A Smooth Variational Principle With Applications to Subdifferentiability and to Differentiability of Convex Functions

DOI10.2307/2000681zbMath0632.49008OpenAlexW4246263434MaRDI QIDQ3769491

Jonathan M. Borwein, David Preiss

Publication date: 1987

Full work available at URL: https://doi.org/10.2307/2000681

zbMATH Keywords

convex functionsEkeland's variational principlesubderivatives``smooth variational principlerenorm

Mathematics Subject Classification ID

Nonsmooth analysis (49J52) Fréchet and Gateaux differentiability in optimization (49J50) Differentiation theory (Gateaux, Fréchet, etc.) on manifolds (58C20)

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Cites Work

• Unnamed Item
• Unnamed Item
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• Unnamed Item
• Unnamed Item
• Stability and regular points of inequality systems
• Norms with locally Lipschitzian derivatives
• Weak local supportability and applications to approximation
• Geometry of Banach spaces. Selected topics
• Martingales with values in uniformly convex spaces
• A new proof of Fréchet differentiability of Lipschitz functions
• On the variational principle
• Lipschitz Smooth Points of Convex Functions and Isomorphic Characterizations of Hilbert Spaces
• Proximal Analysis and Boundaries of Closed Sets in Banach Space, Part I: Theory
• Optimization and nonsmooth analysis
• The Proximal Normal Formula in Banach Space
• Clarke's Gradients and Epsilon-Subgradients in Banach Spaces
• Gateaux Differentiability of Convex Functions on Banach Spaces
• Nonconvex minimization problems
• Proximal Subgradients, Marginal Values, and Augmented Lagrangians in Nonconvex Optimization
• Generic Frechet-Differentiability and Perturbed Optimization Problems in Banach Spaces
• Differentiable Functions and Rough Norms on Banach Spaces