Identifiability, the KL property in metric spaces, and subgradient curves

Identifiability, and the closely related idea of partial smoothness, unify classical active set methods and more general notions of solution structure. Diverse optimization algorithms generate iterates in discrete time that are eventually confined to identifiable sets. We present two fresh perspectives on identifiability. The first distills the notion to a simple metric property, applicable not … Read more

Optimality, identifiability, and sensitivity

Around a solution of an optimization problem, an “identifiable” subset of the feasible region is one containing all nearby solutions after small perturbations to the problem. A quest for only the most essential ingredients of sensitivity analysis leads us to consider identifiable sets that are “minimal”. This new notion lays a broad and intuitive variational-analytic … Read more

Identifying Active Manifolds in Regularization Problems

In this work we consider the problem $\min_x \{ f(x) + P(x) \}$, where $f$ is $\mathcal{C}^2$ and $P$ is nonsmooth, but contains an underlying smooth substructure. Specifically, we assume the function $P$ is prox-regular partly smooth with respect to a active manifold $\M$. Recent work by Tseng and Yun \cite{tseng-yun-2009}, showed that such a … Read more

A Proximal Method for Identifying Active Manifolds

The minimization of an objective function over a constraint set can often be simplified if the “active manifold” of the constraints set can be correctly identified. In this work we present a simple subproblem, which can be used inside of any (convergent) optimization algorithm, that will identify the active manifold of a “prox-regular partly smooth” … Read more

The mathematics of eigenvalue optimization

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