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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

      Theorem 1.2 ([3])

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

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

      Remark 1.3

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

Image described by caption.

      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

      (1.31)equation

      1.3.1.2 Solution of Linear Programming Problem

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

      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 imagesth constraint is redundant if and only if images. Note that this identifies weakly and strongly redundant constraints.

      Remark 1.5

      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

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