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

Ta Van Tu

doi:10.1007/s40306-021-00437-y

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.

Highlights

Convex polyhedrons are widely used in theoretical and practical problems
We first propose a method for determining all minimal representations of a given face of a convex polyhedral set, we show some applications of minimal representations of a face
To reduce computational efforts for 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 obtained by finding only the set of all prime representations of the face corresponding to the maximal descriptor index set for it

Summary

Introduction

Convex polyhedrons are widely used in theoretical and practical problems. representations of a convex polyhedron play an important role in investigating and solving many problems. Is called a prime representation of a face F corresponding to a descriptor index set K for F if (I, J ) is a weak reduction of (K, K ) and contains no redundant indices for S(I, J ). The main difficulty for determining all prime and minimal representations of a face is 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 large. To reduce computational efforts for 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 obtained by finding only the set of all prime representations of the face corresponding to the maximal descriptor index set for it.

Some Properties of Minimal Representations of a Face of a Polyhedron

An Algorithm for Determining All Minimal Representations of a Given Face

Examples

Some Applications of Minimal Representations of a Face

Conclusions