Solving inverse Pareto eigenvalue problems

We compare in this note a variety of methods for solving inverse Pareto eigenvalue problems which are aimed at constructing matrices whose Pareto spectrum contains a prescribed set of distinct reals. We choose to deal with such problems by first formulating them as nonlinear systems of equations which can be smooth or nonsmooth, depending on the chosen approach, and subsequently adopt Newton type methods to solve the corresponding systems. Our smooth approach includes the Squaring Trick (ST) and the so-called Mehrotra Predictor Corrector Method (MPCM), adapted in this context to inverse Pareto eigenvalue complementarity problems. For the nonsmooth approach, we consider the Lattice Projection Method (LPM), and two other nonsmooth methods using complementarity function techniques, namely SNMFB and SNMmin (with Fischer-Burmeister and minimum complementarity functions respectively). We compare the five methods using the performance profiles (Dolan, More), where the average number of iterations and the percentage of failures are the performance measures. Numerical tests show that among the methods considered, SNMFB performs the best in terms of the number of failures whereas LPM surpasses all other methods with respect to the number of iterations. Finally, we point out possible extensions of the discussed methods to the inverse quadratic pencil eigenvalue complementarity problem.