On the P*(κ) horizontal linear complementarity problems over Cartesian product of symmetric cones

Abstract

The aim of this paper is to extend the theoretical framework of the horizontal linear complementarity problem over Cartesian product of symmetric cones (Cartesian-SCHLCP). The concepts of column and row sufficiency are defined for a linear operator on a Euclidean Jordan algebra. Some connections between the property of a linear operator on a Euclidean Jordan algebra and its property as well as its column sufficiency are presented. Then these definitions and connections are generalized to a pair of linear operators determining the Cartesian -SCHLCP. It is also demonstrated that the Cartesian -SCHLCP can be equivalent with a linear complementarity problem over Cartesian product of symmetric cones (Cartesian -SCLCP), in a certain sense. Finally, based on some obtained results and using a suitable potential function, the main task of the paper is done, which is to show that the central trajectory of the Cartesian -SCHLCP exists and is unique. Therefore, we open the way for presenting and extending the interior-point methods for the Cartesian-SCHLCP.