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On an interconnection between the Lipschitz continuity of the solution map and the positive principal minor property in linear complementarity problems over Euclidean Jordan algebras. (English) Zbl 1138.90033
Let $V$ be an Euclidean Jordan algebra, $K$ be a symmetric cone in $V$, $L: V\longrightarrow V$ be a linear transformation and $q\in V$. The linear complementary problem associated to $L$ and $q$, $\text{LCP}(L,q)$, prescribes finding $x\in V$ such that $x\in K$, $Lx+q\in K$ and $\langle x,Lx+q\rangle=0$. It is well known that when $V=\Bbb R^n$ and $L$ is a real matrix, $\text{LCP}(L,q)$ has a unique solution for all $q\in \Bbb R^n$, iff all the principal minors of $L$ are positive. In this case the solution map of the $\text{LCP}(L,q)$ is well defined and Lipschitz continuous in $\Bbb R^n$. The main result of this paper establishes one direction of the analogous property in the general case: if the solution map is Lipschitz continuous and if $L$ has the $Q$-property, then $L$ has the positive principal minor property.

90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
17C30Automorphisms and other operators on Jordan algebras
17C50Jordan structures associated with other structures
Full Text: DOI
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