arXiv subjects

A. S. Lewis

Publications and source records attributed to A. S. Lewis.

10 recordsLinked to original sources

arXiv2022

The cost of nonconvexity in deterministic nonsmooth optimization

We study the impact of nonconvexity on the complexity of nonsmooth optimization, emphasizing objectives such as piecewise linear functions, which may not be weakly convex. We focus on a dimension-independent analysis, slightly modifying a black-box algorithm of Zhang et al. (2020) that approximates an $ε$-stationary point of any directionally differentiable Lipschitz objective using $O(ε^{-4})$ calls to a specialized subgradient oracle and a randomized line search. Our simple black-box deterministic version, achieves $O(ε^{-5})$ for any difference-of-convex objective, and $O(ε^{-4})$ for the weakly convex case. Our complexity bound depends on a natural nonconvexity modulus, related, intriguingly, to the negative part of directional second derivatives of the objective, understood in the distributional sense.

math.OC↗

arXiv2017

OSIRIS-REx Contamination Control Strategy and Implementation

OSIRIS-REx will return pristine samples of carbonaceous asteroid Bennu. This article describes how pristine was defined based on expectations of Bennu and on a realistic understanding of what is achievable with a constrained schedule and budget, and how that definition flowed to requirements and implementation. To return a pristine sample, the OSIRIS- REx spacecraft sampling hardware was maintained at level 100 A/2 and <180 ng/cm2 of amino acids and hydrazine on the sampler head through precision cleaning, control of materials, and vigilance. Contamination is further characterized via witness material exposed to the spacecraft assembly and testing environment as well as in space. This characterization provided knowledge of the expected background and will be used in conjunction with archived spacecraft components for comparison with the samples when they are delivered to Earth for analysis. Most of all, the cleanliness of the OSIRIS-REx spacecraft was achieved through communication among scientists, engineers, managers, and technicians.

astro-ph.IM↗

arXiv2016

Transversality and alternating projections for nonconvex sets

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.

math.OC↗

arXiv2015

Generic minimizing behavior in semi-algebraic optimization

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.

math.OC↗

arXiv2015

A proximal method for composite minimization

We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe an algorithmic framework based on a subproblem constructed from a linearized approximation to the objective and a regularization term. Properties of local solutions of this subproblem underlie both a global convergence result and an identification property of the active manifold containing the solution of the original problem. Preliminary computational results on both convex and nonconvex examples are promising.

math.OC↗

arXiv2013

Orbits of geometric descent

We prove that quasiconvex functions always admit descent trajectories bypassing all non-minimizing critical points.

math.CA↗

arXiv2013

Curves of Descent

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. We moreover show that for semi-algebraic functions --- prototypical nonpathological functions in nonsmooth optimization --- such curves are precisely the solutions of subgradient dynamical systems.

math.OC↗

arXiv2009

Generic identifiability and second-order sufficiency in tame convex optimization

We consider linear optimization over a fixed compact convex feasible region that is semi-algebraic (or, more generally, "tame"). Generically, we prove that the optimal solution is unique and lies on a unique manifold, around which the feasible region is "partly smooth", ensuring finite identification of the manifold by many optimization algorithms. Furthermore, second-order optimality conditions hold, guaranteeing smooth behavior of the optimal solution under small perturbations to the objective.

math.OC↗

arXiv2008

Randomized Methods for Linear Constraints: Convergence Rates and Conditioning

We study randomized variants of two classical algorithms: coordinate descent for systems of linear equations and iterated projections for systems of linear inequalities. Expanding on a recent randomized iterated projection algorithm of Strohmer and Vershynin for systems of linear equations, we show that, under appropriate probability distributions, the linear rates of convergence (in expectation) can be bounded in terms of natural linear-algebraic condition numbers for the problems. We relate these condition measures to distances to ill-posedness, and discuss generalizations to convex systems under metric regularity assumptions.

math.OC↗

arXiv2006

Clarke subgradients of stratifiable functions

We establish the following result: if the graph of a (nonsmooth) real-extended-valued function $f:\mathbb{R}^{n}\to \mathbb{R}\cup\{+\infty\}$ is closed and admits a Whitney stratification, then the norm of the gradient of $f$ at $x\in{dom}f$ relative to the stratum containing $x$ bounds from below all norms of Clarke subgradients of $f$ at $x$. As a consequence, we obtain some Morse-Sard type theorems as well as a nonsmooth Kurdyka-Łojasiewicz inequality for functions definable in an arbitrary o-minimal structure.

math.OC↗