Multi-parametric Optimization and Control. Efstratios N. Pistikopoulos

that satisfy the facet‐to‐facet property are adjacent, the opposite may not be true.

      1.3.1 Approaches for the Removal of Redundant Constraints

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

      Theorem 1.2 ([3])

      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)

      Remark 1.3

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 1.3 A 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 :

      Remark 1.4

      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.

      1.3.1.1 Lower‐Upper Bound Classification

      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)

      1.3.1.2 Solution of Linear Programming Problem

      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.

      Remark 1.5

The solution of problem (1.32) identifies the largest Euclidean ball which on the set , i.e. which lies on the th constraint. Thus, the solution can be understood as the center of the th constraint with respect to .

      1.3.2 Projections

      One of the operations used in this book is the (orthogonal) projection:

      Definition 1.11 (Projection [7])

      Let be a polytope. Then the projection of onto is defined as:

      (1.33)

Projecting polytopes is one of the fundamental operations in computational geometry and has many applications in control