A Hybrid Algorithm for Optimizing Eigenvalues of Symmetric Definite Pencils

Abstract
An algorithm is presented for the optimization of the maximum eigenvalue of a symmetric definite pencil depending affinely on a vector of parameters. The algorithm uses a hybrid approach, combining a scheme based on the method of centers, developed by Boyd and El Ghaoui [Linear Algebra Appl., 188 (1993), pp. 63–112], with a new quadratically convergent local scheme. A convenient expression for the generalized gradient of the maximum eigenvalue of the pencil is also given, expressed in terms of a dual matrix. The algorithm computes the dual matrix that establishes the optimality of the computed solution. An algorithm is presented for the optimization of the maximum eigenvalue of a symmetric definite pencil depending affinely on a vector of parameters. The algorithm uses a hybrid approach, combining a scheme based on the method of centers, developed by Boyd and El Ghaoui [Linear Algebra Appl., 188 (1993), pp. 63–112], with a new quadratically convergent local scheme. A convenient expression for the generalized gradient of the maximum eigenvalue of the pencil is also given, expressed in terms of a dual matrix. The algorithm computes the dual matrix that establishes the optimality of the computed solution.

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