Convex polytope

B. Griinbanm, Convex Polytopes , Interscience Publishers, New York, 1967. 

Schlafli, L. Gesammelte mathematische Abhandlungen. Birkhauser, Basel (1950). Edited by Steiner-Schlafli-Komitee der Schweizerischen Naturforschenden Gesellschaft Tverbeig, H. How to cut a convex polytope into simplices. Geometriae Dedicata 3(2), 239-240(1974)... 

A hyperplane separates a space R" into two half-spaces. This property of hyperplanes is useful in AR construction as any convex polytope, P, that is either closed or unbounded may be constructed from a collection of appropriately orientated hyperplanes in R". When P is described as a set of... 

Hereafter, it is assumed that the reader is comfortable with the idea of the stoichiometric subspace S, which is discussed in Chapter 6. In this section, we shall describe how to numerically calculate the bounds of S (which is a convex polytope that is expressed as a collection of hyperplane constraints). 

Suppose that a system of reactions with associated kinetics is available, obeying a certain reaction stoichiometry. Since expressions for the rate of formation are known and available, it is possible to compute the AR for the system in concentration space. The particular region computed, defined by the kinetics for the system, will exist as a convex polytope residing in R . 

If the positions of the extreme points of S can be identified in extent space, then Equation 8.1 may be invoked to solve for the corresponding points in concentration space. Computing the extreme points of a convex polytope, defined by a set of hyperplane constraints, is termed vertex enumeration. 

A convex polytope, P, in IR" may be described independently, both in terms of its vertices and in terms of its facets. P is said to be given in the vertex representation (the V-representation), when the pol5tiope is described... 

The vertex enumeration problem involves computing the extreme points of the convex polytope, defined by the inequality constraints. 

Here, X is a matrix generated by the solution to the con2vert () function in MATLAB. Each row in X indicates a unique extreme point of the convex polytope, described by the system of inequalities. Therefore, the convex polytope contains four extreme points [0, 0], [0, 0.5], [1, Of, and [0.5, 0.5]. ... 

There are four extreme points to the convex polytope for this system, which indicates that there are four extreme points in extent space for the methane steam reforming reaction [1, 0] , [-0.25, 0.75] , [0, I] , and [0, 0] . ... 

This system describes six inequalities involving four variables the region is a four-dimensional convex polytope in X1-X2-X3-X4 space. Inputting the values into con2vert () in MATLAB gives... 

We find that all six vertices in 8j and 8j belong to 8 j. All three stoichiometric subspaces are shown in Figure 8.5. Note that 8(ot is a convex polytope that resides in a three-dimensional space, whereas 8[ and 82 alone are both two-dimensional subspaces, indicating that the AR from a single feed Cfi or is two-dimensional—they both exist as planes in c -Cb-Cc space. Thus, for multiple feeds, the dimension of the AR could exceed the number of independent reactions, as mixing between different stoichiometric subspaces is possible. 

Basic Idea Construction via two-dimensional slices is intuitive in principle. Candidate ARs are convex polytopes that always reside in the full space defined by... 

Initial polytope computation Given a feed point Cf and reaction stoichiometry, compute the stoichiometric subspace S in the H-representation. Let represent the convex polytope at iteration k of the construction process. Initially, Pq is equal to the stoichiometric subspace S. 

A candidate AR, represented by the shaded region, is given at iteration k of a typical construction process in the shrink-wrap method. The convex polytope, P , represents an approximation to the AR at iteration k in the construction, and is the set of extreme points contained within P. The... 

This reasoning outlines a basic elimination procedure for the construction of candidate ARs. Each extreme point C,- X, on the current convex polytope Pj., is checked for feasibility via backward CSTR solutions and PFR trajectories. If either the backward CSTR or PFR trajectory from C,- intersects the current polytope boundary, then it is retained. Otherwise, C, is removed from the current set of extreme points... 

Definition 2.38. A geometric polyhedral complex P in R is a collection of convex polytopes in R such that... 

P. McMullen and G. C. Shephard, Convex Polytopes and the Upper Bound Conjecture , Cambridge University Press, Cambridge, 1971. 

The main feature of a LP problem is that all functions involved, the objective function and those expressing the constraints, must be linear. The solution space or feasible region of an n-variables LP is geometrically defined by the intersection of the hyperplanes and halfspaces representing each of the constraints. Such a set is characterized as a convex polytope. 

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