Читать книгу Multi-parametric Optimization and Control - Efstratios N. Pistikopoulos - Страница 36

The set is called a ‐dimensional polytope if it satisfies

(1.23)

where is finite.

A schematic representation of a polytope is given in Figure 1.1.

Figure 1.1 A schematic representation of a two‐dimensional polytope .

In addition to Definition (1.10), the following well‐known characteristics of polytopes are considered:

 A polytope is called bounded if and only if there exists a finite and such for all .

 A polytope, which is closed and bounded, is called compact.

 Let be an ‐dimensional polytope. Then, a subset of a polytope is called a face of if it can be represented as(1.24) for some inequality , which holds for all . The faces of polytopes of dimension , 1, and 0 are referred to as facets, edges, and vertices, respectively.

 Two polytopes and are called disjoint if . Similarly, two polytopes and are called overlapping if . Lastly, two polytopes and are called adjacent or neighboring if is a ‐dimensional polytope.

 Let and be two adjacent polytopes. Then the facet‐to‐facet property is said to hold if is a facet of both and (see Figure 1.2 for an illustration).

 Let be an ‐dimensional polytope. Then, there exists a series of vertices such that(1.25)

 Eq. (1.23) is referred to the halfspace (or H) representation, while Eq. (1.25) denotes the vertex (or V) representation. The process of moving from the halfspace to the vertex representation is referred to as vertex enumeration.

 The Chebyshev center of a polytope is given as the largest Euclidean ball that lies in a polytope [2]. It can be determined by solving the following linear programming (LP) problem:(1.26) where the solution denotes the radius of the largest Euclidean ball. Based on the solution of problem (1.26), the following conclusions can be drawn:– Problem (1.26) is infeasible: The polytope is empty.– : The polytope is lower‐dimensional.– : The polytope is full‐dimensional.

Figure 1.2 A schematic representation of the differences between two polytopes and (a), which are adjacent and (b) where the facet‐to‐facet property holds. Clearly, while all polytopes that satisfy the facet‐to‐facet property are adjacent, the opposite may not be true.