Reformulation strategies for eigenvalue optimization using Sylvester's criterion and Cholesky decomposition

• Published in: 23rd European Symposium on Computer Aided Process Engineering Vol. 32; pp. 487 - 492
• Main Authors: Wicaksono, Danan S., Marquardt, Wolfgang
• Format: Book Chapter Reference
• Language: English
• Published: 2013
• Subjects: applied linear algebra; symbolic-numeric computation; applied linear algebra; symbolic-numeric computation
• ISBN: 0444632344; 9780444632340
• ISSN: 1570-7946
• DOI: 10.1016/B978-0-444-63234-0.50082-8

Only a few solvers are currently available to solve eigenvalue optimization problems. In case of nonlinear objective and/or constraints, the capabilities of existing methods are still limited. This contribution addresses two classes of eigenvalue optimization problems: the maximization (minimization) of the smallest (largest) eigenvalue of a real symmetric matrix and optimization subject to inequalities constraining the real parts of all eigenvalues of a real square matrix. This contribution considers the reformulation of such problems into optimization problems subject to the positive definiteness of a suitable matrix to enable the use of efficient and robust off-the-shelf solvers. This contribution revisits the utilization of Sylvester's criterion suggested previously and proposes to alternatively employ Cholesky decomposition to compel the constraints on positive definiteness. The methodology is implemented in an integrated symbolic-numeric computational environment. A comparative computational study demonstrates that the latter performs better than the former, at least in the set of examples studied.