒withx_𝑘 ̸∈𝐹𝑖and(x_𝑘, 𝑓(x_𝑘))∈𝐻^*.

In all cases either𝐻 =𝑇(𝑥₀, 𝑦₀)or(x_𝑘, 𝑓(x_𝑘))∈(𝐻∩𝜕𝒮)∖ 𝒮_𝐹𝑖 ̸=∅. 2 Theorem 5.37

For any𝐿𝑖 ⊂𝑋ˇ^𝑔∩𝜕𝑋 with𝑖∈ {1, . . . , 𝑚₁}and for any(𝑥0, 𝑦0)∈𝐿𝑖 ⊂𝐹𝑖, a hyperplane𝐻^* which fulfills Lemma 5.36 is either a tight valid hyperplane or there exists a tight valid hyperplane 𝐻^**which is parallel to𝐻^*. Furthermore,𝐻^*is always a tight valid hyperplane if the Hessian matrix is negative semidefinite, i.e., it satisfies

𝐻(𝑓)(𝑥, 𝑦) = (︃ _𝜕2

𝜕𝑥²𝑓 _{𝜕𝑥𝜕𝑦}^𝜕2 𝑓

𝜕²

𝜕𝑥𝜕𝑦𝑓 _𝜕𝑦^𝜕22𝑓 )︃

⪯0 (5.20)

for all(𝑥, 𝑦)∈int𝑋.

Proof. Denote 𝐻^* = {(𝑥, 𝑦, 𝑧) | 𝑧 = 𝑎_𝑖𝑥 +𝑏_𝑖𝑦 +𝑐_𝑖}with 𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖 ∈ R. Consider the optimization problem

min

(𝑥,𝑦)∈𝑋𝑓(𝑥, 𝑦)−(𝑎_𝑖𝑥+𝑏_𝑖𝑦+𝑐_𝑖) (OP^min_𝐻^*) that has a minimum𝑧^*, since𝑋is a compact set and𝑓(𝑥, 𝑦)−(𝑎𝑖𝑥+𝑏𝑖𝑦+𝑐𝑖)is a continuous function. Note that𝑧^* ≤0since𝒮 ∩𝐻^*̸=∅. The hyperplane𝐻^*is valid if and only if𝑧^* = 0. The maximally valid subtangent plane𝑇_𝐻^max(𝑥₀, 𝑦₀)isŁ_𝑖(𝑥₀, 𝑦₀)and there exists another point (𝑥1, 𝑦1, 𝑧1)∈ 𝒮 ∩𝐻^*with(𝑥1, 𝑦1, 𝑧1)̸∈aff{Ł_𝑖(𝑥0, 𝑦0)}. Recalling the definition of tight valid

hyperplanes in Section 5.3, we have

aff{Ł_𝑖(𝑥0, 𝑦0),{(𝑥₁, 𝑦1, 𝑧1)}} ⊂aff^{︁𝑇_𝐻^max* (x⁰) : for allx⁰∈𝑋^𝐻*}︁=:𝑆^𝐻* which implies

2 = dim aff{Ł_𝑖(𝑥0, 𝑦0),{(𝑥₁, 𝑦1, 𝑧1)}} ≤dim𝑆^𝐻* ≤2.

Hence𝑆^𝐻* is a hyperplane which implies𝐻^*is a tight valid hyperplane. Otherwise we have 𝑧^* <0with solution(𝑥^*, 𝑦^*). Note that(𝑥^*, 𝑦^*)must be an interior point of𝑋 since𝐻^*is below𝒮over𝑋ˇ^𝑔∩𝜕𝑋. We have then𝑇(𝑥^*, 𝑦^*) =𝑎_𝑖𝑥+𝑏_𝑖𝑦+𝑐_𝑖+𝑧^*which is a tight valid hyperplane and is parallel to𝐻^*.

Every globally convex interior point is also a locally convex interior point. According to [Edw94] and Lemma 5.5, (𝑥0, 𝑦0)satisfies𝐻(𝑓)(𝑥0, 𝑦0) ⪰ 0. If (5.20) is satisfied for all (𝑥, 𝑦) ∈ int𝑋 then we have𝑋ˇ^𝑔 ∩int𝑋 = ∅ which implies that 𝐻^* is below 𝒮 over𝑋ˇ^𝑔. Lemma 5.20 implies that𝐻^*is valid. As we discussed above,𝐻^* is tight if it is valid. 2

The hyperplanes fulfilling Lemma 5.36 arepotentiallytight valid hyperplanes since we only need to check if the corresponding optimization problem (OP^min_𝐻^*) has the minimum𝑧^* = 0. Until now we have only consideredpotentiallytight valid hyperplanes that containŁ_𝑖(𝑥0, 𝑦0)for an(𝑥₀, 𝑦₀)in an𝐿_𝑖. In order to show that there exists otherpotentiallytight valid hyperplanes 𝐻which are below𝑓(𝑥, 𝑦)over𝑋ˇ^𝑔∩𝜕𝑋we give the following definition.

Definition 5.38 (Potentially tight valid hyperplanes of type𝐴and type𝐵)

A hyperplane which fulfills Lemma 5.36 is apotentially tightvalid hyperplanes oftype𝐴. A hyperplane𝐻is apotentially tightvalid hyperplane oftype𝐵if

• 𝐻 is below𝑓(𝑥, 𝑦)over𝑋ˇ^𝑔∩𝜕𝑋,

• it satisfies𝜋x(𝐻∩𝜕𝒮)⊂𝑋^𝑒and|𝐻∩𝜕𝒮| ≥3and

• there does not exist𝐿𝑖 withx^𝑒∈𝐿𝑖andx^𝑒∈𝜋x(𝐻∩𝜕𝒮)such that𝐻⊃Ł_𝑖(x^𝑒). Due to the last condition in the definition of potentially tight valid hyperplanes of type𝐵, the set of potentially tight valid hyperplanes of type𝐴and the set of type𝐵are disjoint.

Corollary 5.39

Let𝐻^*be a potentially tight valid hyperplane of type𝐵. Then𝐻^*is either a tight valid hyperplane or there exists a tight valid hyperplane𝐻^**which is parallel to𝐻^*. Furthermore,𝐻^*is always a tight valid hyperplane if every(𝑥, 𝑦)∈int𝑋satisfies (5.20).

Proof. The proof is the same as the proof of Theorem 5.37. 2 Now we discuss how to compute potentially tight valid hyperplanes𝐻^**of type𝐵 algo-rithmically. Note that every such𝐻satisfies𝜋x(𝐻∩𝜕𝒮)⊂𝑋^𝑒and|𝐻∩𝜕𝒮| ≥3. As every three pointsx^𝑖,x^𝑗,x^𝑘 ∈𝑋^𝑒 with1 ≤𝑖 < 𝑗 < 𝑘 ≤ 𝑚do not lie on a same line, it implies that𝐻^𝑖𝑗𝑘= aff{(x^𝑖, 𝑓(x^𝑖),(x^𝑗, 𝑓(x^𝑗),(x^𝑘, 𝑓(x^𝑘)}is a hyperplane. There are^(︀𝑚₃^)︀such hyper-planes𝐻^𝑖𝑗𝑘. We can easily prove that a given𝐻^𝑖𝑗𝑘is a potentially tight valid hyperplane𝐻^**

of type𝐵if and only if

• for everyx^𝑙 ∈𝑋^𝑒,𝐻^𝑖𝑗𝑘is below(x^𝑙, 𝑓(x^𝑙));

• for every𝐿𝑖,𝐻^𝑖𝑗𝑘is below𝒮_𝐿𝑖, for this we need only to compare the curve𝒮_𝐿𝑖defined by a polynomial of degree up to3and the line segment𝐻^𝑖𝑗𝑘∩ {(x, 𝑧)|x∈𝐿_𝑖};

• for any𝐿𝑘containingx^𝑠,𝑠∈ {𝑖, 𝑗, 𝑘}, check if it fulfillsŁ_𝑖(x^𝑙)̸⊂𝐻^𝑖𝑗𝑘.

All the three conditions above can be checked easily. Thus we design Algorithm 5.3 to compute potentially tight valid hyperplanes of type𝐵.

The proof of Lemma 5.36 describes an algorithm to compute the unique potentially tight valid hyperplane𝐻^* of type𝐴that containsŁ_𝑖(x₀), denoted by𝐻^*(x₀, 𝐿𝑖). Note that we cannot omit𝐿_𝑖 in the notation since there may existx_𝑘 ∈𝑋^𝑒withx_𝑘 ∈𝐿_𝑖,x_𝑘 ∈𝐿_𝑗,𝑖̸=𝑗

Im Dokument Optimal Operation of Water Supply Networks by Mixed Integer Nonlinear Programming and Algebraic Methods (Seite 115-129)