Efstratios N. Pistikopoulos - Multi-parametric Optimization and Control

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 1.2A 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.

A concept that is very important in multi‐parametric programming is the aspect of redundancy:

Consider an ‐dimensional compact polytope in halfspace representation. A constraint is redundant if and only if

(1.27)

Additionally, a constraint is strongly redundant if and only if

(1.28)

A constraint is called weakly redundant if it is redundant but not strongly redundant, i.e. Eq. (1.27)but not Eq. (1.28)holds. A schematic representation of weakly and strongly redundant constraints is shown in Figure 1.3.

If a polytope does not feature any redundant constraints, it is said to be in minimal representation.

Figure 1.3A schematic representation of (a) strongly and (b) weakly redundant constraints.

Consider an ‐dimensional compact polytope , where and . The following strategies aim at identifying the minimal representation of :

Here, two of the most common approaches used are reported. The field of the removal of redundant constraints has been widely studied, and its review is beyond the scope of this book. The reader is referred to [3,4] for an interesting treatment of the matter.

Given the bounds , , a constraint is redundant if

(1.29)

where

(1.30)

where and . This approach relies on the identification of the worst‐case scenario given the lower and upper bounds [5]. If these bounds are not available, they can be calculated by solving the following LP problems [6]:

(1.31)

Consider the following constraint‐specific version of problem (1.26):

(1.32)

where denotes the element‐wise square of . Note that is assumed to be normalized such that for all . Then the th constraint is redundant if and only if . Note that this identifies weakly and strongly redundant constraints.