We consider the problem of minimizing over an affine set of square matrices the maximum of the real parts of the eigenvalues. Such problems are prototypical in robust control and stability analysis. Under nondegeneracy conditions, we show that the multiplicities of the active eigenvalues at a critical matrix remain unchanged under small perturbations of the … Read more
Given an affine subspace of square matrices, we consider the problem of minimizing the spectral abscissa (the largest real part of an eigenvalue). We give an example whose optimal solution has Jordan form consisting of a single Jordan block, and we show, using nonlipschitz variational analysis, that this behaviour persists under arbitrary small perturbations to … Read more
Many interesting real functions on Euclidean space are differentiable almost everywhere. All Lipschitz functions have this property, but so, for example, does the spectral abscissa of a matrix (a non-Lipschitz function). In practice, the gradient is often easy to compute. We investigate to what extent we can approximate the Clarke subdifferential of such a function … Read more
Consider the affine matrix family $A(x) = A_0 + \sum_{k=1}^m x_k A_k$, mapping a design vector $x\in\Rl^m$ into the space of $n \times n$ real matrices. Consider the affine matrix family $A(x) = A_0 + \sum_{k=1}^m x_k A_k$, mapping a design vector $x\in\Rl^m$ into the space of $n \times n$ real matrices. We are interested … Read more
This paper presents and analyzes a pattern search method for general constrained optimization based on filter methods for step acceptance. Roughly, a filter method accepts a step that either improves the objective function value or the value of some function that measures the constraint violation. The new algorithm does not compute or approximate any derivatives, … Read more
We consider minimizing a nonsmooth objective subject to nonsmooth constraints. The nonsmooth functions are approximated by a bundle of subgradients. The novel idea of a filter is used to promote global convergence. Citation NA\195, Department of Mathematics, University of Dundee, UK, December, 1999 Article Download View A bundle filter method for nonsmooth nonlinear optimization