A new infeasible interior‐point method based on Darvay’s technique for symmetric optimization

We present a full Nesterov and Todd step primal‐dual infeasible interior‐point algorithm for symmetric optimization based on Darvay’s technique by using Euclidean Jordan algebras. The search directions are obtained by an equivalent algebraic transformation of the centering equation. The algorithm decreases the duality gap and the feasibility residuals at the same rate. During this algorithm we construct strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. Each main iteration of the algorithm consists of a feasibility step and some centering steps. The starting point in the first iteration of the algorithm depends on a positive number ξ and it is strictly feasible for a perturbed pair. The feasibility steps find strictly feasible iterates for the next perturbed pair. By using centering steps for the new perturbed pair, we obtain strictly feasible iterates close to the central path of the new perturbed pair. The algorithm finds an ϵ‐optimal solution or detects infeasibility of the given problem. Moreover, we derive the currently best known iteration bound for infeasible interior‐point methods.