77 research outputs found
Nonsmooth Analysis
This survey of nonsmooth analysis sets out to prove an inverse function theorem for set-valued maps. The inverse function theorem for the more usual smooth maps plays a very important role in the solution of many problems in pure and applied analysis, and we can expect such an adaptation of this theorem also to be of great value. For example, it can be used to solve convex minimization problems and to prove the Lipschitz behavior of its solutions when the natural parameters vary--a very important problem in marginal theory in economics
Scalar Representation and Conjugation of Set-Valued Functions
To a function with values in the power set of a pre-ordered, separated
locally convex space a family of scalarizations is given which completely
characterizes the original function. A concept of a Legendre-Fenchel conjugate
for set-valued functions is introduced and identified with the conjugates of
the scalarizations. Using this conjugate, weak and strong duality results are
proven.Comment: arXiv admin note: substantial text overlap with arXiv:1012.435
Regularization of Linear Ill-posed Problems by the Augmented Lagrangian Method and Variational Inequalities
We study the application of the Augmented Lagrangian Method to the solution
of linear ill-posed problems. Previously, linear convergence rates with respect
to the Bregman distance have been derived under the classical assumption of a
standard source condition. Using the method of variational inequalities, we
extend these results in this paper to convergence rates of lower order, both
for the case of an a priori parameter choice and an a posteriori choice based
on Morozov's discrepancy principle. In addition, our approach allows the
derivation of convergence rates with respect to distance measures different
from the Bregman distance. As a particular application, we consider sparsity
promoting regularization, where we derive a range of convergence rates with
respect to the norm under the assumption of restricted injectivity in
conjunction with generalized source conditions of H\"older type
Regularity of a kind of marginal functions in Hilbert spaces
We study well-posedness of some mathematical programming problem depending on a parameter that generalizes in a certain sense the metric projection onto a closed nonconvex set. We are interested in regularity of the set of minimizers as well as of the value function, which can be seen, on one hand, as the viscosity solution to a Hamilton-Jacobi equation, while, on the other, as the minimal time in some related optimal time control problem. The regularity includes both the Fréchet differentiability of the value function and the Hölder continuity of its (Fréchet) gradient
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