Multi-parametric Optimization and Control. Efstratios N. PistikopoulosЧитать онлайн книгу.

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

(1.27) equation

Additionally, a constraint images is strongly redundant if and only if

(1.28) equation

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

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 images , images , a constraint images is redundant if

(1.29) equation

where

(1.30) equation

where images and images . 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 images LP problems [6]:

(1.31) equation

1.3.1.2 Solution of Linear Programming Problem

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

(1.32) equation

where images denotes the element‐wise square of images . Note that images is assumed to be normalized such that images for all images . Then the images th constraint is redundant if and only if images . 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 images , i.e. which lies on the images th constraint. Thus, the solution can be understood as the center of the images th constraint with respect to images .

1.3.2 Projections

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

Definition 1.11 (Projection [7])

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

(1.33) equation

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

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