Minimal Representations of a Face of a Convex Polyhedron and Some Applications

Abstract

In this paper, we propose a method for determining all minimal representations of a face of a polyhedron defined by a system of linear inequalities. Main difficulties for determining prime and minimal representations of a face are that the deletion of one redundant constraint can change the redundancy of other constraints and the number of descriptor index pairs for the face can be huge. To reduce computational efforts in finding all minimal representations of a face, we prove and use properties that deleting strongly redundant constraints does not change the redundancy of other constraints and all minimal representations of a face can be found only in the set of all prime representations of the face corresponding to the maximal descriptor index set for it. The proposed method is based on a top-down search strategy, is easy to implement, and has many computational advantages. Based on minimal representations of a face, a reduction of degeneracy degrees of the face and ideas to improve some known methods for finding all maximal efficient faces in multiple objective linear programming are presented. Numerical examples are given to illustrate the method.

A combined top-down and bottom-up search method for determining all maximal efficient faces in multiple objective linear programming

“…The method given in Tohidi and Hassasi [16] uses the Vrepresentation of a face (a representation by finitely many points and directions). Since every face can be represented by a descriptor index set for it (see Tu [21] for more information), it is easily seen that the method given in Tohidi and Hassasi [16] is a local top-down search method for finding the efficient set of an MOLP problem. A subset M of {1, • • • , m} is eliminated from further consideration for the efficiency in a TDS method if there is a subset M 1 of {1, • • • , m} that has been considered for the efficiency in the TDS method such that M 1 ⊂ M and S(M 1 ) is empty or is efficient.…”

“…Remark 7.5 The maximal descriptor index set for an arbitrary face is a unique set among all descriptor index sets for it that can be used to identify this face. By this special property, investigations of degenerate or non-degenerate faces can be based on only the maximal descriptor index sets for them, for example, comparing descriptor sets for faces (Property 2.3), determining the dimensions of descriptor sets for faces, determining all faces of a convex polyhedron, the efficiency of descriptor sets for faces (Property 2.7), finding all maximal efficient faces, finding all minimal representations of a face [21]. The proposed method works with the maximal descriptor index sets for faces in which the maximal descriptor index sets for all faces whose dimensions are larger than 1 can be immediately found based on only the maximal descriptor index sets for edges and extreme rays without any computations (Properties 2.1 and 4.1).…”