Optimization of the solution of the parameter-dependent Sylvester equation and applications

This paper deals with an efficient algorithm for optimization of the solution of the parameter-dependent Sylvester equation (A0−vC1C2T)X(v)+X(v)(B0−vD1D2T)=E, where A0A0, B0B0 are m×mm×m and n×nn×n matrices, respectively. Further, C1C1 and C2C2 are m×r1m×r1, D1D1 and D2D2 are n×r2n×r2 and XX, EE are m×nm×n matrices, while vv is real parameter. For optimization we use the following two optimization criteria: Tr(X(v))→min and ‖X(v)‖F→min‖X(v)‖F→min. We present an efficient algorithm based on derived formulas for the trace and for the Frobenius norm of the solution XX as functions v→Tr(X(v)) and v→‖X(v)‖Fv→‖X(v)‖F as well as for derivatives of these functions. That ensures fast optimization of these functions via standard optimization methods like Newton’s method. A special case of this problem is a very important problem of damper viscosity optimization in mechanical systems.