About

Adrian S. Lewis is the Samuel B. Eckert Professor of Engineering at Cornell University in the School of Operations Research and Information Engineering. He received his B.A., M.A., and Ph.D. degrees from Cambridge University, U.K., and has held faculty positions at the University of Waterloo and Simon Fraser University before joining Cornell in 2004. His research focuses on nonsmooth optimization and variational analysis, with particular interest in the mathematical theory underlying these areas and their practical applications in science and engineering. His work includes the design and analysis of computational algorithms for nonsmooth optimization, especially problems involving eigenvalues such as robust control and pseudospectral sensitivity. Recently, his research has expanded into semi-algebraic geometry as a model for generic structure in nonsmooth optimization, blending variational analysis, classical mathematics, numerical computation, and applied modeling.

Research topics

• Artificial Intelligence
• Computer Science
• Mathematics
• Algorithm
• Mathematical optimization

Selected publications

• Minimal enclosing balls via geodesics

  ArXiv.org · 2026-03-16

  articleOpen accessSenior author

  Algorithms for minimal enclosing ball problems are often geometric in nature. To highlight the metric ingredients underlying their efficiency, we focus here on a particularly simple geodesic-based method. A recent subgradient-based study proved a complexity result for this method in the broad setting of geodesic spaces of nonpositive curvature. We present a simpler, intuitive and self-contained complexity analysis in that setting, which also improves the convergence rate. We furthermore derive the first complexity result for the algorithm on geodesic spaces with curvature bounded above.

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• Stochastic and incremental subgradient methods for convex optimization on Hadamard spaces

  Mathematical Programming · 2026-03-04

  preprintOpen access
  PublisherOA PDFDOI

• Minimal enclosing balls via geodesics

  arXiv (Cornell University) · 2026-03-16

  preprintOpen accessSenior author

  Algorithms for minimal enclosing ball problems are often geometric in nature. To highlight the metric ingredients underlying their efficiency, we focus here on a particularly simple geodesic-based method. A recent subgradient-based study proved a complexity result for this method in the broad setting of geodesic spaces of nonpositive curvature. We present a simpler, intuitive and self-contained complexity analysis in that setting, which also improves the convergence rate. We furthermore derive the first complexity result for the algorithm on geodesic spaces with curvature bounded above.

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• Stochastic and incremental subgradient methods for convex optimization on Hadamard spaces

  Mathematical Programming · 2026-03-04

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• Convex optimization on CAT(0) cubical complexes

  Advances in Applied Mathematics · 2025-01-21 · 1 citations

  articleCorresponding
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• Recognizing Weighted Means in Geodesic Spaces

  Foundations of Computational Mathematics · 2025-09-26

  articleCorresponding
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• Lipschitz minimization and the Goldstein modulus

  Mathematical Programming · 2025-07-29

  articleSenior author
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• Identifiability, the KL Property in Metric Spaces, and Subgradient Curves

  Foundations of Computational Mathematics · 2024-05-28 · 2 citations

  article1st authorCorresponding
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• Lipschitz minimization and the Goldstein modulus

  arXiv (Cornell University) · 2024-05-21 · 1 citations

  preprintOpen accessSenior author

  Goldstein's 1977 idealized iteration for minimizing a Lipschitz objective fixes a distance - the step size - and relies on a certain approximate subgradient. That "Goldstein subgradient" is the shortest convex combination of objective gradients at points within that distance of the current iterate. A recent implementable Goldstein-style algorithm allows a remarkable complexity analysis (Zhang et al. 2020), and a more sophisticated variant (Davis and Jiang, 2022) leverages typical objective geometry to force near-linear convergence. To explore such methods, we introduce a new modulus, based on Goldstein subgradients, that robustly measures the slope of a Lipschitz function. We relate near-linear convergence of Goldstein-style methods to linear growth of this modulus at minimizers. We illustrate the idea computationally with a simple heuristic for Lipschitz minimization.

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• Basic Convex Analysis in Metric Spaces with Bounded Curvature

  SIAM Journal on Optimization · 2024-01-19 · 3 citations

  article1st authorCorresponding

  .Differentiable structure ensures that many of the basics of classical convex analysis extend naturally from Euclidean space to Riemannian manifolds. Without such structure, however, extensions are more challenging. Nonetheless, in Alexandrov spaces with curvature bounded above (but possibly positive), we develop several basic building blocks. We define subgradients via projection and the normal cone, prove their existence, and relate them to the classical affine minorant property. Then, in what amounts to a simple calculus or duality result, we develop a necessary optimality condition for minimizing the sum of two convex functions.Keywordssubdifferentialnormal coneAlexandrov spacesMSC codes65K1053C20

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Recent grants

• Variational Analysis for Practical Optimization

  NSF · $388k · 2008–2012

• Semi-Structured Optimization: Geometry and Nonsmooth Algorithms

  NSF · $351k · 2020–2024

• Geometry in nonsmooth optimization

  NSF · $413k · 2012–2016

• Nonsmooth Optimization: Structure, Geometry, and Conditioning

  NSF · $349k · 2016–2020

• Applied Variational Analysis: Structure, Regularity, and Algorithms

  NSF · $275k · 2005–2008

Frequent coauthors

• Jonathan M. Borwein

  University of Newcastle Australia

  69 shared

• Dmitriy Drusvyatskiy

  41 shared

• Michael L. Overton

  38 shared

• James V. Burke

  22 shared

• Aris Daniilidis

  TU Wien

  17 shared

• A. D. Ioffe

  Technion – Israel Institute of Technology

  12 shared

• Adriana Nicolae

  8 shared

• Genaro López-Acedo

  Universidad de Sevilla

  7 shared

Awards & honors

• 1995 Aisenstadt Prize
• 2003 Lagrange Prize
• 2005 Outstanding Paper Prize from SIAM
• Section Lecturer at the 2014 International Congress of Mathe…