Representation of Polytopes as Polynomial Zonotopes
RRepresentation of Polytopes as Polynomial Zonotopes
Niklas Kochdumper · Matthias AlthoffAbstract
We prove that each bounded polytope can be represented as a polynomial zonotope, whichwe refer to as the Z-representation of polytopes. Previous representations are the vertex representation(V-representation) and the halfspace representation (H-representation). Depending on the polytope, theZ-representation can be more compact than the V-representation and the H-representation. In addition,the Z-representation enables the computation of linear maps, Minkowski addition, and convex hull with acomputational complexity that is polynomial in the representation size. The usefulness of the new repre-sentation is demonstrated by range bounding within polytopes.
Keywords
Polytopes · Polynomial zonotopes · Z-representation. facet enumeration problem , and the inverse problem offinding the V-representation given the H-representation is known as the vertex enumeration problem . Bothtransformations can be computed in polynomial time with respect to the number of polytope vertices [14].One of the major criteria for a comparison of different set representations is the computational efficiency ofoperations on them. Of special interest are operations that are closed for polytopes: Linear transformation,Minkowski sum, convex hull, and intersection.Let us first consider the H-representation. Linear transformations are simple if the matrix M definingthe transformation is invertible; otherwise, the computational complexity is exponential in the dimension n [13]. The complexity of computing the Minkowski sum of two H-polytopes is also exponential in n since theworst-case number of resulting halfspaces is exponential in n [9,25]. The calculation of the convex hull oftwo polytopes in H-representation is NP-hard [25]. Computation of the intersection of two polytopes in H- representation can be easily realized by a concatenation of the inequality constraints. A possible subsequentelimination of redundant halfspaces can be implemented with linear programming.For the V-representation, the situation is different: The linear transformation is trivial to calculate,even for cases where the transformation matrix M is not invertible. Computation of the Minkowski sumof two polytopes in V-representation has exponential complexity in the number of dimensions [6,7]. Thesame holds for the convex hull operation, because constructing a redundant V-representation of the convexhull is trivial, and the redundant points can be eliminated by solving an exponential number of linearprograms [25]. However, computation of the intersection of two polytopes in V-representation is NP-hard[25]. Overall, neither the H-representation nor the V-representation has polynomial complexity for all fourdiscussed operations. Niklas KochdumperTechnical University of MunichE-mail: [email protected] AlthoffTechnical University of MunichE-mail: althoff@tum.de a r X i v : . [ m a t h . C O ] O c t Niklas Kochdumper, Matthias Althoff
Two other set representations which are able to represent any bounded polytope are constrained zono-topes [23] and zonotope bundles [4]. Constrained zonotopes are zonotopes with additional linear equalityconstraints on the zonotope factors. Since the implementation of the linear transformation, the Minkowskisum, and the intersection operation only involves matrix multiplications, additions, and concatenations [23,Prop. 1], their computational complexity is at most O ( n ). Furthermore, there exist efficient techniques forthe removal of redundant constraints and zonotope generators [23, Sec. 4.2]. However, at present there isno known algorithm for the computation of the convex hull of two constraint zonotopes. Zonotope bundlesdefine a set implicitly as the intersection of multiple zonotopes. While the computation of the linear trans-formation and the intersection is trivial, efficient algorithms for the Minkowski sum [4, Prop. 2] and theconvex hull [4, Prop. 5] only exist for the calculation of an over-approximation.The Z-representation that we present in this paper is based on polynomial zonotopes, a non-convexset representation first introduced in [2]. In particular, we use the sparse representation of polynomialzonotopes from [17]. Taylor Models are a type of set representation closely related to polynomial zonotopes[18]. However, while Taylor models, like polynomial zonotopes, are able to represent non-convex sets, theycan not represent arbitrary polytopes.The Z-representation is based on polynomials; the interplay between polytopes and polynomials hasa long history in computational geometry: the Newton polytope of a polynomial is the convex hull ofits exponent vectors [16]. Newton polytopes are, among other things, useful for analyzing the roots ofmultivariate polynomials [24]. The Ehrhart polynomial of a polytope specifies the number of integer pointsthe polytope contains [10]. Furthermore, it is well-known that polytopes can be equivalently represented bypolynomial inequalities [8]. The classical result of Minkowski implies that a polytope can be approximatedarbitrary closely by a single polynomial inequality [19]. In this work, we introduce the Z-representation for bounded polytopes. This new representation enables the computation of the linear transformation, Minkowski sum, and convex hull with polynomial complexitywith respect to the representation size. In addition, the Z-representation can be more compact than theV- and the H-representation. We further provide algorithms for the conversion from V-representation toZ-representation and from Z-representation to V-representation. One application of the Z-representation isrange bounding, for which we demonstrate the advantages resulting from the new representation.1.2 NotationIn the remainder of this paper, we will use the following notations: Sets and tuples are always denoted bycalligraphic letters, matrices by uppercase letters, and vectors by lowercase letters. Given a vector b ∈ R n , b ( i ) refers to the i -th entry. Given a matrix A ∈ R n × m , A ( i, · ) represents the i -th matrix row, A ( · ,j ) the j -thcolumn, and A ( i,j ) the j -th entry of matrix row i . The empty matrix is denoted by [ ]. Given two matrices C and D , [ C, D ] denotes the concatenation of the matrices. The symbols and represent matrices of zerosand ones of proper dimension. Given a n -tuple L = ( l , . . . , l n ), |L| = n denotes the cardinality of the tupleand L ( i ) refers to the i -th entry of tuple L . Given a tuple H = ( h , . . . , h n ) with h i ∈ R m i ∀ i ∈ { , . . . , n } ,notation H ( i,j ) = h i ( j ) refers to the j -th entry in the i -th element of H . The empty tuple is denotedby ∅ . Given two tuples L = ( l , . . . , l n ) and K = ( k , . . . , k m ), ( L , K ) = ( l , . . . , l n , k , . . . , k m ) denotesthe concatenation of the tuples. We further introduce an n-dimensional interval as I := [ l, u ] , ∀ i l ( i ) ≤ u ( i ) , l, u ∈ R n . The ceil operator (cid:100) x (cid:101) and the floor operator (cid:98) x (cid:99) round a scalar number x ∈ R to the nexthigher and lower integer, respectively. The binomial coefficient is denoted by (cid:0) rz (cid:1) , r, z ∈ N . For the derivationof computational complexity, we consider all binary operations, except concatenations and initializations since their computational time is negligible. We first provide some definitions that are important for the remainder of the paper.
Definition 1 (V-Representation) Given the q polytope vertices v i ∈ R n , the vertex representation of P ⊂ R n is defined as P := (cid:40) q (cid:88) i =1 β i v i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β i ≥ , q (cid:88) i =1 β i = 1 (cid:41) . For a concise notation we introduce the shorthand P = (cid:104) [ v , . . . , v q ] (cid:105) V for the V-representation. The V-representation is called redundant if the matrix [ v , . . . , v q ] contains points that are not vertices of P . epresentation of Polytopes as Polynomial Zonotopes 3 Definition 2 (H-Representation) Given a matrix A ∈ R h × n and a vector b ∈ R h , the halfspace represen-tation of P ⊂ R n is defined as P := { x | Ax ≤ b } . Polynomial zonotopes are a non-convex set representation that was first introduced in [2]. We use aslight modification of the sparse representation of polynomial zonotopes from [17]:
Definition 3 (Polynomial Zonotope) Given a generator matrix G ∈ R n × h and an exponent matrix E ∈ Z p × h ≥ , a polynomial zonotope PZ ⊂ R n is defined as PZ := (cid:40) h (cid:88) i =1 (cid:32) p (cid:89) k =1 α E ( k,i ) k (cid:33) G ( · ,i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α k ∈ [ − , (cid:41) . (1)The variables α k are called factors, and the vectors G ( · ,i ) generators. The number of factors α k is p , andthe number of generators is h .The set operations considered in this paper are linear transformation, Minkowski sum, convex hull, andintersection. Given two sets S , S ⊂ R n , these operations are defined as follows: − Linear transformation : M S = { M s | s ∈ S } , M ∈ R m × n (2) − Minkowski sum : S ⊕ S = { s + s | s ∈ S , s ∈ S } (3) − Convex hull : conv ( S , S ) = (cid:26)
12 (1 + λ ) s + 12 (1 − λ ) s | s ∈ S , s ∈ S , λ ∈ [ − , (cid:27) (4) − Intersection : S ∩ S = { s | s ∈ S , s ∈ S } (5) In this section, we introduce the Z-representation of bounded polytopes, we derive the equations for setoperations applied in Z-representation, and we provide algorithms for the conversion between P- and V-representation.3.1 DefinitionWe first introduce the Z-representation of bounded polytopes:
Definition 4 (Z-Representation) Given a starting point c ∈ R n and a generator matrix G ∈ R n × h , theZ-representation defines the set P := (cid:40) c + h (cid:88) i =1 (cid:32) m i (cid:89) k =1 α E ( i,k ) (cid:33) G ( · ,i ) (cid:12)(cid:12) (cid:12) (cid:12)(cid:12) α E ( i,k ) ∈ [ − , (cid:41) E = ( e , . . . , e h ) , ∀ i ∈ { , . . . , h } : e i ∈ N m i ≤ p , and ∀ i ∈ { , . . . , h } ∀ j, k : j (cid:54) = k ⇒ e i ( j ) (cid:54) = e i ( k ) , (6)where the tuple E stores the factor indices, m i is the length of the i -th element of E , p is the number offactors α E ( i,k ) , and h is the number of generators. The overall number of entries in E is µ = h (cid:88) i =1 m i . (7)The Z-representation is regular if ∀ j, k : j (cid:54) = k ⇒ e k (cid:54) = e j . (8) Niklas Kochdumper, Matthias Althoff -3 -2 -1 0 1 2 3 x -2-10123 x -3 -2 -1 0 1 2 3 x -2-10123 x Fig. 1: Visualization of the set defined by the Z-representation from Example 1 (left) and from Example 2(right).For a concise notation we introduce the shorthand P = (cid:104) c, G, E(cid:105) Z . All components of a set (cid:3) i have theindex i , e.g., p i , h i , µ i , and m i,j , j = 1 . . . h i belong to P i = (cid:104) c i , G i , E i (cid:105) Z . In the remainder of this work,we call the term α E ( i, · . . . · α E ( i,mi ) · G ( · ,i ) in (6) a monomial and α E ( i, · . . . · α E ( i,mi ) the variable part ofthe monomial. The Z-representation defines a special type of polynomial zonotope (see Def. 3) where theexponents of the factors α k are restricted to the values 0 and 1. Therefore, every set in Z-representation canbe equivalently represented as a polynomial zonotope. It further holds that every bounded polytope can beequivalently represented by the Z-representation, which we prove later in Theorem 1. The converse does nothold: not every set defined by a Z-representation is a bounded polytope. We illustrate the Z-representationof polytopes with two simple examples: Example 1
The Z-representation P = (cid:28)(cid:20) − . (cid:21) , (cid:20) . − . − . − . − . (cid:21) , (cid:18) , , (cid:20) (cid:21)(cid:19)(cid:29) Z defines the polytope P = (cid:26) (cid:20) − . (cid:21) + α (cid:20) . − . (cid:21) + α (cid:20) − . − (cid:21) + α α (cid:20) − . . (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) α , α ∈ [ − , (cid:27) , which is visualized in Fig. 1 (left). Example 2
The Z-representation P = (cid:28)(cid:20) − . (cid:21) , (cid:20) − . − . . . − − . (cid:21) , (cid:18) , , (cid:20) (cid:21)(cid:19)(cid:29) Z defines the set P = (cid:26) (cid:20) − . (cid:21) + α (cid:20) − . . (cid:21) + α (cid:20) − . − (cid:21) + α α (cid:20) . − . (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) α , α ∈ [ − , (cid:27) , which is not a polytope as can be seen in Fig. 1 (right).3.2 OperationsIn this subsection, we derive and prove implementations for the set operations in (2)-(5). In addition, wedetermine the computational complexity for each operation. For further derivations, let us establish thataccording to the definition of the Z-representation in (6) we can write (cid:40) c + h (cid:88) i =1 (cid:32) m ,i (cid:89) k =1 α E i,k ) (cid:33) G · ,i ) + h (cid:88) i =1 (cid:32) m ,i (cid:89) k =1 α E i,k ) (cid:33) G · ,i ) (cid:12)(cid:12)(cid:12)(cid:12) α E i,k ) , α E i,k ) ∈ [ − , (cid:41) = (cid:10) c, [ G , G ] , (cid:0) E , E (cid:1)(cid:11) Z . (9) epresentation of Polytopes as Polynomial Zonotopes 5 Proposition 1 (Linear Transformation) Given a set in Z-representation P = (cid:104) c, G, E(cid:105) Z ⊂ R n and amatrix M ∈ R m × n , the linear transformation is computed as M P = (cid:104) M c, M G,
E(cid:105) Z , (10) which has complexity O ( mnh ) , where h is the number of generators.Proof The result follows directly from inserting the definition of the Z-representation in (6) into the defi-nition of the linear transformation (2).
Complexity
The complexity results from the matrix multiplication with the starting point c and the gen-erator matrix G and is therefore O ( mn ) + O ( mnh ) = O ( mnh ). (cid:3) Proposition 2 (Minkowski Sum) Given two sets in Z-representation P = (cid:104) c , G , E (cid:105) Z and P = (cid:104) c , G , E (cid:105) Z , the Minkowski sum is computed as P ⊕ P = (cid:10) c + c , [ G , G ] , (cid:0) E , E (cid:1)(cid:11) Z with E = (cid:0) E + p , . . . , E h ) + p (cid:1) , (11) which has complexity O ( n + µ ) , where p is the number of factors of P and µ is the number of entriesin tuple E .Proof The proposition follows from inserting the definition of the Z-representation in (6) into the definitionof the Minkowski sum (3) and using (9): P ⊕ P = (cid:40) c + c (cid:124) (cid:123)(cid:122) (cid:125) c + h (cid:88) i =1 (cid:32) m ,i (cid:89) k =1 α E i,k ) (cid:33) G · ,i ) + h (cid:88) i =1 (cid:32) m ,i (cid:89) k =1 α E i,k ) + p (cid:124) (cid:123)(cid:122) (cid:125) α E i,k ) (cid:33) G · ,i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α E i,k ) , α E i,k ) ∈ [ − , (cid:41) (9) = (cid:10) c + c , [ G , G ] , (cid:0) E , E (cid:1)(cid:11) Z . (12) Complexity
The complexity for the addition of the center vectors is O ( n ), and the complexity for theconstruction of the tuple E is O ( µ ) with µ defined as in (7). Thus, the overall complexity is O ( n + µ ). (cid:3) Proposition 3 (Convex Hull) Given two sets in Z-representation P = (cid:104) c , G , E (cid:105) Z and P = (cid:104) c , G , E (cid:105) Z ,their convex hull is computed as conv ( P , P ) = (cid:28) c + c ) , c − c ) , G , G , G , − G ] , (cid:16) ( p ) , E , (cid:98) E , E , (cid:98) E (cid:17)(cid:29) Z with (cid:98) E = (cid:16) [ E T , p ] T , . . . , [ E T h ) , p ] T (cid:17) , E = (cid:0) E + p , . . . , E h ) + p (cid:1) , (cid:98) E = (cid:16) [ E T , p ] T , . . . , [ E T h ) , p ] T (cid:17) . (13) It holds that p = p + p + 1 h = 2 h + 2 h + 1 µ = 2 µ + 2 µ + h + h + 1 , (14) where p is the number of factors, h the number of generators, and µ the number of tuple entries ( see (7)) of conv ( P , P ) . The complexity is O ( n + µ ) , where µ is the number of entries in the tuple E . Niklas Kochdumper, Matthias Althoff
Proof
The result follows from inserting the definition of the Z-representation in (6) into the definition ofthe convex hull (4) and using (9): conv ( P , P ) = (cid:40)
12 ( c + c ) (cid:124) (cid:123)(cid:122) (cid:125) c + 12 ( c − c ) λ + 12 h (cid:88) i =1 (cid:32) m ,i (cid:89) k =1 α E i,k ) (cid:33) G · ,i ) +12 h (cid:88) i =1 λ (cid:32) m ,i (cid:89) k =1 α E i,k ) (cid:33) G · ,i ) + 12 h (cid:88) i =1 (cid:32) m ,i (cid:89) k =1 α E i,k ) + p (cid:124) (cid:123)(cid:122) (cid:125) α E i,k ) (cid:33) G · ,i ) − h (cid:88) i =1 λ (cid:32) m ,i (cid:89) k =1 α E i,k ) + p (cid:124) (cid:123)(cid:122) (cid:125) α E i,k ) (cid:33) G · ,i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α E i,k ) ∈ [ − , , α E i,k ) ∈ [ − , , λ ∈ [ − , (cid:41) (9)and α p := λ = (cid:28)
12 ( c + c ) ,
12 [( c − c ) , G , G , G , − G ] , (cid:16) ( p ) , E , (cid:98) E , E , (cid:98) E (cid:17)(cid:29) Z , (15)where p = p + p + 1 (see (14)). For the transformation in the last line of (15), we substitute λ with anadditional factor α p . With this substitution and ˆ E and ˆ E defined as in (13), it holds that λ m ,i (cid:89) k =1 α E i,k ) = α p m ,i (cid:89) k =1 α E i,k ) = ˆ m ,i = m ,i +1 (cid:89) k =1 α ˆ E i,k ) ,λ m ,i (cid:89) k =1 α E i,k ) = α p m ,i (cid:89) k =1 α E i,k ) = ˆ m ,i = m ,i +1 (cid:89) k =1 α ˆ E i,k ) . Since λ ∈ [ − ,
1] and α p ∈ [ − , λ with α p does not change the set.The number of generators h in (14) directly follows from the construction of the generator matrix0 . c − c , G , G , G , − G ]. The number of tuple entries µ in (14) results from the construction of thetuple (( p ) , E , (cid:98) E , E , (cid:98) E ), since E has µ entries, ˆ E has µ + h entries, E has µ entries, and ˆ E has µ + h entries. Complexity
The complexity for the addition and subtraction of the center vectors is O (2 n ), and the com-plexity for the construction of the set E is O ( µ ) with µ defined as in (7). Thus, the overall complexityis O ( n + µ ). (cid:3) Contrary to previous set operations where the computation in Z-representation is straightforward, thecalculation of the intersection is non-trivial. At present, there exists no algorithm to compute the intersectiondirectly in Z-representation without conversion to another polytope representation.
Theorem 1
Every bounded polytope can be equivalently represented in Z-representation.Proof
If the polytope is bounded, then the set can be described as the convex hull of its vertices (see Def.1). Each vertex v i ∈ R n can be equivalently represented by the Z-representation v i = (cid:104) v i , [ ] , ∅(cid:105) Z . Since theZ-representation is closed under the convex hull operation as shown in Prop. 3, computation of the convexhull results in a Z-representation of the polytope. (cid:3) An algorithm for the conversion of a polytope in V-representation to a polytope in Z-representation isprovided in the next subsection. epresentation of Polytopes as Polynomial Zonotopes 7
Algorithm 1
Conversion from V-representation to Z-representation
Require:
Bounded polytope in V-representation P = (cid:104) [ v , . . . , v q ] (cid:105) V Ensure:
Z-representation P = (cid:104) c, G, E(cid:105) Z of the polytope1: K = ∅ for i ← q do K ← (cid:0) K , (cid:104) v i , [ ] , ∅(cid:105) Z (cid:1) end for while |K| > do (cid:98) K ← ∅ while |K| ≥ do (cid:98) K ← (cid:0) (cid:98) K , conv ( K (1) , K (2) ) (cid:1) if |K| > then K ← (cid:0) K (3) , . . . , K ( |K| ) (cid:1) else K ← ∅ end if end while if |K| == 1 then (cid:98) K ← (cid:0) (cid:98) K , K (1) (cid:1) end if K ← (cid:98) K end while (cid:104) c, G, E(cid:105) Z ← K (1) The algorithm is structured as follows: First, all vertices of the polytope are converted to Z-representationduring the for-loop in lines 2-4, and the result is stored in the tuple K . The remainder of the algorithmcan then be viewed as the exploration of a binary tree as it is shown in Fig. 3, where the nodes of the treeare polytopes in Z-representation. Each iteration of the outer while-loop in lines 5-19 of Alg. 1 representsthe exploration of one level of the tree. For each of these levels, the inner while-loop in lines 7-14 visits allnodes at one level and computes the convex hull of two nodes to form one node of the next higher level ofthe tree. If the root node of the binary tree is reached, all polytope vertices have been united by the convexhull, and the root element is the desired Z-representation of the polytope P . We demonstrate Alg. 1 witha short example: Example 3
The conversion of the polytope P = (cid:28)(cid:20)(cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21)(cid:21)(cid:29) V , from V-representation to Z-representation with Alg. 1 is visualized in Fig. 2. The algorithm terminatesafter 3 iterations. The Z-representation of a polytope is not unique. If Alg. 1 is used for the conversion of a polytopegiven in V-representation, then the resulting Z-representation depends on the order of the vertices in thematrix [ v , . . . , v q ], since this order defines which vertices are combined by the convex hull operation.Therefore, it might be meaningful to sort the matrix [ v , . . . , v q ] before applying Alg. 1 in order to obtaina Z-representation with desirable properties. For example, to minimize the length of the vectors in thegenerator matrix of the Z-representation, the vertices have to be sorted so that vertices located close toeach other are combined first. The computational complexity of the algorithm can be derived as follows: Proposition 4
The computational complexity of the conversion from V-representation to Z-representationusing Alg. 1 is O ( q log ( q )+ nq ) with respect to q , where n is the dimension and q is the number of polytopevertices.Proof The for-loop in lines 2-4 of Alg. 1 can be ignored since it only involves initializations. Let us firstconsider the case where q = 2 k , k ∈ N : each iteration of the outer while-loop in lines 5-19 of Alg. 1corresponds to one level of a perfect binary tree with depth k = log ( q ) (see Fig. 3). Each node at level i = 0 . . . k is a polytope in Z-representation P ( i ) = (cid:104) c ( i ) , G ( i ) , E ( i ) (cid:105) Z with G ( i ) ∈ R n × h ( i ) , and µ ( i ) denoting Niklas Kochdumper, Matthias Althoff -1 0 1 2 3 4 5 6 x -101234567 x -1 0 1 2 3 4 5 6 x -101234567 x -1 0 1 2 3 4 5 6 x -101234567 x Fig. 2: First (left), second (middle), and third (right) iteration of Alg. 1 applied to the polytope P fromExample 3.the number of entries in the list E ( i ) (see (7)). From (14) we can derive the number of factors p ( i ) , thenumber of generators h ( i ) , and the number of tuple entries µ ( i ) of a node at level i of the binary tree; thevalues for a perfect binary tree on level i are: p ( i ) = 2 p ( i − + 1 = 2 i p (0) + i − (cid:88) j =0 j h ( i ) = 4 h ( i − + 1 = 4 i h (0) + i − (cid:88) j =0 j µ ( i ) = 4 µ ( i − + 2 h ( i − + 1 = 4 µ (0) + i − (cid:88) j =0 j (cid:16) h ( i − − j ) (cid:17) . (16)The nodes at the bottom level of the binary tree are the polytope vertices v l represented in Z-representation v l = (cid:104) v l , [ ] , ∅(cid:105) Z , so that p (0) = 0, h (0) = 0, and µ (0) = 0. Inserting these values into (16) and using thefinite sum of the geometric series z (cid:88) j =0 r j = 1 − r z +1 − r , r ∈ R , one obtains p ( i ) = i − (cid:88) j =0 j geom . series = 2 i − h ( i ) = i − (cid:88) j =0 j geom . series = i − µ ( i ) = i − (cid:88) j =0 j (cid:18) i − − j − (cid:19) = 13 i − (cid:88) j =0 j + 23 i − (cid:88) j =0 i − . series = 4 i (cid:18) i + 19 (cid:19) − . (17)Each level i of the tree contains 2 k − i nodes, where k is the depth of the tree. Consequently, at each level2 k − i convex hull operations have to be computed, where each operation involves 2 n + µ ( i − additions (seeProp. 3). The number of necessary basic operations O required for the conversion with Alg. 1 is therefore O = k (cid:88) i =1 k − i (cid:16) n + µ ( i − (cid:17) . (18)In the general case the binary tree explored by Alg. 1 is not a perfect binary tree. However, the numberof operations required for the general case is obviously smaller than the number of operations required for epresentation of Polytopes as Polynomial Zonotopes 9 Fig. 3: Example of a perfect binary tree as explored by Alg. 1.the exploration of a perfect binary tree with depth (cid:100) log ( q ) (cid:101) . Since it holds that (cid:100) log ( q ) (cid:101) = log ( q ) + a , a ∈ [0 , k = (cid:100) log ( q ) (cid:101) = log ( q ) + a for the tree depth in (18) yields O = log ( q )+ a (cid:88) i =1 log ( q )+ a − i (cid:16) n + µ ( i − (cid:17) = 2 a log ( q )+ a (cid:88) i =1 q i (cid:16) n + µ ( i − (cid:17) µ ( i − from (17) = 2 a (cid:18) n − (cid:19) q log ( q )+ a (cid:88) i =1 i (cid:124) (cid:123)(cid:122) (cid:125) geom . series = 1 − aq − a q log ( q )+ a (cid:88) i =1 i (cid:124) (cid:123)(cid:122) (cid:125) geom . series = 2(2 a q − + 2 a q log ( q )+ a (cid:88) i =1 i · i (cid:124) (cid:123)(cid:122) (cid:125) (see [12, Chapter 1.2.2.3]) =2(1+2 a q ( a − a log ( q ) q ) = 4 a q log ( q ) + 4 a
36 (3 a − q + 2 a +1 nq − n + 19 . The complexity of Alg. 1 is therefore O ( q log ( q ) + nq ) with respect to q . (cid:3) Proposition 5
Given a polytope P = (cid:104) c, G, E(cid:105) Z in Z-representation, the polytope vertices are a subset ofthe finite set K defined as K = (cid:40) c + h (cid:88) i =1 (cid:32) m i (cid:89) k =1 (cid:98) α E ( i,k ) (cid:33) G ( · ,i ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:98) α = [ (cid:98) α , . . . , (cid:98) α p ] T ∈ vertices ( I ) (cid:41) , where the operation vertices returns the p vertices of the hypercube I = [ − , ] ⊂ R p .Proof We have to show that each vertex v ( j ) of the polytope P corresponds to one vertex (cid:98) α ( j ) of thehypercube I : v ( j ) = c + h (cid:88) i =1 (cid:32) m i (cid:89) k =1 (cid:98) α ( j ) E ( i,k ) (cid:33) G ( · ,i ) . (19)As shown in [20, Chapter 7.2(d)], for each vertex v ( j ) of a polytope P there exists a vector d j ∈ R n suchthat v ( j ) = argmax s ∈P d Tj s. (20) By using (19), (20) can be equivalently formulated as v ( j ) = c + h (cid:88) i =1 (cid:32) m i (cid:89) k =1 α ∗E ( i,k ) (cid:33) G ( · ,i ) with [ α ∗ , . . . , α ∗ p ] T = argmax [ α ,...,α p ] T ∈I d Tj (cid:32) c + h (cid:88) i =1 (cid:32) m i (cid:89) k =1 α E ( i,k ) (cid:33) G ( · ,i ) (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) f ( α ,...,α p ) . (21)We therefore have to show that the point α ∗ = [ α ∗ , . . . , α ∗ p ] T where the function f ( · ) from (21) reaches itsextremum within the domain [ α , . . . , α p ] T ∈ I is identical to a vertex (cid:98) α ( j ) of the hypercube I . Since thefunction f ( · ) from (21) does not contain polynomial exponents greater than 1 (see (6)), it holds that thepartial derivative of f ( · ) with respect to variable α i does not depend on α i : ∀ i ∈ { , . . . , p } : ∂f ( α , . . . , α p ) ∂α i = g i ( α , . . . , α i − , α i +1 , . . . , α p ) . (22)According to (22) the function f ( · ) therefore reaches its extremum on the domain α i ∈ [ − ,
1] at either α ∗ i = 1 or α ∗ i = −
1. Since this holds for all α i , i = 1 , . . . , p , function f ( · ) reaches its extremum withinthe domain [ α , . . . , α p ] T ∈ I at the point α ∗ = [ α ∗ , . . . , α ∗ p ] T with α ∗ j ∈ {− , } , j = 1 , . . . , p , which is avertex of the hypercube I . (cid:3) Algorithm 2
Conversion from Z-representation to V-representation
Require:
Bounded polytope in Z-representation P = (cid:104) c, G, E(cid:105) Z Ensure:
V-representation P = (cid:104) [ v , . . . , v q ] (cid:105) V of the polytope1: I ← [ − , ] ⊂ R p (cid:8)(cid:98) α (1) , . . . , (cid:98) α (2 p ) (cid:9) ← vertices ( I )3: K ← ∅ for j ← p do v ← c + (cid:80) hi =1 (cid:16)(cid:81) m i k =1 (cid:98) α ( j ) E ( i,k ) (cid:17) G ( · ,i ) if v / ∈ K then K ← K ∪ v end if end for
10: [ v , . . . , v q ] ← convexHull ( K )11: P ← (cid:104) [ v , . . . , v q ] (cid:105) V Alg. 2 shows, based on Prop. 5, the conversion from Z-representation to V-representation. The operation vertices in lines 2 of Alg. 2 returns the vertices of the hypercube I . In the for-loop in line 4-9 of Alg.2 we compute the potential polytope vertices for each of these hypercube vertices according to Prop. 5.We check if the potential vertex v is already part of the set K in line 6 since this decreases the averageruntime of the algorithm. The points stored in K define a redundant V-representation of the polytope P .Redundant points are removed by computation of the convex hull in line 10 of Alg. 2, where the operation convexHull returns the vertices of the convex hull. For Z-representations that define non-convex sets (seee.g., Example 2), Alg. 2 returns the convex hull of the set.Since it is required for the derivation of the computation complexity, we first derive a formula for themaximum number of generators and the maximum number of tuple entries for a regular Z-representationwith a fixed number of factors: Proposition 6
Given a regular Z-representation P = (cid:104) c, G, E(cid:105) Z with p factors, it holds that h ( p ) ≤ p − µ ( p ) ≤ p p − (23) are upper bounds for the number of generators h ( p ) and the number of entries µ ( p ) in the tuple E .Proof For a regular Z-representation (see (8)), the maximum number of generators is equal to the numberof different monomial variable parts α E ( i, · . . . · α E ( i,mi ) that can be constructed with p factors α E ( i,k ) . Given epresentation of Polytopes as Polynomial Zonotopes 11 a fixed m i , there exist (cid:0) pm i (cid:1) different monomial variable parts since there are (cid:0) pm i (cid:1) possible combinations tochoose m i from the p factors α E ( i,k ) without order. Summation over all m i ∈ { , . . . , p } therefore yields h ( p ) ≤ p (cid:88) m i =1 (cid:32) pm i (cid:33) = (cid:32) p (cid:88) m i =0 (cid:32) pi (cid:33)(cid:33) − [22, Chapter 8.6 (7.)] = 2 p − . For each monomial, m i entries have to be stored in the tuple E . Consequently, µ ( p ) ≤ p (cid:88) m i =1 m i (cid:32) pm i (cid:33) = p (cid:88) m i =0 m i (cid:32) pm i (cid:33) [22, Chapter 8.6 (8.)] = p p − is an upper bound for the number of entries µ in the tuple E . (cid:3) Next, we derive the computational complexity of Alg. 2:
Proposition 7
The computational complexity of the conversion of a polytope
P ⊂ R n from a regular Z-representation to V-representation with Alg. 2 is O ((2 p ) (cid:98) n/ (cid:99) +1 + 4 p ( p + n )) with respect to p , where n isthe dimension and p is the number of factors of the Z-representation.Proof Upper bounds for the number of generators h ( p ) and the number of entries µ ( p ) in the tuple E for a regular Z-representation with p factors are given by Prop. 6. The hypercube I = [ − , ] ⊂ R p has2 p vertices. Therefore, the for-loop in lines 4-9 of Alg. 2 consists of 2 p iterations. In each iteration, thepotential vertex v has to be calculated according to Prop. 5. This requires nh ( p ) additions and at most µ ( p ) − nh ( p ) multiplications, resulting for all iterations in O = 2 p (2 nh ( p ) + µ ( p ) − (23) ≤ p (cid:18) p + 2 n (cid:19) − n p − K storing the potentialvertices in line 6 of Alg. 2 requires, in the worst case, O = p ( q ) (cid:88) i =1 n ( i − [12, Chapter 1.2.2.1] = n p − p ) (25)comparisons of scalar numbers. The complexity of the computations in the for-loop in lines 4-9 of Alg. 2is therefore O ( O + O ) = O (4 p ( p + n )) with respect to p . The Beneath-Beyond algorithm [15] for thecomputation of the convex hull of a n -dimensional point cloud with w points has complexity O ( w (cid:98) n/ (cid:99) +1 )[21, Theorem 3.16]. For Alg. 2, the point cloud stored in the set K consists of w = 2 p points in the worstcase, resulting in complexity O ((2 p ) (cid:98) n/ (cid:99) +1 ) with respect to p for the computation of the convex hull inline 10 of Alg. 2. Combining the complexity from the for-loop and from the convex hull computation resultsin an overall complexity of O ( O + O ) + O ((2 p ) (cid:98) n/ (cid:99) +1 ) = O (4 p ( p + n )) + O ((2 p ) (cid:98) n/ (cid:99) +1 ) (26)with respect to p . (cid:3) For general Z-representations, it is not possible to specify a relation between the number of polytopevertices and the number of factors. However, under the assumption that the Z-representation is obtainedby conversion from V-representation with Alg. 1, the number of factors can be expressed as a function ofthe number of vertices, which enables us to derive the computation complexity with respect to the numberof polytope vertices:
Proposition 8
The computational complexity of the conversion of a polytope P from Z-representation toV-representation with Alg. 2, where P is computed by conversion from V-representation with Alg. 1, is O (16 q n + (0 . · q ) (cid:98) n/ (cid:99) +1 ) with respect to q , where n is the dimension and q is the number of polytopevertices.Proof We first express the number of factors p , the number of generators h , and the number of list elements µ (see (7)) of a polytope in Z-representation as functions of the number of polytope vertices q . As statedin the proposition, we assume that the Z-representation of the polytope is obtained by conversion fromV-representation using Alg. 1. As shown in Sec. 3.4, we obtain over-approximations for p , h , and µ if we view Alg. 1 as the exploration of a perfect binary tree with depth k = (cid:100) log ( q ) (cid:101) . Since it holds that (cid:100) log ( q ) (cid:101) = log ( q ) + a , a ∈ [0 , ( q ) + a into (17) to obtain p ( q ) = p (log ( q )+ a ) = 2 a q − h ( q ) = h (log ( q )+ a ) = 4 a q − µ ( q ) = µ (log ( q )+ a ) = 4 a q (cid:18)
16 log ( q ) + 16 a + 19 (cid:19) − . (27)The overall complexity for the conversion with Alg. 2 is given by (26). According to (24) and (25), it holdsthat O = 2 p ( q ) (2 nh ( q ) + µ ( q ) − (27) = (4 a ) q (cid:18) a q (cid:18) n + 112 a + 118 (cid:19) + 112 4 a q log ( q ) − n − (cid:19) (28)and O = n (cid:16) p ( q ) − p ( q ) (cid:17) (27) = n a ) q − n a ) q . (29)Inserting (28) and (29) into (26) yields O ( O + O ) + O ((2 p ( q ) ) (cid:98) n/ (cid:99) +1 ) (27) , (28) , (29) = O ((16 a ) q n ) + O ((0 . a ) q ) (cid:98) n/ (cid:99) +1 )with respect to q , which is O (16 q n + (0 . · q ) (cid:98) n/ (cid:99) +1 ). (cid:3) In this section, we compare the representation complexity in V-representation, H-representation, and Z-representation for two special classes of polytopes. Given a polytope P , we denote by N V ( P ), N H ( P ) and N Z ( P ) the number of values required for V-, H-, and Z-representation, respectively. Furthermore, we denotethe number of i -dimensional polytope faces by F i ( P ). We first derive the representation complexity for V-,H-, and Z-representation: Proposition 9
The representation complexity of a polytope P with q vertices in V-representation is N V ( P ) = nq. Proof
For each of the q vertices, a vector with n entries has to be stored. (cid:3) Proposition 10
The representation complexity of an n -dimensional polytope P in H-representation is N H ( P ) = ( n + 1) F n − ( P ) . Proof
Each of the F n − ( P ) ( n − n + 1 values. (cid:3) Proposition 11
The representation complexity of a polytope P = (cid:104) c, G, E(cid:105) Z in Z-representation is N Z ( P ) = n ( h + 1) + µ, where h is the number of generators and µ is the number of entries in the tuple E .Proof The center vector c ∈ R n consists of n values, the generator matrix G ∈ R n × h of nh values, and thetuple E of µ values (see (7)). (cid:3) C = conv ( Z , d ) that can be described by the convex hull ofa zonotope Z ⊂ R n and a single point d ∈ R n . A zonotope is a special type of a polytope or a polynomialzonotope [26]. Using the shorthand for the Z-representation, a zonotope is Z = (cid:104) c, G, (1 , . . . , m ) (cid:105) Z , where c ∈ R n is the center vector, G ∈ R n × m is the generator matrix, and m is the number of zonotope generators.Next, we derive the representation complexity of the polytope C in V-, H-, and Z-representation. epresentation of Polytopes as Polynomial Zonotopes 13 V-Representation
The number of vertices q n ( m ) of an n -dimensional zonotope with m generators is [7, Prop. 2.1.2] q n ( m ) = 2 min( n,m ) − (cid:88) i =0 (cid:32) m − i (cid:33) . (30)The polytope C defined by the convex hull of a zonotope and a point has, in the worst case, q n ( m ) + 1vertices. According to Prop. 9, N V ( C ) ≤ n ( q n ( m ) + 1) (30) = n min( n,m ) − (cid:88) i =0 (cid:32) m − i (cid:33) is an upper bound for the representation complexity. H-Representation An n -dimensional zonotope Z with m generators has at most 2 (cid:0) mn − (cid:1) facets [1, Chapter 2.2.1]. The maximumnumber of facets for the set C = conv ( Z , d ) is obtained in the case where d (cid:54)∈ Z , and all facets of Z exceptfor one facet ˆ F are facets of C . It therefore holds that F n − ( C ) ≤ F n − ( Z ) − (cid:124) (cid:123)(cid:122) (cid:125) F A + F n − ( conv ( ˆ F , d )) − (cid:124) (cid:123)(cid:122) (cid:125) F B , (31) where F A is the number of facets of Z that are facets of C , and F B is the number of additional facetsresulting from the convex hull of the facet ˆ F with the point d . The number of facets for the convex hullof facet ˆ F and point d is identical to F n − ( conv ( ˆ F , d )) = F n − ( ˆ F ) + 1, where F n − ( ˆ F ) is the number of( n − F . An upper bound for F n − ( ˆ F ) is 2 (cid:0) mn − (cid:1) , which corresponds to thecase where ˆ F is an ( n − m generators. Inserting F n − ( Z ) ≤ (cid:0) mn − (cid:1) and F n − ( conv ( ˆ F , d )) ≤ (cid:0) mn − (cid:1) + 1 into (31) yields F n − ( C ) ≤ (cid:32) mn − (cid:33) − (cid:32) mn − (cid:33) . (32)According to Prop. 10, an upper bound for the representation complexity is N H ( C ) = ( n + 1) F n − ( C ) (32) ≤ ( n + 1) (cid:32) (cid:32) mn − (cid:33) − (cid:32) mn − (cid:33)(cid:33) . Z-Representation
For the zonotope Z = (cid:104) c, G, (1 , . . . , m ) (cid:105) Z , we have h z = m and µ z = m . The point d can be representedwith the Z-representation d = (cid:104) d, [ ] , ∅(cid:105) Z with h d = 0 and µ d = 0. Computation of the convex hull C = conv ( Z , d ) therefore results according to (14) in a Z-representation with h c = 2 h z + 2 h d + 1 = 2 m + 1and µ c = 2 µ z + 2 µ d + h z + h d + 1 = 3 m + 1. The representation complexity for C according to Prop. 11 is N Z ( C ) = n ( h c + 1) + µ c = 2 n + 2 mn + 3 m + 1 . C = conv ( Z , Z ) defined by the convex hull of a full-dimensionalzonotope Z = (cid:104) c , G , (1 , . . . , m ) (cid:105) Z with m generators and a full-dimensional zonotope Z = (cid:104) c , G , (1 , . . . , m ) (cid:105) Z with m generators. For the Z-representation, the exact representation complexity can becomputed. Since the number of facets and vertices of C depends on the shape of the two zonotopes, theexact representation complexity for the V-representation and the H-representation cannot be computed forthe case of general zonotopes. We therefore consider the case where the zonotope with fewer generatorsencloses the second zonotope, which results in a minimum number of facets and vertices for C , and thereforerepresents the best case for the V- and H-representation. m N N V N H N Z Fig. 4: Representation complexity of a 3-dimensional polytope defined by the convex hull of a zonotopewith m generators and a single point. Exact values are visualized by solid lines, and upper bounds arevisualized by dashed lines. V-Representation
The number of vertices q n ( m ) of an n -dimensional zonotope with m generators is given by (30). A lowerbound for the number of vertices of the polytope C is therefore q n (min( m , m )). According to Prop. 9, N V ( C ) ≥ n q n (min( m , m )) = 2 n min( n,m ,m ) − (cid:88) i =0 (cid:32) min( m , m ) − i (cid:33) is a lower bound for the representation complexity. H-Representation An n -dimensional zonotope with m generators has at most 2 (cid:0) mn − (cid:1) facets [1, Chapter 2.2.1]. Therefore,2 (cid:0) min( m ,m ) n − (cid:1) is a lower bound for the number of facets of the set C . According to Prop. 10, N H ( C ) ≥ (cid:32) min( m , m ) n − (cid:33) ( n + 1)is a lower bound for the representation complexity. Z-Representation
For the two zonotopes Z = (cid:104) c , G , (1 , . . . , m ) (cid:105) Z and Z = (cid:104) c , G , (1 , . . . , m ) (cid:105) Z , we have h = m , µ = m , h = m , and µ = m . Computation of the convex hull C = conv ( P , P ) therefore resultsaccording to (14) in a Z-representation with h c = 2 h + 2 h + 1 = 2 m + 2 m + 1 and µ c = 2 µ + 2 µ + h + h + 1 = 3 m + 3 m + 1. The representation complexity for C according to Prop. 11 is N Z ( C ) = n c ( h c + 1) + µ c = 2 n ( m + m + 1) + 3 m + 3 m + 1 . Convex Hull of a Zonotope and a Point
The visualization of the representation complexity in Fig. 4 shows that depending on the polytope the Z-representation can be significantly more compact than the V- and H-representation, especially for zonotopeswith many generators. We further demonstrate this with a numerical example:
Example 4
The polytope C = conv ( I , d ) corresponding to the convex hull of the point d = [2 , ] T ∈ R and the hypercube I = [ − , ] ⊂ R has representation size N V ( C ) = 10485770, N H ( C ) = 1617, and N Z ( C ) = 901. epresentation of Polytopes as Polynomial Zonotopes 15 V-repH-repZ-rep
Fig. 5: Representation with the smallest representation complexity for an n -dimensional polytope C resultingfrom the convex hull of a zonotope with m generators and a zonotope with m generators. Convex Hull of Two Zonotopes
A visualization of the representation with the minimal representation complexity for different value combi-nations of n , m , and m is shown in Fig. 5. The figure demonstrates that for high-dimensional zonotopeswith a large number of generators, the Z-representation has the smallest representation complexity, eventhough we used lower bounds for the representation complexity in V- and H-representation.For general polytopes, the V- and H-representation are usually more compact than the Z-representation. One application of the introduced Z-representation of polytopes is range bounding of nonlinear functions,which is defined as follows:
Definition 5 (Range Bounding) Given a function f : R n → R and a set S ⊂ R n , the range boundingoperation bound ( f ( x ) , S ) ⊇ (cid:20) min x ∈S f ( x ) , max x ∈S f ( x ) (cid:21) returns an over-approximation of the exact bounds.Two common approaches for range bounding are interval arithmetic [11] and affine arithmetic [5].But due to the dependency problem [11], these techniques often result in large over-approximations. Inaddition, both approaches require the over-approximation of the set S by an axis-aligned interval, whichfurther increases the conservatism in the calculated bounds. One can also use Taylor models [18] for rangebounding, which often enable the computation of significantly tighter bounds [3]. However, the Taylor modelapproach still requires over-approximating the set S by an axis-aligned interval. Due to the similaritybetween polynomial zonotopes and Taylor models, the principles behind the Taylor model approach forrange bounding can easily be transferred to polynomial zonotopes, and consequently also to polytopesin Z-representation. The main advantage of using polynomial zonotopes for range bounding is that it isnot required to over-approximate the set S with an axis-aligned interval. This enables the computation of significantly tighter function bounds, as the following example demonstrates: Example 5
We consider the nonlinear function f ( x , x ) = − ( x − . − ( x − + 4 cos( x ) sin( x )and the polytope in Z-representation P = (cid:28)(cid:20) − . (cid:21) , (cid:20) − . . − . (cid:21) , (cid:18) , , (cid:20) (cid:21)(cid:19)(cid:29) Z . The function and the polytope are visualized in Fig. 6. For comparison, we determined the under-approximation [ − . , . I ⊇ P of the polytope, we obtain bound ( f ( x , x ) , I ) = [ − . , bound ( f ( x , x ) , I ) = [ − . , . bound ( f ( x , x ) , P ) = [ − . , . -202-15-10 1 2 f ( x , x ) -5 1 x x x -2-1.5-1-0.500.511.52 x Fig. 6: Visualization of the function f ( x , x ) (left) and the polytope P (red, right) from Example 5. Inaddition, the figure on the right shows isoclines of the function f ( x , x ). In this work, we introduced the novel Z-representation of bounded polytopes. Contrary to all other knownrepresentations of polytopes, this new representation enables the computation of linear transformation,Minkowski sum, and convex hull with a computational complexity that is polynomial in the representa-tion size. We further specified algorithms for the conversion between Z-representation and other polytoperepresentations. The complexity of the conversion from the V-representation to Z-representation is poly-nomial in the number of polytope vertices, and the conversion from Z-representation to V-representationhas exponential complexity. In addition, we show that depending on the polytope, the Z-representation canbe more compact than the H- and the V-representation. One application for the Z-representation is rangebounding, which enables tighter bounds as demonstrated by a numerical example.
Acknowledgements
The authors gratefully acknowledge financial support by the German Research Foundation (DFG)project faveAC under grant AL 1185/5 1.
References
1. Althoff, M.: Reachability analysis and its application to the safety assessment of autonomous cars. Dissertation,Technische Universit¨at M¨unchen (2010)2. Althoff, M.: Reachability analysis of nonlinear systems using conservative polynomialization and non-convex sets. In:Hybrid Systems: Computation and Control, pp. 173–182 (2013)3. Althoff, M., Grebenyuk, D., Kochdumper, N.: Implementation of taylor models in CORA 2018. In: Proc. of the 5thInternational Workshop on Applied Verification for Continuous and Hybrid Systems (2018)4. Althoff, M., Krogh, B.H.: Zonotope bundles for the efficient computation of reachable sets. In: Proc. of the 50th IEEEConference on Decision and Control, pp. 6814–6821 (2011)5. de Figueiredo, L.H., Stolfi, J.: Affine arithmetic: Concepts and applications. Numerical Algorithms (1-4), 147–158(2004)6. Fukuda, K.: From the zonotope construction to the Minkowski addition of convex polytopes. Journal of SymbolicComputation (4), 1261–1272 (2004)7. Gritzmann, P., Sturmfels, B.: Minkowski addition of polytopes: computational complexity and applications to Gr¨obnerbases. SIAM Journal on Discrete Mathematics , 246–269 (1993)8. Gr¨otschel, M., Henk, M.: The representation of polyhedra by polynomial inequalities. Discrete & ComputationalGeometry (4), 485–504 (2003)9. Hagemann, W.: Efficient geometric operations on convex polyhedra, with an application to reachability analysis ofhybrid systems. Mathematics in Computer Science , 283–325 (2015)10. Hibi, T.: Some results on ehrhart polynomials of convex polytopes. Discrete mathematics (1), 119–121 (1990)11. Jaulin, L., Kieffer, M., Didrit, O.: Applied Interval Analysis. Springer (2006)12. Jeffrey, A., Dai, H.H.: Handbook of mathematical formulas and integrals. Elsevier (2008)13. Jones, C.N., Kerrigan, E.C., Maciejowski, J.M.: On polyhedral projection and parametric programming. Journal ofOptimization Theory and Applications , 207–220 (2008)14. Kaibel, V., Pfetsch, M.E.: Algebra, Geometry and Software Systems, chap. Some Algorithmic Problems in PolytopeTheory, pp. 23–47. Springer (2003)15. Kallay, M.: Convex hull algorithms for higher dimensions (1981)16. Khovanskii, A.G.: Newton polyhedra and toroidal varieties. Functional analysis and its applications (4), 289–296(1977)17. Kochdumper, N., Althoff, M.: Sparse polynomial zonotopes: A novel set representation for reachability analysis (2019)epresentation of Polytopes as Polynomial Zonotopes 1718. Makino, K., Berz, M.: Taylor models and other validated functional inclusion methods. International Journal of Pureand Applied Mathematics (4), 379–456 (2003)19. Minkowski, H.: Volumen und oberfl¨ache. Math. Ann. , 447–495 (1903)20. Padberg, M.: Linear optimization and extensions (2013)21. Preparata, F.P., Shamos, M.I.: Computational geometry: an introduction. Springer Science & Business Media (2012)22. Rade, L., Westergren, B.: Mathematics Handbook for Science and Engineering, 5 edn. Springer Science & BusinessMedia (2013)23. Scott, J.K., Raimondo, D.M., Marseglia, G.R., Braatz, R.D.: Constrained zonotopes: A new tool for set-based estima-tion and fault detection. Automatica , 126–136 (2016)24. Sturmfels, B.: Polynomial equations and convex polytopes. The American mathematical monthly (10), 907–922(1998)25. Tiwary, H.R.: On the hardness of computing intersection, union and Minkowski sum of polytopes. Discrete andComputational Geometry40