arXiv subjects
Publications and source records attributed to Dmitriy Drusvyatskiy.
We show that any semi-algebraic sweeping process admits piecewise absolutely continuous solutions, and any such bounded trajectory must have finite length. Analogous results hold more generally for sweeping processes definable in o-minimal structures. This extends previous work on (sub)gradient dynamical systems beyond monotone sweeping sets.
Minimizing the Euclidean distance to a set arises frequently in applications. When the set is algebraic, a measure of complexity of this optimization problem is its number of critical points. In this paper we provide a general framework to compute and count the real smooth critical points of a data matrix on an orthogonally invariant set of matrices. The technique relies on "transfer principles" that allow calculations to be done in the space of singular values of the matrices in the orthogonally invariant set. The calculations often simplify greatly and yield transparent formulas. We illustrate the method on several examples, and compare our results to the recently introduced notion of Euclidean distance degree of an algebraic variety.
We consider the problem of constructing quantum operations or channels, if they exist, that transform a given set of quantum states $\{ρ_1, \dots, ρ_k\}$ to another such set $\{\hatρ_1, \dots, \hatρ_k\}$. In other words, we must find a {\em completely positive linear map}, if it exists, that maps a given set of density matrices to another given set of density matrices. This problem, in turn, is an instance of a positive semi-definite feasibility problem, but with highly structured constraints. The nature of the constraints makes projection based algorithms very appealing when the number of variables is huge and standard interior point-methods for semi-definite programming are not applicable. We provide emperical evidence to this effect. We moreover present heuristics for finding both high rank and low rank solutions. Our experiments are based on the \emph{method of alternating projections} and the \emph{Douglas-Rachford} reflection method.
We prove that quasiconvex functions always admit descent trajectories bypassing all non-minimizing critical points.
Orthogonally invariant functions of symmetric matrices often inherit properties from their diagonal restrictions: von Neumann's theorem on matrix norms is an early example. We discuss the example of "identifiability", a common property of nonsmooth functions associated with the existence of a smooth manifold of approximate critical points. Identifiability (or its synonym, "partial smoothness") is the key idea underlying active set methods in optimization. Polyhedral functions, in particular, are always partly smooth, and hence so are many standard examples from eigenvalue optimization.
Using a geometric argument, we show that under a reasonable continuity condition, the Clarke subdifferential of a semi-algebraic (or more generally stratifiable) directionally Lipschitzian function admits a simple form: the normal cone to the domain and limits of gradients generate the entire Clarke subdifferential. The characterization formula we obtain unifies various apparently disparate results that have appeared in the literature. Our techniques also yield a simplified proof that closed semialgebraic functions on $\R^n$ have a limiting subdifferential graph of uniform local dimension $n$.
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 foundation for optimality conditions, sensitivity, and active set methods.
We prove that uniform second order growth, tilt stability, and strong metric regularity of the limiting subdifferential --- three notions that have appeared in entirely different settings --- are all essentially equivalent for any lower-semicontinuous, extended-real-valued function.
We prove that the subdifferential of any semi-algebraic extended-real-valued function on $\R^n$ has $n$-dimensional graph. We discuss consequences for generic semi-algebraic optimization problems.
Consider a face F in an arrangement of n Jordan curves in the plane, no two of which intersect more than s times. We prove that the combinatorial complexity of F is O(λ_s(n)), O(λ_{s+1}(n)), and O(λ_{s+2}(n)), when the curves are bi-infinite, semi-infinite, or bounded, respectively; λ_k(n) is the maximum length of a Davenport-Schinzel sequence of order k on an alphabet of n symbols. Our bounds asymptotically match the known worst-case lower bounds. Our proof settles the still apparently open case of semi-infinite curves. Moreover, it treats the three cases in a fairly uniform fashion.
Examples exist of extended-real-valued closed functions on ${\bf R}^n$ whose subdifferentials (in the standard, limiting sense) have large graphs. By contrast, if such a function is semi-algebraic, then its subdifferential graph must have everywhere constant local dimension $n$. This result is related to a celebrated theorem of Minty, and surprisingly may fail for the Clarke subdifferential.
We show that minimizers of convex functions subject to almost all linear perturbations are nondegenerate. An analogous result holds more generally, for lower-C^2 functions.