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.
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