Optimal time-domain moment matching with partial placement of poles and zeros

In this paper we consider a minimal, linear, time-invariant (LTI) system of order n, large. Our goal is to compute an approximation of order ν < n that simultaneously matches ν moments, has ℓ poles and k zeros fixed, with ℓ + k < ν, and achieves minimal H2 norm of the approximation error. For this, in the family of ν order parametrized models that match ν moments we impose ℓ+k linear constraints yielding a subfamily of models with ℓ poles and k zeros imposed. Then, in the subfamily of ν order models matching ν moments, with ℓ poles and k zeros imposed we propose an optimization problem that provides the model yielding the minimal H2-norm of the approximation error. We analyze the first-order optimality conditions of this optimization problem and compute explicitly the gradient of the objective function in terms of the controllability and the observability Gramians of the error system. We then propose a gradient method that finds the (optimal) stable model, with fixed ℓ poles and k zeros

Fast Primal-Dual Gradient Method for Strongly Convex Minimization Problems with Linear Constraints

In this paper we consider a class of optimization problems with a strongly convex objective function and the feasible set given by an intersection of a simple convex set with a set given by a number of linear equality and inequality constraints. A number of optimization problems in applications can be stated in this form, examples being the entropy-linear programming, the ridge regression, the elastic net, the regularized optimal transport, etc. We extend the Fast Gradient Method applied to the dual problem in order to make it primal-dual so that it allows not only to solve the dual problem, but also to construct nearly optimal and nearly feasible solution of the primal problem. We also prove a theorem about the convergence rate for the proposed algorithm in terms of the objective function and the linear constraints infeasibility.Comment: Submitted for DOOR 201