Showing papers by "Dmitriy Drusvyatskiy published in 2015"

Transversality and Alternating Projections for Nonconvex Sets

Abstract: We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear convergence. When the two sets are semi-algebraic and bounded, but not necessarily transversal, we nonetheless prove subsequence convergence.

Coordinate Shadows of Semidefinite and Euclidean Distance Matrices

Abstract: We consider the projected semidefinite and Euclidean distance cones onto a subset of the matrix entries. These two sets are precisely the input data defining feasible semidefinite and Euclidean distance completion problems. We classify when these sets are closed and use the boundary structure of these two sets to elucidate the Krislock--Wolkowicz facial reduction algorithm. In particular, we show that under a chordality assumption, the “minimal cones” of these problems admit combinatorial characterizations. As a by-product, we record a striking relationship between the complexity of the general facial reduction algorithm (singularity degree) and facial exposedness of conic images under a linear mapping.

Quadratic growth and critical point stability of semi-algebraic functions

Abstract: We show that quadratic growth of a semi-algebraic function is equivalent to strong metric subregularity of the subdifferential--a kind of stability of generalized critical points. In contrast, this equivalence can easily fail outside of the semi-algebraic setting. As a consequence, we derive necessary conditions and sufficient conditions for optimality in subdifferential terms.

Curves of Descent

Abstract: Steepest descent is central in variational mathematics. We present a new transparent existence proof for curves of near-maximal slope---an influential notion of steepest descent in a nonsmooth setting. The existence theory is further amplified for semialgebraic functions, prototypical nonpathological functions in nonsmooth optimization: such functions always admit nontrivial descent curves emanating from any (even critical) nonminimizing point. We moreover show that curves of near-maximal slope of semialgebraic functions have a more classical description as solutions of subgradient dynamical systems.

Variational analysis of spectral functions simplified

Abstract: Spectral functions of symmetric matrices -- those depending on matrices only through their eigenvalues -- appear often in optimization. A cornerstone variational analytic tool for studying such functions is a formula relating their subdifferentials to the subdifferentials of their diagonal restrictions. This paper presents a new, short, and revealing derivation of this result. We then round off the paper with an illuminating derivation of the second derivative of twice differentiable spectral functions, highlighting the underlying geometry. All of our arguments have direct analogues for spectral functions of Hermitian matrices, and for singular value functions of rectangular matrices.

Generic minimizing behavior in semi-algebraic optimization

Abstract: We present a theorem of Sard type for semi-algebraic set-valued mappings whose graphs have dimension no larger than that of their range space: the inverse of such a mapping admits a single-valued analytic localization around any pair in the graph, for a generic value parameter. This simple result yields a transparent and unified treatment of generic properties of semi-algebraic optimization problems: "typical" semi-algebraic problems have finitely many critical points, around each of which they admit a unique "active manifold" (analogue of an active set in nonlinear optimization); moreover, such critical points satisfy strict complementarity and second-order sufficient conditions for optimality are indeed necessary.

Extreme point inequalities and geometry of the rank sparsity ball

Abstract: We investigate geometric features of the unit ball corresponding to the sum of the nuclear norm of a matrix and the $$l_1$$l1 norm of its entries--a common penalty function encouraging joint low rank and high sparsity. As a byproduct of this effort, we develop a calculus (or algebra) of faces for general convex functions, yielding a simple and unified approach for deriving inequalities balancing the various features of the optimization problem at hand, at the extreme points of the solution set.

Projection methods for quantum channel construction

Abstract: We consider the problem of constructing quantum channels, if they exist, that transform a given set of quantum states $$\{\rho _1, \ldots , \rho _k\}$${?1,?,?k} to another such set $$\{\hat{\rho }_1, \ldots , \hat{\rho }_k\}$${?^1,?,?^k}. In other words, we must find a completely positive linear map, if it exists, that maps a given set of density matrices to another given set of density matrices, possibly of different dimension. Using the theory of completely positive linear maps, one can formulate the problem as an instance of a positive semidefinite feasibility problem 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 semidefinite programming are not applicable. We provide empirical evidence to this effect. We moreover present heuristics for finding both high-rank and low-rank solutions. Our experiments are based on the method of alternating projections and the Douglas---Rachford reflection method.

Counting real critical points of the distance to orthogonally invariant matrix sets

Abstract: 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.

Clarke Subgradients for Directionally Lipschitzian Stratifiable Functions

Abstract: 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 Rn have a limiting subdifferential graph of uniform local dimension n.

Orbits of geometric descent

Abstract: We prove that quasiconvex functions always admit descent trajectories bypassing all nonminimizing critical points

Counting real critical points of the distance to spectral matrix sets

Abstract: 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 (spectral) 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 spectral 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.

Counting real critical points of the distance to orthogonally invariant matrix sets

Abstract: 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.