Convex Hull of Graphs of Polynomial Functions
5.4 Bivariate polynomial functions: a case study
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
๐(๐ฅ, ๐ฆ) = โ๏ธ
0โค๐,๐โค3 0โค๐+๐โค3
๐๐๐๐ฅ๐๐ฆ๐, (5.17)
where all๐๐๐ โRare constants and๐ โR2is the domain which is a polytope. Then๐is a convex polygon with๐โฅ3edges and vertices. Every edge is a line segment as well as a facet of๐and every vertex is an extreme point of๐.
Again, we only consider the downward closed part. Recall that๐ห๐is the set of all globally convex domain points and๐ห๐is the set of all locally convex domain points. Theoretically, for anyx0 โint๐we need only to check if๐(x0)is valid. However, in practice, this is not easy even for๐ 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 function๐ on a facet of๐is isomorphic to the graph of a univariate polynomial function on a corresponding projected domain. We show later that finding๐ห๐ for univariate polynomial functions with
degreeโค 3is tractable. Thus we can easily find the set๐ห๐โฉ๐๐ for bivariate polynomial functions with degreeโค3. In the following we design algorithms which first compute a few hyperplanes that are below๐ฎover๐ห๐โฉ๐๐. For each of the hyperplanes which are below the boundary of๐ฎ, 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(๐ฅ, ๐ฆ, ๐ง)to denote a point inR3. Similar to Section 5.3, we usex= (๐ฅ, ๐ฆ)โR2to denote domain points and use e.g.,x0 = (๐ฅ0, ๐ฆ0)โR2 to denote a certain domain point.
For a boundary point(๐ฅ, ๐ฆ) โ๐๐ there exists at least one facet๐น๐ of๐with(๐ฅ, ๐ฆ) โ๐น๐. Since๐น๐is a line segment, it must be contained in a line denoted by
{(๐ฅ, ๐ฆ)|๐๐๐ฅ+๐๐๐ฆ+๐๐ = 0}=: aff{๐น๐},
where๐๐, ๐๐, ๐๐ โ Rare constants and at least one of๐๐ and๐๐ is nonzero. Without loss of generality we assume๐๐ ฬธ= 0(otherwise permute๐ฅand๐ฆ) and set๐๐ = 1(otherwise scale๐๐, ๐๐
and๐๐). Facet๐น๐can be then be represented as
๐น๐ ={(๐ฅ, ๐ฆ)|๐ฆ=โ๐๐๐ฅโ๐๐, ๐ฅโ[๐ฅmin๐ , ๐ฅmax๐ ]},
where๐ฅmin๐ , ๐ฅmax๐ โRare constants with๐ฅmin๐ < ๐ฅmax๐ . Recalling the definitions in Section 5.3 and using the same notations, we have the projection map
๐๐: aff{๐น๐} โR,(๐ฅ, ๐ฆ)โฆโ๐ฅ and its inverse map
๐๐โ1 :Rโaff{๐น๐}, ๐ฅโฆโ
(๏ธ ๐ฅ
โ๐๐๐ฅโ๐๐
)๏ธ
as well as
๐๐(๐ฅ) =๐(๐ฅ,โ๐๐๐ฅโ๐๐) = โ๏ธ
0โค๐,๐โค3 0โค๐+๐โค3
๐๐๐๐ฅ๐(โ๐๐๐ฅโ๐๐)๐ =๐๐ฅ3+๐๐ฅ2+๐๐ฅ+๐,
where๐, ๐, ๐, ๐are constants depending on๐๐,๐๐and all๐๐๐. 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(๐ฅ0, ๐ฆ0)on facet๐น๐of๐is globally convex for๐ฎ if and only if๐ฅ0is globally convex for the graph of๐๐(๐ฅ)over[๐ฅmin๐ , ๐ฅmax๐ ].
Proof. The result is a special case of Theorem 5.12. 2 Let๐ห๐๐ โ[๐ฅmin๐ , ๐ฅmax๐ ]denote the set of all locally convex domain points for the graph of ๐๐(๐ฅ)and๐ห๐๐ โ๐ห๐๐the set of the globally convex domain points. Note that๐ห๐โฉ๐น๐ =๐๐โ1( ห๐๐๐). Thus, finding๐ห๐๐for every๐โ {1, . . . , ๐}will find๐ห๐โฉ๐๐.
Lemma 5.33
The set of globally convex domain points๐ห๐๐ โ[๐ฅmin๐ , ๐ฅmax๐ ]has one of the four following forms 1. {๐ฅmin๐ , ๐ฅmax๐ },
2. [๐ฅmin๐ , ๐ฅmax๐ ],
3. [๐ฅmin๐ , ๐ฅmid๐ ]โช {๐ฅmax๐ }, 4. {๐ฅmin๐ } โช[๐ฅmid๐ , ๐ฅmax๐ ].
In the two latter cases,๐ฅmid๐ is a constant with๐ฅmin๐ < ๐ฅmid๐ < ๐ฅmax๐ .
Proof. If๐= 0, then๐๐(๐ฅ)is a convex function (if๐โฅ0) or a concave function (if๐โค0). For this reason, we only need to consider the case๐ฬธ= 0. We first seek the locally convex points๐ฅ0 since every globally convex point is also locally convex. Let๐๐(๐)denote the๐th derivative of ๐๐. We have
๐๐(1)(๐ฅ) = 3๐๐ฅ2+ 2๐๐ฅ+๐, ๐๐(2)(๐ฅ) = 6๐๐ฅ+ 2๐, ๐๐(3)(๐ฅ) = 6๐ฬธ= 0, ๐๐(๐)(๐ฅ) = 0for all๐โฅ4.
Similar to the proof of Lemma 5.5, using Taylorโs Formula, we can easily prove that for any ๐ฅโ(๐ฅmin๐ , ๐ฅmax๐ ),๐ฅis locally convex if๐๐(2)(๐ฅ) = 6๐๐ฅ+ 2๐ >0. Note that the extreme points ๐ฅmin๐ and๐ฅmax๐ are globally convex and thus locally convex which is implied by Corollary 5.22.
Since๐๐(2)(๐ฅ) = 6๐๐ฅ+ 2๐is a monotonic function and has at most one root, depending on the value of๐,๐ ๐ฅmin๐ and๐ฅmax๐ , the set of locally convex domain๐ห๐๐has one of the following four forms:
1. {๐ฅmin๐ , ๐ฅmax๐ }, 2. [๐ฅmin๐ , ๐ฅmax๐ ],
3. [๐ฅmin๐ ,โ๐/3๐)โช {๐ฅmax๐ },
4. {๐ฅmin๐ } โช(โ๐/3๐, ๐ฅmax๐ ].
We now discuss the set๐ห๐๐with the four cases above.
1. It is clear that๐ห๐๐ ={๐ฅmin๐ , ๐ฅmax๐ }if๐ห๐๐={๐ฅmin๐ , ๐ฅmax๐ }.
2. If๐ห๐๐ = [๐ฅmin๐ , ๐ฅmax๐ ], then๐๐(2)(๐ฅ)โฅ0for all๐ฅโ[๐ฅmin๐ , ๐ฅmax๐ ]which implies that๐๐(๐ฅ) is a convex function with domain[๐ฅmin๐ , ๐ฅmax๐ ]. Since๐๐(๐ฅ)is differentiable, the tangent plane{(๐ฅ, ๐ฆ)|๐ฆ= (3๐๐ฅ20+ 2๐๐ฅ0+๐)(๐ฅโ๐ฅ0) +๐๐(๐ฅ0)}at point(๐ฅ0, ๐๐(๐ฅ0))for every ๐ฅ0โ๐น๐ is below the graph of๐๐over๐น๐. Hence๐ห๐๐ = [๐ฅmin๐ , ๐ฅmax๐ ].
3. Examples of this case can be seen in Figure 5.14. Note that๐ห๐๐ โ[๐ฅmin๐ ,โ๐/3๐)โช {๐ฅmax๐ } and{๐ฅmin๐ , ๐ฅmax๐ } โ๐ห๐๐. Theorem 5.8 implies that๐ฅ0โ(๐ฅmin๐ ,โ๐/3๐)is globally convex if and only if the corresponding tangent plane
๐๐(๐ฅ0) ={(๐ฅ, ๐ฆ)|๐ฆ= (3๐๐ฅ20+ 2๐๐ฅ0+๐)(๐ฅโ๐ฅ0) +๐๐(๐ฅ0)}
is valid. Note that ๐๐(๐ฅ0) is below the graph of ๐๐ in [๐ฅmin๐ ,โ๐/3๐]. With ๐ห๐๐ โ [๐ฅmin๐ ,โ๐/3๐)โช {๐ฅmax๐ }, Lemma 5.20 implies that๐ฅ0 โ(๐ฅmin๐ ,โ๐/3๐)is globally convex if and only if๐๐(๐ฅ0)is below the point(๐ฅmax๐ , ๐๐(๐ฅmax๐ )). Define
๐max(๐ฅ) = (3๐๐ฅ2+ 2๐๐ฅ+๐)(๐ฅmax๐ โ๐ฅ) +๐๐(๐ฅ)
such that point (๐ฅmax๐ , ๐max(๐ฅ)) โ ๐๐(๐ฅ) for any ๐ฅ โ [๐ฅmin๐ ,โ๐/3๐]. The tangent plane ๐๐(๐ฅ0)for๐ฅ0 โ [๐ฅmin๐ ,โ๐/3๐]is below the point(๐ฅmax๐ , ๐๐(๐ฅmax๐ ))if and only if๐max(๐ฅ0) โค ๐๐(๐ฅmax๐ ). Thus we only need to compare๐max(๐ฅ0)and๐๐(๐ฅmax๐ ). Con-sider the first derivative of๐max(๐ฅ)
๐(1)max(๐ฅ) = (๐ฅmax๐ โ๐ฅ)(6๐๐ฅ+ 2๐๐ฅ)โ(3๐๐ฅ2+ 2๐๐ฅ+๐) +๐๐(1)(๐ฅ)
= (๐ฅmax๐ โ๐ฅ)(6๐๐ฅ+ 2๐๐ฅ).
Thus,๐maxis strictly increasing on[๐ฅmin๐ ,โ๐/3๐)since we have๐(1)max(๐ฅ)>0for any๐ฅโ [๐ฅmin๐ ,โ๐/3๐); similarly๐maxis strictly decreasing on(โ๐/3๐, ๐ฅmax๐ )since๐max(1) (๐ฅ)<0 for any๐ฅโ[๐ฅmin๐ ,โ๐/3๐). It is then clear that
๐max(โ๐/3๐)> ๐max(๐ฅmax๐ ) =๐๐(๐ฅmax๐ ).
Now we compare๐max(๐ฅmin๐ )and๐๐(๐ฅmax๐ ). If๐max(๐ฅmin๐ )> ๐๐(๐ฅmax๐ ), see an example in Figure 5.14a, we have๐max(๐ฅ)> ๐๐(๐ฅmax๐ )for all๐ฅโ(๐ฅmin๐ ,โ๐/3๐). Hence no point in(๐ฅmin๐ ,โ๐/3๐)is globally convex, which implies๐ห๐๐ ={๐ฅmin๐ , ๐ฅmax๐ }.
Otherwise we have๐max(๐ฅmin๐ )โค๐๐(๐ฅmax๐ ), see an example in Figure 5.14b. Consider the strictly increasing function๐max(๐ฅ)โ๐๐(๐ฅmax๐ )over(๐ฅmin๐ ,โ๐/3๐), with๐max(๐ฅmin๐ )โ ๐๐(๐ฅmax๐ )โค0and๐max(โ๐/3๐)โ๐๐(๐ฅmax๐ )>0. This function has exactly one real root over[๐ฅmin๐ ,โ๐/3๐), say๐ฅmid๐ . Then we have๐ห๐๐ = [๐ฅmin๐ , ๐ฅmid๐ ]โช {๐ฅmax๐ }if๐ฅmin๐ < ๐ฅmid๐ and๐ห๐๐ ={๐ฅmin๐ , ๐ฅmax๐ }if๐ฅmin๐ =๐ฅmid๐ .
5.4 Bivariate polynomial functions: a case study
4. Similar to case3, we need only to know whether the polynomial function of๐ฅ (3๐๐ฅ2+ 2๐๐ฅ+๐)(๐ฅmin๐ โ๐ฅ) +๐๐(๐ฅ)
โ โ
=:๐min(๐ฅ)
โ๐๐(๐ฅmin๐ )
has a real root over ๐ฅ โ (โ๐/3๐, ๐ฅmax๐ ). If the root exists, say ๐ฅmid๐ , then we have ๐ห๐๐ ={๐ฅmin๐ } โช[๐ฅmid๐ , ๐ฅmax๐ ]; otherwise, we have๐ห๐๐ ={๐ฅmin๐ , ๐ฅmax๐ }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 points๐ห๐๐ โ[๐ฅmin๐ , ๐ฅmax๐ ]has either two points, or is an interval plus a point, or an interval. Since the projection function๐โ1๐ is bijective, the set of globally convex points on๐น๐, denoted by๐ห๐โฉ๐น๐ =๐โ1๐ ( ห๐๐๐), 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 ๐is 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 ๐ห๐ โฉ๐๐ of globally convex boundary domain points for the graph of ๐(๐ฅ, ๐ฆ) over the polytope ๐ โ R2 is a union of ๐1 line segments and ๐2 isolated extreme points with ๐1, ๐2 โN0, ๐1 โค๐and๐2 โค๐.
Let๐ฟ1, ๐ฟ2, . . . , ๐ฟ๐1 be the๐1line segments andx๐1,x๐2, . . . ,x๐๐2 be the๐2isolated extreme points. With this notation we have
๐ห๐โฉ๐๐ =๐ฟ1โช ยท ยท ยท โช๐ฟ๐1 โช {x๐1, . . . ,x๐๐
2}.
Furthermore, let๐ฎ๐ฟ๐be the graph of๐(๐ฅ, ๐ฆ)on๐ฟ๐with
๐ฎ๐ฟ๐ ={(๐ฅ, ๐ฆ, ๐ง)|๐ง=๐(๐ฅ, ๐ฆ),(๐ฅ, ๐ฆ)โ๐ฟ๐}
Figure 5.15:Hyperplane that intersects๐ฎ๐ฟ๐,๐ฎ๐ฟ๐ and below them
for every๐โ {1, . . . , ๐}and let
๐ฎ๐๐ ={(๐ฅ, ๐ฆ, ๐ง)|๐ง=๐(๐ฅ, ๐ฆ),(๐ฅ, ๐ฆ)โ๐๐}.
In the following, for any(๐ฅ0, ๐ฆ0)โ๐ฟ๐, we show that there exists a hyperplane๐ปthrough (๐ฅ0, ๐ฆ0, ๐(๐ฅ0, ๐ฆ0))such that๐ปis below๐ฎ๐ฟ๐over๐ห๐โฉ๐๐. In Lemma 5.36 we have more details included. We show later in Theorem 5.37 that either๐ป is a tight valid hyperplane or a tight valid hyperplane๐ป*can be found very easily which is parallel to๐ป.
To find the hyperplane๐ปwith the properties described above, we first prove Lemma 5.35, which implies that for any๐ฟ๐ there exists a hyperplane๐ป๐๐ which is below๐ฎ over๐ฟ๐โช๐ฟ๐
with ๐, ๐ โ {1, . . . , ๐1}, ๐ ฬธ= ๐. Using this result, we show that a hyperplane ๐ป through (๐ฅ0, ๐ฆ0, ๐(๐ฅ0, ๐ฆ0))exists such that๐ปis below๐ฎ๐ฟ๐ for any๐โ {1, . . . , ๐1}, ๐ฬธ=๐. In addition,
a hyperplane๐ปcan be found that it is below(x๐๐, ๐(x๐๐))for any๐โ {1, . . . , ๐1}. Lemma 5.35
For any๐ฟ๐ and๐ฟ๐ with๐, ๐ โ {1, . . . , ๐1}, ๐ ฬธ= ๐ and for any (๐ฅ0, ๐ฆ0) โ ๐ฟ๐, there exists a hyperplane๐ปthrough(๐ฅ0, ๐ฆ0, ๐(๐ฅ0, ๐ฆ0))with๐ปโฉ ๐ฎ๐ฟ๐ ฬธ=โ and๐ปis below๐ฎ๐ฟ๐and๐ฎ๐ฟ๐.
Moreover, such a hyperplane๐ปis unique for any(๐ฅ0, ๐ฆ0)โ๐ฟ๐โ๐๐.
5.4 Bivariate polynomial functions: a case study
Proof. An example is shown in Figure 5.15. The two blue curves are๐ฎ๐ฟ๐ and๐ฎ๐ฟ๐. We need to find a hyperplane๐ปthrough(๐ฅ0, ๐ฆ0, ๐(๐ฅ0, ๐ฆ0))that intersects both๐ฎ๐ฟ๐ and๐ฎ๐ฟ๐ and at the same time๐ปis below them.
For the special case(๐ฅ0, ๐ฆ0)โ๐ฟ๐, we can easily check that๐ป =๐(๐ฅ0, ๐ฆ0), i.e., the tangent plane at(๐ฅ0, ๐ฆ0)fulfills all the conditions. In this case๐ป is not unique.
Assume that(๐ฅ0, ๐ฆ0)ฬธโ๐ฟ๐. We discuss the case(๐ฅ0, ๐ฆ0)โ๐ฟ๐โ๐๐, e.g.,(๐ฅ0, ๐ฆ0) = (๐ฅ1, ๐ฆ1) in Figure 5.15. Corollary 5.13 implies that a hyperplane๐ปthrough(๐ฅ0, ๐ฆ0, ๐(๐ฅ0, ๐ฆ0))which is below๐ฎ๐ฟ๐ contains the subtangent plane
ล(๐ฅ0, ๐ฆ0) =๐(๐ฅ0, ๐ฆ0)โฉ {(๐ฅ, ๐ฆ, ๐ง)|(๐ฅ, ๐ฆ)โaff{๐ฟ๐}} (5.18) which is the lower left green line in Figure 5.15. Denote๐0 = (๐ฅ0, ๐ฆ0, ๐(๐ฅ0, ๐ฆ0)). For every point๐๐ = (๐ฅ๐, ๐ฆ๐, ๐ง๐) โ ๐ฎ๐ฟ๐, we define๐ป(ล(๐ฅ0, ๐ฆ0), ๐๐) = aff{ล(๐ฅ0, ๐ฆ0),{๐๐}}which is a hyperplane below๐๐. Similar to the proof of Theorem 5.27, there exists a point๐*โ ๐ฎ๐ฟ๐ such that๐ป* =๐ป(ล(๐ฅ0, ๐ฆ0), ๐*)is below๐ฎ๐ฟ๐. Note that๐ป* 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(๐ฅ0, ๐ฆ0) โ ๐ฟ๐ โฉ๐๐, e.g., (๐ฅ0, ๐ฆ0) = (๐ฅ2, ๐ฆ2) in Figure 5.15.
Consider a lineล๐(๐ฅ0, ๐ฆ0)โ {(๐ฅ, ๐ฆ, ๐ง)|(๐ฅ, ๐ฆ)โaff{๐ฟ๐}}through(๐ฅ0, ๐ฆ0, ๐(๐ฅ0, ๐ฆ0))which is belowล(๐ฅ0, ๐ฆ0)defined by (5.18) such that(๐ฅ, ๐ฆ)โ๐ฟ๐. In the example in Figure 5.15,ล๐(๐ฅ0, ๐ฆ0) is the red line andล(๐ฅ0, ๐ฆ0)is the upper right green line. Every hyperplane๐ปwhich contains ล๐(๐ฅ0, ๐ฆ0)is through(๐ฅ0, ๐ฆ0, ๐(๐ฅ0, ๐ฆ0))and below๐ฎ๐ฟ๐. Similar to the discussion above, there exists a point๐* โ ๐ฎ๐ฟ๐such that๐ป*= aff{ล๐(๐ฅ0, ๐ฆ0),{๐๐}}is below๐ฎ๐ฟ๐. Note that for every fixed chosenล๐(๐ฅ0, ๐ฆ0)there exists a unique๐ป*. However, we have infinitely manyล๐(๐ฅ0, ๐ฆ0)
to choose. 2
Now we discuss how to algorithmically find๐ป which fulfills Lemma 5.35. Note that for (๐ฅ0, ๐ฆ0)โ๐ฟ๐โฉ๐๐we may chooseล๐(๐ฅ0, ๐ฆ0) = ล(๐ฅ0, ๐ฆ0)which can be computed easily. For any (๐ฅ0, ๐ฆ0)โ๐ฟ๐we compute a hyperplane๐ปthat fulfills Lemma 5.35 and satisfies๐ปโล(๐ฅ0, ๐ฆ0) which is a line defined in (5.18). This is equivalent to finding a point(๐ฅ*, ๐ฆ*)โ๐ฟ๐ such that ๐ป = aff{ล(๐ฅ0, ๐ฆ0),{((๐ฅ*, ๐ฆ*), ๐(๐ฅ*, ๐ฆ*))}}is below๐ฎ๐ฟ๐. Consider the two linesaff{๐ฟ๐}and aff{๐ฟ๐}. They are either not parallel or parallel. Examples for both cases are in Figure 5.16.
As mentioned before, for the case(๐ฅ0, ๐ฆ0) โ ๐ฟ๐, we set ๐ป =๐(๐ฅ0, ๐ฆ0) and we are done.
Otherwise, let
๐ฟ๐ ={(๐ฅ, ๐ฆ)|๐ฆ=๐๐๐ฅ+๐๐, ๐ฅโ[๐ฅmin๐ , ๐ฅmax๐ ]}
and
๐ฟ๐ ={(๐ฅ, ๐ฆ)|๐ฆ=๐๐๐ฅ+๐๐, ๐ฅโ[๐ฅmin๐ , ๐ฅmax๐ ]}.
Define ๐๐ฟ๐(๐ฅ) = ๐(๐ฅ, ๐๐๐ฅ +๐๐) for ๐ฅ โ [๐ฅmin๐ , ๐ฅmax๐ ]and define๐๐ฟ๐(๐ฅ) = ๐(๐ฅ, ๐๐๐ฅ +๐๐) for๐ฅ โ [๐ฅmin๐ , ๐ฅmax๐ ]. ๐๐ฟ๐(๐ฅ)and๐๐ฟ๐(๐ฅ) are univariate functions with degree up to3. The
(a)aff{๐ฟ๐}andaff{๐ฟ๐}are not parallel (b)aff{๐ฟ๐}andaff{๐ฟ๐}are parallel Figure 5.16:Two linesaff{๐ฟ๐}andaff{๐ฟ๐}can be parallel or not parallel
lineล๐(๐ฅ0, ๐ฆ0) = ๐(๐ฅ0, ๐ฆ0)โฉ {(๐ฅ, ๐ฆ, ๐ง) |(๐ฅ, ๐ฆ) โaff{๐ฟ๐}}for๐ฅ0 โ [๐ฅmin๐ , ๐ฅmax๐ ]with๐ฆ0 = ๐๐๐ฅ0+๐๐ can also be represented as
ล๐(๐ฅ0, ๐ฆ0) ={(๐ฅ, ๐ฆ, ๐ง)|๐ฅโR, ๐ฆ=๐๐๐ฅ+๐๐, ๐ง =๐๐ฟโฒ๐(๐ฅ0)(๐ฅโ๐ฅ0) +๐๐ฟ๐(๐ฅ0)}. (5.19) Analogously, for every๐ฅ1 โ[๐ฅmin๐ , ๐ฅmax๐ ]and๐ฆ1 =๐๐๐ฅ1+๐๐, we get
ล๐(๐ฅ1, ๐ฆ1) ={(๐ฅ, ๐ฆ, ๐ง)|๐ฅโR, ๐ฆ=๐๐๐ฅ+๐๐, ๐ง=๐๐ฟโฒ๐(๐ฅ1)(๐ฅโ๐ฅ1) +๐๐ฟ๐(๐ฅ1)}.
First we discuss the case thataff{๐ฟ๐}andaff{๐ฟ๐}are not parallel, see an example in Fig-ure 5.16a. Since๐๐ ฬธ=๐๐, the intersection ofaff{๐ฟ๐}andaff{๐ฟ๐}is(๐ฅ๐๐, ๐ฆ๐๐)with
๐ฅ๐๐ = ๐๐โ๐๐
๐๐โ๐๐ and๐ฆ๐๐ = ๐๐๐๐โ๐๐๐๐
๐๐โ๐๐ . Consider the point๐๐๐ = (๐ฅ๐๐, ๐ฆ๐๐, ๐ง๐๐)with๐ง๐๐ =๐๐ฟโฒ
๐(๐ฅ0)(๐ฅ๐๐ โ๐ฅ0) +๐๐ฟ๐(๐ฅ0). We can check that๐๐๐ โล๐(๐ฅ0, ๐ฆ0)which implies that๐๐๐ โ๐ปfor every๐ปthat fulfills Lemma 5.35. Since๐ป also intersects๐ฎ๐ฟ๐, finding๐ปfulfilling Lemma 5.35 is equivalent to finding a point๐* โ ๐ฎ๐ฟ๐ such thataff{ล๐(๐ฅ0, ๐ฆ0),{๐*}}is below๐ฎ๐ฟ๐. Consider the function
๐๐(๐ฅ) =๐๐ฟโฒ๐(๐ฅ)(๐ฅ๐๐โ๐ฅ) +๐๐ฟ๐(๐ฅ)
for๐ฅ โ [๐ฅmin๐ , ๐ฅmax๐ ]. Note that the point (๐ฅ1, ๐๐๐ฅ1 +๐๐, ๐๐(๐ฅ1))lies in lineล๐(๐ฅ1, ๐ฆ1) for ๐ฅ1 โ [๐ฅmin๐ , ๐ฅmax๐ ]. As we analyzed before by considering the sign of the first derivative, ๐๐(๐ฅ)is a strictly decreasing function if ๐ฅ๐๐ < ๐ฅmin๐ and is a strictly increasing function if
5.4 Bivariate polynomial functions: a case study
(a) For case๐ง๐๐< ๐๐(๐ฅmin๐ )< ๐๐(๐ฅmax๐ ) (b) For case๐๐(๐ฅmax๐ )< ๐๐(๐ฅmin๐ )< ๐ง๐๐
(c) For case๐ง๐๐< ๐๐(๐ฅmax๐ )< ๐๐(๐ฅmin๐ ) (d) For case๐๐(๐ฅmin๐ )< ๐๐(๐ฅmax๐ )< ๐ง๐๐
Figure 5.17:Example for the4cases in the proof of Lemma 5.35
๐ฅ๐๐ > ๐ฅmax๐ . There are no other cases since๐ฅ๐๐ ฬธโ[๐ฅmin๐ , ๐ฅmax๐ ]. In both cases we have๐๐(๐ฅmin๐ )ฬธ=
๐๐(๐ฅmax๐ ). We compare๐ง๐๐,๐๐(๐ฅmin๐ )and๐๐(๐ฅmax๐ ). If๐ง๐๐ is between๐๐(๐ฅmin๐ )and๐๐(๐ฅmax๐ ), i.e., ๐๐(๐ฅmin๐ ) โค๐ง๐๐ โค๐๐(๐ฅmax๐ )or๐๐(๐ฅmax๐ ) โค๐ง๐๐ โค๐๐(๐ฅmin๐ ), then the increasing or decreasing function๐๐(๐ฅ)โ๐ง๐๐ has a unique root๐ฅ* โ[๐ฅmin๐ , ๐ฅmax๐ ]with๐๐(๐ฅ*)โ๐ง๐๐ = 0. Note that since ๐๐(๐ฅ)is a polynomial function of degree up to3,๐ฅ*can be computed very easily. It implies that the lineล๐(๐ฅ*, ๐ฆ*)with๐ฆ* =๐๐๐ฅ*+๐๐contains also๐๐๐ and
๐ป = aff{ล๐(๐ฅ0, ๐ฆ0),ล๐(๐ฅ*, ๐ฆ*)}
is a hyperplane which fulfills Lemma 5.35.
Otherwise, if๐ง๐๐ is not between๐๐(๐ฅmin๐ )and๐๐(๐ฅmax๐ ), we have the following four cases 1. ๐ง๐๐ < ๐๐(๐ฅmin๐ )< ๐๐(๐ฅmax๐ ),
2. ๐๐(๐ฅmax๐ )< ๐๐(๐ฅmin๐ )< ๐ง๐๐,
3. ๐ง๐๐ < ๐๐(๐ฅmax๐ )< ๐๐(๐ฅmin๐ ), 4. ๐๐(๐ฅmin๐ )< ๐๐(๐ฅmax๐ )< ๐ง๐๐.
Examples for the four cases are shown in Figure 5.17. In the first two cases we set ๐*= (๐ฅmin๐ , ๐๐๐ฅmin๐ +๐๐, ๐๐ฟ๐(๐ฅmin๐ )).
In the last two cases we set
๐* = (๐ฅmax๐ , ๐๐๐ฅmax๐ +๐๐, ๐๐ฟ๐(๐ฅmax๐ )).
For all four cases, the lineaff{๐*, ๐๐๐}, shown as the red line in the corresponding subgraphs, is below๐ฎ๐ฟ๐ (the corresponding blue curve). We then have ๐ป = aff{ล๐(๐ฅ0, ๐ฆ0),{๐*}} โ aff{๐*, ๐๐๐}which fulfills Lemma 5.35 and is the hyperplane we are looking for.
Now it only remains to discuss the case thataff{๐ฟ๐}andaff{๐ฟ๐}are parallel, see an example in Figure 5.16b. Note that ๐๐ = ๐๐ and for any ๐ฅ1 โ [๐ฅmin๐ , ๐ฅmax๐ ] with ๐ฆ1 = ๐๐๐ฅ1 +๐๐, the red line in the corresponding subgraphs, is below๐ฎ๐ฟ๐ (the corresponding blue curve). Thus ๐ป= aff{ล๐(๐ฅ0, ๐ฆ0),{๐*}}fulfills the requirements of Lemma 5.35 and is the hyperplane we solving a system of linear equations. The algorithm is summarized in Algorithm 5.1.
Define๐๐ฎ = {(๐ฅ, ๐ฆ, ๐ง) |(๐ฅ, ๐ฆ) โ ๐๐, ๐ง = ๐(๐ฅ, ๐ฆ)}and for each๐โ {1, . . . , ๐}and the corresponding facet๐น๐we further define๐ฎ๐น๐ ={(๐ฅ, ๐ฆ, ๐ง)|(๐ฅ, ๐ฆ)โ๐น๐, ๐ง=๐(๐ฅ, ๐ฆ)}.
Lemma 5.36
For any๐ฟ๐ โ ๐ห๐โฉ๐๐ with๐โ {1, . . . , ๐1}and for any(๐ฅ0, ๐ฆ0) โ๐ฟ๐ โ ๐น๐, there exists a hyperplane๐ป
5.4 Bivariate polynomial functions: a case study
Algorithm 5.1:Algorithm that computes a hyperplane that intersects๐ฎ๐ฟ๐,๐ฎ๐ฟ๐and is below them
(a) For the case๐๐ฟโฒ๐(๐ฅ0)< ๐๐ฟโฒ๐(๐ฅmin๐ ) (b) For the case๐๐ฟโฒ๐(๐ฅ0)> ๐๐ฟโฒ๐(๐ฅmax๐ ) Figure 5.18:Two cases by๐๐ฟโฒ
๐(๐ฅ0)ฬธโ[๐๐ฟโฒ
๐(๐ฅmin๐ ), ๐๐ฟโฒ
๐(๐ฅmax๐ )] in Algorithm 5.1
1. either with๐ป=๐(๐ฅ0, ๐ฆ0) 2. or with(๐ปโฉ๐๐ฎ)โ ๐ฎ๐น๐ ฬธ=โ
such that๐ป โล๐(๐ฅ0, ๐ฆ0). In addition,๐ปis below๐ฎover(๐ฅ, ๐ฆ)โ๐ห๐โฉ๐๐. Proof. We develop an algorithm to find the hyperplane๐ป.
Denote aff{๐น๐} = {(๐ฅ, ๐ฆ) | ๐๐๐ฅ+๐๐๐ฅ = ๐๐} with three constants ๐๐, ๐๐, ๐๐ โ R. Since ๐น๐ is a facet of๐, we have๐ โ {(๐ฅ, ๐ฆ) | ๐๐๐ฅ+๐๐๐ฅ โฅ ๐๐}or๐ โ {(๐ฅ, ๐ฆ) | ๐๐๐ฅ+๐๐๐ฅ โค ๐๐}. Lemma 5.18 implies that for any two nonvertical ๐ป1, ๐ป2 with ๐ป1 โ ล๐(๐ฅ0, ๐ฆ0) and ๐ป2 โ ล๐(๐ฅ0, ๐ฆ0), either ๐ป1 is below ๐ป2 over ๐ or๐ป2 is below ๐ป1 over ๐. Recall that ๐ห๐โฉ๐๐ =๐ฟ1โช ยท ยท ยท โช๐ฟ๐1โช {x๐1, . . . ,x๐๐2}. For every๐ฟ๐ with๐ฬธ=๐and๐ โ {1, . . . , ๐1}, compute๐ป๐๐ฟ = ๐ป((๐ฅ0, ๐ฆ0), ๐ฟ๐, ๐ฟ๐)as output of Algorithm 5.1. For everyx๐๐ with x๐๐ ฬธโ ๐น๐
and๐ โ {1, . . . , ๐2}, compute๐ป๐๐ = aff{ล๐(๐ฅ0, ๐ฆ0),{(x๐๐, ๐(x๐๐))}}which is a nonvertical hyperplane since(x๐๐, ๐(x๐๐))ฬธโล๐(๐ฅ0, ๐ฆ0)andx๐๐ ฬธโ๐น๐. Consider the set
โ(๐ฅ0, ๐ฆ0) ={๐ป๐๐ฟ|๐ฬธ=๐, ๐โ {1, . . . , ๐1}} โช {๐ป๐๐|x๐๐ ฬธโ๐น๐, ๐ โ {1, . . . , ๐2}}
of finitely many hyperplanes that all containล๐(๐ฅ0, ๐ฆ0). There exists๐ป* โ โ(๐ฅ0, ๐ฆ0)such that๐ป*is below๐ปover๐for every๐ป โ โ(๐ฅ0, ๐ฆ0). The hyperplane๐ป*is below๐ฎ๐น๐ since ล๐(๐ฅ0, ๐ฆ0)โ๐ป*which is below๐ฎ๐น๐. The hyperplane๐ป*is below every๐ฎ๐น๐ with๐ ฬธ=๐since ๐ป*is below๐ป๐๐ฟover๐น๐ โ๐and๐ป๐๐ฟis below๐ฎ๐น๐. Similarly,๐ป*is below every(x๐๐, ๐(x๐๐)) since๐ป*is below๐ป๐๐over๐ โx๐๐. Thus๐ป*is below๐ฎover(๐ฅ, ๐ฆ)โ๐ห๐โฉ๐๐.
5.4 Bivariate polynomial functions: a case study
If๐ป* =๐ป๐๐ฟfor some๐โ {1, . . . , ๐1}, Algorithm 5.1 implies that either๐ป* =๐(๐ฅ0, ๐ฆ0) or there exists a pointx๐โ๐ฟ๐withx๐ฬธโ๐น๐and(x๐, ๐(x๐))โ๐ป*. Otherwise if๐ป* =๐ป๐๐, there exists a pointx๐โ๐๐withx๐ ฬธโ๐น๐and(x๐, ๐(x๐))โ๐ป*.
In all cases either๐ป =๐(๐ฅ0, ๐ฆ0)or(x๐, ๐(x๐))โ(๐ปโฉ๐๐ฎ)โ ๐ฎ๐น๐ ฬธ=โ . 2 Theorem 5.37
For any๐ฟ๐ โ๐ห๐โฉ๐๐ with๐โ {1, . . . , ๐1}and for any(๐ฅ0, ๐ฆ0)โ๐ฟ๐ โ๐น๐, a hyperplane๐ป* which fulfills Lemma 5.36 is either a tight valid hyperplane or there exists a tight valid hyperplane ๐ป**which is parallel to๐ป*. Furthermore,๐ป*is always a tight valid hyperplane if the Hessian matrix is negative semidefinite, i.e., it satisfies
๐ป(๐)(๐ฅ, ๐ฆ) = (๏ธ ๐2
๐๐ฅ2๐ ๐๐ฅ๐๐ฆ๐2 ๐
๐2
๐๐ฅ๐๐ฆ๐ ๐๐ฆ๐22๐ )๏ธ
โชฏ0 (5.20)
for all(๐ฅ, ๐ฆ)โint๐.
Proof. Denote ๐ป* = {(๐ฅ, ๐ฆ, ๐ง) | ๐ง = ๐๐๐ฅ +๐๐๐ฆ +๐๐}with ๐๐, ๐๐, ๐๐ โ R. Consider the optimization problem
min
(๐ฅ,๐ฆ)โ๐๐(๐ฅ, ๐ฆ)โ(๐๐๐ฅ+๐๐๐ฆ+๐๐) (OPmin๐ป*) that has a minimum๐ง*, since๐is a compact set and๐(๐ฅ, ๐ฆ)โ(๐๐๐ฅ+๐๐๐ฆ+๐๐)is a continuous function. Note that๐ง* โค0since๐ฎ โฉ๐ป*ฬธ=โ . The hyperplane๐ป*is valid if and only if๐ง* = 0. The maximally valid subtangent plane๐๐ปmax(๐ฅ0, ๐ฆ0)isล๐(๐ฅ0, ๐ฆ0)and there exists another point (๐ฅ1, ๐ฆ1, ๐ง1)โ ๐ฎ โฉ๐ป*with(๐ฅ1, ๐ฆ1, ๐ง1)ฬธโaff{ล๐(๐ฅ0, ๐ฆ0)}. Recalling the definition of tight valid
hyperplanes in Section 5.3, we have
aff{ล๐(๐ฅ0, ๐ฆ0),{(๐ฅ1, ๐ฆ1, ๐ง1)}} โaff{๏ธ๐๐ปmax* (x0) : for allx0โ๐๐ป*}๏ธ=:๐๐ป* which implies
2 = dim aff{ล๐(๐ฅ0, ๐ฆ0),{(๐ฅ1, ๐ฆ1, ๐ง1)}} โคdim๐๐ป* โค2.
Hence๐๐ป* is a hyperplane which implies๐ป*is a tight valid hyperplane. Otherwise we have ๐ง* <0with solution(๐ฅ*, ๐ฆ*). Note that(๐ฅ*, ๐ฆ*)must be an interior point of๐ since๐ป*is below๐ฎover๐ห๐โฉ๐๐. We have then๐(๐ฅ*, ๐ฆ*) =๐๐๐ฅ+๐๐๐ฆ+๐๐+๐ง*which is a tight valid hyperplane and is parallel to๐ป*.
Every globally convex interior point is also a locally convex interior point. According to [Edw94] and Lemma 5.5, (๐ฅ0, ๐ฆ0)satisfies๐ป(๐)(๐ฅ0, ๐ฆ0) โชฐ 0. If (5.20) is satisfied for all (๐ฅ, ๐ฆ) โ int๐ then we have๐ห๐ โฉint๐ = โ which implies that ๐ป* is below ๐ฎ over๐ห๐. Lemma 5.20 implies that๐ป*is valid. As we discussed above,๐ป* 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 (OPmin๐ป*) has the minimum๐ง* = 0. Until now we have only consideredpotentiallytight valid hyperplanes that containล๐(๐ฅ0, ๐ฆ0)for an(๐ฅ0, ๐ฆ0)in an๐ฟ๐. In order to show that there exists otherpotentiallytight valid hyperplanes ๐ปwhich are below๐(๐ฅ, ๐ฆ)over๐ห๐โฉ๐๐we give the following definition.
Definition 5.38 (Potentially tight valid hyperplanes of type๐ดand type๐ต)
A hyperplane which fulfills Lemma 5.36 is apotentially tightvalid hyperplanes oftype๐ด. A hyperplane๐ปis apotentially tightvalid hyperplane oftype๐ตif
โข ๐ป is below๐(๐ฅ, ๐ฆ)over๐ห๐โฉ๐๐,
โข it satisfies๐x(๐ปโฉ๐๐ฎ)โ๐๐and|๐ปโฉ๐๐ฎ| โฅ3and
โข there does not exist๐ฟ๐ withx๐โ๐ฟ๐andx๐โ๐x(๐ปโฉ๐๐ฎ)such that๐ปโล๐(x๐). Due to the last condition in the definition of potentially tight valid hyperplanes of type๐ต, the set of potentially tight valid hyperplanes of type๐ดand the set of type๐ตare disjoint.
Corollary 5.39
Let๐ป*be a potentially tight valid hyperplane of type๐ต. Then๐ป*is either a tight valid hyperplane or there exists a tight valid hyperplane๐ป**which is parallel to๐ป*. Furthermore,๐ป*is always a tight valid hyperplane if every(๐ฅ, ๐ฆ)โint๐satisfies (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 hyperplanes๐ป**of type๐ต algo-rithmically. Note that every such๐ปsatisfies๐x(๐ปโฉ๐๐ฎ)โ๐๐and|๐ปโฉ๐๐ฎ| โฅ3. As every three pointsx๐,x๐,x๐ โ๐๐ with1 โค๐ < ๐ < ๐ โค ๐do not lie on a same line, it implies that๐ป๐๐๐= aff{(x๐, ๐(x๐),(x๐, ๐(x๐),(x๐, ๐(x๐)}is a hyperplane. There are(๏ธ๐3)๏ธsuch hyper-planes๐ป๐๐๐. We can easily prove that a given๐ป๐๐๐is a potentially tight valid hyperplane๐ป**
of type๐ตif and only if
โข for everyx๐ โ๐๐,๐ป๐๐๐is below(x๐, ๐(x๐));
โข for every๐ฟ๐,๐ป๐๐๐is below๐ฎ๐ฟ๐, for this we need only to compare the curve๐ฎ๐ฟ๐defined by a polynomial of degree up to3and the line segment๐ป๐๐๐โฉ {(x, ๐ง)|xโ๐ฟ๐};
โข for any๐ฟ๐containingx๐ ,๐ โ {๐, ๐, ๐}, check if it fulfillsล๐(x๐)ฬธโ๐ป๐๐๐.
All the three conditions above can be checked easily. Thus we design Algorithm 5.3 to compute potentially tight valid hyperplanes of type๐ต.
The proof of Lemma 5.36 describes an algorithm to compute the unique potentially tight valid hyperplane๐ป* of type๐ดthat containsล๐(x0), denoted by๐ป*(x0, ๐ฟ๐). Note that we cannot omit๐ฟ๐ in the notation since there may existx๐ โ๐๐withx๐ โ๐ฟ๐,x๐ โ๐ฟ๐,๐ฬธ=๐
5.5 Computational results
and๐ป*(x๐, ๐ฟ๐) ฬธ= ๐ป*(x๐, ๐ฟ๐). For anyx1 โ ๐ฟ๐ withx1 ฬธ= x0,๐ป*(x0, ๐ฟ๐) = ๐ป*(x1, ๐ฟ๐) if and only if๐ป*(x0, ๐ฟ๐) โ ล๐(x1). On the other hand, for every three pointsx๐,x๐,x๐ โ ๐๐, we use the algorithm above to check if๐ป๐๐๐ = aff{(x๐, ๐(x๐),(x๐, ๐(x๐),(x๐, ๐(x๐)}
is a potentially tight valid hyperplane ๐ป** of type๐ต. If yes, denote it by๐ป**(๐, ๐, ๐). Let x๐โฒ,x๐โฒ,x๐โฒ โ ๐๐be three points such that๐ป๐โฒ๐โฒ๐โฒ is a potentially tight valid hyperplane of type๐ตwith{x๐โฒ,x๐โฒ,x๐โฒ} ฬธ={x๐,x๐,x๐}. Then๐ป**(๐, ๐, ๐) =๐ป**(๐โฒ, ๐โฒ, ๐โฒ)if and only if all the points{(x๐, ๐(x๐))|๐โ {๐, ๐, ๐, ๐โฒ, ๐โฒ, ๐โฒ}}are on a same hyperplane.
Definition 5.40 (Tight valid hyperplanes of type๐ดand type๐ต)
A tight valid hyperplane๐ป*is a tight valid hyperplane of type๐ดif๐ป* is also a potentially tight valid hyperplane of type๐ดor๐ป* is parallel to a potentially tight valid hyperplanes of type๐ด. Similarly, a tight valid hyperplane๐ป**is a tight valid hyperplane of type๐ตif๐ป**is also a potentially tight valid hyperplane of type๐ตor๐ป**is parallel to a potentially tight valid hyperplane of type๐ต.
For any๐โ {1, . . . , ๐1}, let๐๐ โ๐ฟ๐be a set of finitely many points. Algorithm 5.2 computes a set of tight valid hyperplanes of type๐ด. Let๐2 โNbe the upper bound of the number of tight valid hyperplanes of type๐ตwe want to have. Algorithm 5.3 computes a set of tight valid hyperplanes of type๐ต.
5.5 Computational results
Recall the complete MINLP model (2.26) introduced in Section 2.1. All nonlinearities and integrality conditions can be handled by the solver SCIP directly. Note that in that model, we just consider pumps with fixed speed because of the two real-world instances introduced in Section 3.3.
In many water supply networks, there are variable speed pumps. For them the character-istic diagrams often involve the relative speed๐. In [Hae08; Kol11], the pressure increase is approximated by
ฮโ๐๐ก=๐2๐๐ก๐ผ0๐โ๐๐๐ก๐ผ1๐๐๐๐กโ๐ผ2๐๐2๐๐ก, (5.21) where๐ผ0๐,๐ผ1๐and๐ผ2๐are constants derived from the characteristic curve for pump๐.
Figure 5.19 shows the characteristic curves for pump๐with variable speed, in cases of๐1= 1, ๐1 = 0.8and๐1 = 0.6.
Similar to (2.23), the power consumption of pump๐can be approximated as
๐ถ๐๐ก= ๐ ๐ก๐๐ฮโ๐๐ก๐๐๐ก
๐๐๐ก =
๐ ๐ก๐๐(๏ธ๐๐๐ก2๐ผ0๐๐๐๐กโ๐๐๐ก๐ผ1๐๐2๐๐กโ๐ผ2๐๐3๐๐ก)๏ธ
๐๐๐ก(๐๐๐ก, ๐๐๐ก) =:๐(๐๐๐ก, ๐๐๐ก) (5.22) Note that the efficiency๐๐๐กalso depends on๐๐๐กand๐๐๐กand there exists a function to present it, hence there exists a function๐(๐๐๐ก, ๐๐๐ก)to approximate๐ถ๐๐ก.
Algorithm 5.2:Algorithm that computes a set of tight valid hyperplanes of type๐ด Input: A polynomial function in form (5.17), polytope๐ โR2as the domain set, the
corresponding๐ฟ1, ๐ฟ2, . . . , ๐ฟ๐1 and{x๐1, . . . ,x๐๐
2}, point sets๐1, ๐2, . . . , ๐๐1 Output: Setโ๐ดof tight valid hyperplanes for๐ฎ
1 Initializeโ๐ด=โ* =โ
21 ifeveryxint๐satisfies (5.20)then
22 Setโ๐ด=โ*
5.5 Computational results
Algorithm 5.3:Algorithm that computes a set of tight valid hyperplanes of type๐ต Input: A polynomial function in form (5.17), domain set๐ โR2and๐2โN Output: A setโof up to๐2tight (downward closed) valid hyperplanes for๐ฎ
1 Initializeโ๐ต=โ**=โ
Figure 5.19:Example of characteristic curve for a pump with variable speed
Our solver SCIP can solve general nonconvex MIQCP [VG18; BHV12]. As a consequence, constraints consisting of any polynomial function can be handled by SCIP, e.g., by substituting them recursively until they contain only nonlinear terms in form of๐ฅยท๐ฆor๐ฅ2.1
To enable that SCIP can handle the constraints like (5.22), we try to approximate function๐ with a polynomial function. Note that characteristic diagrams are usually given by the vendor with a set of measured points.
For polynomial fitting, on the one hand, we want to keep the degree of polynomials as low as possible. This is very helpful for the outer-approximation algorithms. On the other hand, the degree of polynomials should be high enough so that the approximation error is acceptable.
For the computation we got a third real-world instance from Tsinghua University, Department of Hydraulic Engineering. Figure 5.20 shows a small water supply networkn9p3a11in the
For the computation we got a third real-world instance from Tsinghua University, Department of Hydraulic Engineering. Figure 5.20 shows a small water supply networkn9p3a11in the