On Implementing a Homogeneous Interior-Point Algorithm for Nonsymmetric Conic Optimization

Based on earlier work by Nesterov, an implementation of a homogeneous infeasible-start interior-point algorithm for solving nonsymmetric conic optimization problems is presented. Starting each iteration from (the vicinity of) the central path, the method computes (nearly) primal-dual symmetric approximate tangent directions followed by a purely primal centering procedure to locate the next central primal-dual point. Features of the algorithm include that it makes use only of the primal barrier function, that it is able to detect infeasibilities in the problem and that no phase-I method is needed. The method further employs quasi- Newton updating both to generate (pseudo) higher order directions and to reduce the number of factorizations needed in the centering process while still retaining the ability to exploit sparsity. Extensive and promising computational results are presented for the p-cone problem, the facility location problem, entropy problems and geometric programs; all formulated as nonsymmetric conic optimization problems.