On theP_*(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 P * (κ) horizontal linear complementarity problem over Cartesian product of symmetric cones (Cartesian P * (κ)-SCHLCP). The concepts of column and row sufficiency are defined for a linear operator on a Euclidean Jordan algebra. Some connections between the P * (κ) property of a linear operator on a Euclidean Jordan algebra and its P 0 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 P * (κ)-SCHLCP. It is also demonstrated that the Cartesian P * (κ)-SCHLCP can be equivalent with a P * (κ) linear complementarity problem over Cartesian product of symmetric cones (Cartesian P * (κ)-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 P * (κ)-SCHLCP exists and is unique. Therefore, we open the way for presenting and extending the interior-point methods for the Cartesian P * (κ)-SCHLCP.

A full Nesterov–Todd step infeasible-interior-point algorithm for CartesianP*(κ) horizontal linear complementarity problems over symmetric cones

“…By this assumption, in [27] it is proved that the perturbed system (2) has a unique solution for each μ > 0. These solutions for the so-called central trajectory (central path) for the Cartesian P * (κ)−SCHLCP.…”

“…The form of the SCHLCP is as follows [27]: Find a vector pair (x, s) ∈ J × J such that Qx + Rs = q, x, s = 0, x, s ∈ K, (P) where K is the symmetric cone of the (Cartesian product space) Euclidean Jordan algebra J and Q, R : J → J are linear operators and q ∈ J .…”

On sufficient properties of sufficient matrices

“…Recently, new new interior-point algorithms were developed, which are suitable for solving (LCP) with sufficient matrices and use the method of algebraically equivalent transformation of the nonlinear system which defines the central path [7,8]. Extension of (LCP) to Cartesian product of symmetric cones was studied in [2]. Additionally, they show that the central trajectory exists and is unique, which is important to develop interior-point methods for the extended version of (LCP).…”

Feasible Corrector-Predictor Interior-Point Algorithm for $P_{*} (\kappa)$-Linear Complementarity Problems Based on a New Search Direction

“…Potra [59] generalized this method to LCP with a P * (κ) matrix. The IPAs for solving sufficient LCPs (i.e., LCP with sufficient matrix M -see Definition (2.3)) have been also extended to general LCPs [38,39] and to P * (κ)-LCPs over symmetric cones [7,45,63].…”