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In this paper we compute families of reduced order models that match aprescribed set of moments of a highly dimensional linear time-invariant system.First, we fully parametrize the models in the interpolation points and in thefree parameters, and then we fix the set of interpolation points andparametrize the models only in the free parameters. Based on these twoparametrizations and using as objective function the H2-norm of the errorapproximation we derive non-convex optimization problems, i.e., we search forthe optimal free parameters and even the interpolation points to determine theapproximation model yielding the minimal H2-norm error. Further, we provide thenecessary first-order optimality conditions for these optimization problemsgiven explicitly in terms of the controllability and the observability Gramiansof a minimal realization of the error system. Using the optimality conditions,we propose gradient type methods for solving the corresponding optimizationproblems, with mathematical guarantees on their convergence. We also deriveconvex SDP relaxations for these problems and analyze when the convexrelaxations are exact. We illustrate numerically the efficiency of our resultson several test examples.

Optimal H2 moment matching-based model reduction for linear systems by (non)convex optimization