Journal of Global Optimization

, Volume 73, Issue 1, pp 153–169 | Cite as

The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra

  • Xiaoni Chi
  • M. Seetharama GowdaEmail author
  • Jiyuan Tao


A weighted complementarity problem is to find a pair of vectors belonging to the intersection of a manifold and a cone such that the product of the vectors in a certain algebra equals a given weight vector. If the weight vector is zero, we get a complementarity problem. Examples of such problems include the Fisher market equilibrium problem and the linear programming and weighted centering problem. In this paper we consider the weighted horizontal linear complementarity problem in the setting of Euclidean Jordan algebras and establish some existence and uniqueness results. For a pair of linear transformations on a Euclidean Jordan algebra, we introduce the concepts of \(\mathbf{R}_0\), \(\mathbf{R}\), and \(\mathbf{P}\) properties and discuss the solvability of wHLCPs under nonzero (topological) degree conditions. A uniqueness result is stated in the setting of \({\mathbb {R}}^{n}\). We show how our results naturally lead to interior point systems.


Weighted horizontal linear complementarity problem Euclidean Jordan algebra Degree \(\mathbf{R}_0\)-pair 

Mathematics Subject Classification




The work of the first author is supported by the National Natural Science Foundation of China (No. 11401126) and Guangxi Natural Science Foundation (Nos. 2016GXNSFBA380102, 2014GXNSFFA118001), China. The third author was supported by Loyola Summer Research Grant 2017.


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Authors and Affiliations

  1. 1.School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and ComputationGuilin University of Electronic TechnologyGuilinPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsUniversity of Maryland, Baltimore CountyBaltimoreUSA
  3. 3.Department of Mathematics and StatisticsLoyola University MarylandBaltimoreUSA

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