Variational Analysis

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Springer Science & Business Media, Jul 17, 2009 - Mathematics - 736 pages
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From its origins in the minimization of integral functionals, the notion of 'variations' has evolved greatly in connection with applications in optimization, equilibrium, and control. It refers not only to constrained movement away from a point, but also to modes of perturbation and approximation that are best describable by 'set convergence', variational convergence of functions and the like. This book develops a unified framework and, in finite dimension, provides a detailed exposition of variational geometry and subdifferential calculus in their current forms beyond classical and convex analysis. Also covered are set-convergence, set-valued mappings, epi-convergence, duality, maximal monotone mappings, second-order subderivatives, measurable selections and normal integrands.

The changes in this 3rd printing mainly concern various typographical corrections, and reference omissions that came to light in the previous printings. Many of these reached the authors' notice through their own re-reading, that of their students and a number of colleagues mentioned in the Preface. The authors also included a few telling examples as well as improved a few statements, with slightly weaker assumptions or have strengthened the conclusions in a couple of instances.


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Max and Min
Cones and Cosmic Closure
Set Convergence
SetValued Mappings
Variational Geometry
Epigraphical Limits
Subderivatives and Subgradients
Subdifferential Calculus
Monotone Mappings
SecondOrder Theory
Index of Statements
Index of Notation

Lipschitzian Properties

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Page 709 - Proto-differentiability of set-valued mappings and its applications in optimization, in Analyse Non Lineaire, edited by H. Attouch, J.-P.
Page 706 - Mathematical Programming 34, 142-162. Moulin. H. (1982), Game Theory for the Social Sciences, New York University Press, New York. Pang, JS, and Chan, D. (1982), "Iterative methods for variational and complementary problems". Mathematical Programming 24. 284-313. Robinson, SM (1980), "Strongly regular generalized equations" Mathematics of Operations Research 5, 43-62.
Page 709 - Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives, Mathematics of Operations Research, 14(3), 462-484.
Page 709 - A. loffe, M. Marcus and S. Reich, Pitman Research Notes in Math.
Page 697 - On the convergence of algorithms with implications for stochastic and nondifferentiable optimization, Mathematics of Operations Research, 17, pp.
Page 689 - Quantitative stability of variational systems. II. A framework for nonlinear conditioning, SIAM J.
Page 689 - Aubin, JP and Ekeland, I. (1976) Estimates of the duality gap in nonconvex optimization, Mathematics of Operations Research 1, 225-245.

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About the author (2009)

Both authors have long worked with applications of convex, and later nonconvex, analysis to problems in optimization. Both are recipients of the Dantzig Prize (awarded by SIAM and the Mathematical Programming Society): Rockafellar in 1982 and Wets in 1994.

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