Linear Systems Arising In Interior Methods For Convex Optimization: A Symmetric Formulation With Bounded Condition Number

We provide eigenvalues bounds for a new formulation of the step equations in interior methods for convex quadratic optimization. The matrix of our formulation, named K-2.5, has bounded condition number, converges to a well-defined limit under strict complementarity, and has the same size as the traditional, ill-conditioned, saddle-point formulation. We evaluate the performance in the context of a Matlab object-oriented implementation of PDCO, an interior-point solver for minimizing a smooth convex function subject to linear constraints. The main benefit of our implementation, named PDCOO, is to separate the logic of the interior-point method from the formulation of the system used to compute a step at each iteration and the method used to solve the system. Thus, PDCOO allows easy addition of a new system formulation and/or solution method for experimentation. Our numerical experiments indicate that the K-2.5 formulation has the same storage requirements as the traditional ill-conditioned saddle-point formulation, and its condition is often more favourable than the unsymmetric block 3 x 3 formulation.