The $\epsilon$-pseudospectral abscissa $\alpha_\epsilon$ and radius $\rho_\epsilon$ of an n x n matrix are respectively the maximal real part and the maximal modulus of points in its $\epsilon$-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang. Variational analysis of pseudospectra. SIAM Journal on Optimization, 19:1048-1072, 2008] that for fixed … Read more
To minimize or upper-bound the value of a function “robustly”, we might instead minimize or upper-bound the “epsilon-robust regularization”, defined as the map from a point to the maximum value of the function within an epsilon-radius. This regularization may be easy to compute: convex quadratics lead to semidefinite-representable regularizations, for example, and the spectral radius … Read more
Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for … Read more
Given a real function on a Euclidean space, we consider its “robust regularization”: the value of this new function at any given point is the maximum value of the original function in a fixed neighbourhood of the point in question. This construction allows us to impose constraints in an optimization problem *robustly*, safeguarding a constraint … Read more