Preservation of convergence of convex sets and functions in finite dimensions
L. McLinden,Roy C. Bergstrom +1 more
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In this paper, the convergence of sets, functions and subdifferentials is studied in finite-dimensional settings and conditions under which this convergence is preserved under various basic operations, including those of addition and infimal convolution in the case of functions.Abstract:
We study a convergence notion which has particular relevance for convex analysis and lends itself quite naturally to successive approximation schemes in a variety of areas. Motivated particularly by problems in optimization subject to constraints, we develop technical tools necessary for systematic use of this convergence in finite-dimensional settings. Simple conditions are established under which this convergence for sequences of sets, functions and subdifferentials is preserved under various basic operations, including, for example, those of addition and infimal convolution in the case of functions.read more
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Robust regression through the Huber’s criterion and adaptive lasso penalty
TL;DR: This paper proposes to combine the Huber's criterion and adaptive penalty as lasso, and shows that the estimator associated with this procedure enjoys the oracle properties.
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Operations on convergent families of sets and functions
Dominique Azé,Jean-Paul Penot +1 more
TL;DR: In this paper, the lower limit of the images by a convex set-valued mapping of a family of convex sets is studied, and applications to variational convergence are presented.
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On convergence of closed sets in a metric space and distance functions
TL;DR: In this article, the convergence of a nonempty closed subsets of a metric space X to a non-empty closed set C is shown to be equivalent to the pointwise convergence of the distance functions for the sets in the sequence to the distance function for C. The convergence theorems for equicontinuous families of real valued functions are studied.
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Equivalent Subgradient Versions of Hamiltonian and Euler--Llagrange Equations in Variational Analysis
TL;DR: In this article, it was shown that new, tighter subgradient versions of the Euler-Lagrange equations are actually equivalent to each other and the theory of epi-convergence of convex functions provides the technical basis for this development.
References
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Book
Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert
TL;DR: In this article, Operateurs Maximaux Monotones: Et Semi-Groupes De Contractions Dans Les Espaces De Hllbert are described and discussed. But the focus is not on the performance of the operators.
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Conjugate Duality and Optimization
TL;DR: The Role of Convexity and Duality Examples of CONVEX Optimization Problems ConjugateConvex Functions in Paired Spaces Dual Problems and Lagrangians Examples of Duality Schemes Continuity and Derivatives of ConVEx Functions Solutions to Optimization problems Calculating Conjugates and Subgradients Integral Functionals
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