Eigenvalue And Singular Value Optimization

Abstract

Eigenvalues, singular values and condition number of matrices, play an important
role in many fields of applied mathematics to engineering. Among the plethora of applications
of eigenvalues in mathematics and engineering we can mention numerical analysis, structural
design, quantum mechanics and system dynamics (physical and biological models). For some
applications it may be desirable to choose the parameters of a model in order to optimize an
objective function and/or to verify constraints that involve eigenvalues or singular values of a
certain matrix. In general, the elements of the matrix depend in a nonlinear fashion on the
optimization parameters. The purpose of this contribution is to introduce recent formulations
of eigenvalue and singular value optimization as well as techniques to include condition
numbers within the optimization problem. A chemical engineering design problem is
presented to illustrate the proposed techniques.