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
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 \(V=\mathbb R^n\) and \(L\) is a real matrix, \(\text{LCP}(L,q)\) has a unique solution for all \(q\in \mathbb R^n\), 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 \(\mathbb R^n\). 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.
scientific article; zbMATH DE number 5190762