A Calculation Approach to Scalarization for Polyhedral Sets by Means of Set Relations

Abstract

In this paper, we focus on certain functions as scalarization for six types of set relations and discuss calculation algorithms for them between polyhedral sets, while those between polytopes have been already investigated. A major difference between polyhedral sets and polytopes is in boundedness. Polyhedral sets are no longer necessarily bounded. Methods for calculating types (1), (2), (4), (6) are easily available by a similar way to existing ideas. However, those for types (3) and (5), which are actually the most famous and long-standing types, require some technical ways approaching to the value of them by using the fact that finitely generatedness and polyhedrality coincide and can be algorithmically switched in finite-dimensional spaces. As a result, we show all types are reduced to a finite number of linear programming problems. Also, we demonstrate our methods through an example and give detailed calculation process.