Linear systems in Jordan algebras and primal-dual interior-point algorithms (Q1378999)

The author discusses a possibility of the extension of a primal-dual interior-point algorithm for optimization problems of the form \[ \langle a,x\rangle \rightarrow \min X \in (b+X) \cap \Omega, \] where \(V\) is an Euclidean Jordan algebra, \(X\) is a vector subspace of \(V\), \(\Omega\) is the cone of squares of \(V\), \(a\in X, b \in X^\perp\). The question of solvability of a linear system arising in the implementation of the primal-dual algorithm is analyzed and a nondegeneracy theory for the considered class of problems is developed.