Partially smooth variational principles and applications (original) (raw)

On the Equivalence of Some Basic Principles in Variational Analysis

Journal of Mathematical Analysis and Applications, 1999

The primary goal of this paper is to study relationships between certain basic principles of variational analysis and its applications to nonsmooth calculus and optimization. Considering a broad class of Banach spaces admitting smooth renorms with respect to some bornology, we establish an equivalence between useful versions of a smooth variational principle for lower semicontinuous functions, an extremal principle for nonconvex sets, and an enhanced fuzzy sum rule formulated in terms of viscosity normals and subgradients with controlled ranks. Further re nements of the equivalence result are obtained in the case of a Fr echet di erentiable renorm. Based on the new enhanced sum rule, we provide a simplied proof for the re ned sequential description of approximate normals and subgradients in smooth spaces. 1991 Mathematical Subject Classi cation. Primary 49J52; Secondary 58C20, 46B20.

On Smooth Variational Principles in Banach Spaces

Journal of Mathematical Analysis and Applications, 1996

A new smooth variational principle for spaces admitting Frechet differentiablé bump functions is proved. Further it is shown that each proper lower semicontinuous bounded below function can be supported by a smooth function with locally Holder derivative if and only if the space is superreflexive. Some geometrical refinements of the Borwein᎐Preiss smooth variational principle using Deville's techniques are obtained. ᮊ

The smooth variational principle and generic differentiability

Bulletin of the Australian Mathematical Society, 1991

A modified version of the smooth variational principle of Borwein and Preiss is proved. By its help it is shown that in a Banach space with uniformly Gâteaux differentiable norm every continuous function, which is directionally differentiable on a dense Gδ subset of the space, is Gâteaux differentiable on a dense Gδ subset of the space.

A Generalization of Ekeland's ϵ-Variational Principle and Its Borwein–Preiss Smooth Variant

Journal of Mathematical Analysis and Applications, 2000

We give a generalization of Ekeland's ⑀-Variational Principle and of its Bor-wein᎐Preiss smooth variant, replacing the distance and the norm by a ''gauge-type'' lower semi-continuous function. As an application of this generalization, we show that if on a Banach space X there exists a Lipschitz ␤-smooth ''bump function,'' then every continuous convex function on an open subset U of X is densely ␤-differentiable in U. This generalizes the Borwein᎐Preiss theorem on the differentiability of convex functions.

Vector-valued variational principles

Nonlinear Analysis: Theory, Methods & Applications, 2003

In the context of vector-valued extensions of variational principles we are dealing with functions taking values in a Banach space partially ordered by a closed convex pointed cone. We introduce and study a new notion of semi-continuity connected with the order and we improve the vector-valued extensions of Deville-Godefroy-Zizler perturbed minimization principle.

Variational principles in Banach spaces and their parametrizations

Ekeland's variational principle and its smooth analogues are now classical tools for investigations of many non-linear problems in various areas in mathematics (see for instance [8], [9], [1], [2], [5], [6]). In this paper we present parametric versions of the Ekeland variational principle [8], [9], [1], stating that the minimum point of the perturbffi function, under some conditions, can be mathrmcmathrmhmathrmomathrmsmathrmemathrmfmathrml\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{f}\mathrm{l}mathrmcmathrmhmathrmomathrmsmathrmemathrmfmathrml to depend continuously on a parameter. We introduce a new smooth variational principle mathrmimathrmnmathrmvmathrmomathrmlmathrmvmathrmimathrmnmathrmf3mathrmr\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{i}\mathrm{n}_{\mathrm{f}_{3}^{\mathrm{r}}}mathrmimathrmnmathrmvmathrmomathrmlmathrmvmathrmimathrmnmathrmf3mathrmr bump functions, called here modified smooth variational principle, which unifies Borwein-Preiss' variational principle [2] and Deville-Godefroy-Zizler's variational principle [5] (concerning only existence of arbitrarily small mmooth perturbations producing a point of minimum of the perturbed function). We present also a parametric variant of this principle. The tool for proving the parametric analogue of the Ekeland variational principle is a parametric version of a Phelps' lemma [18]. This parfrmetric version produces 'extremal selections': this is, in fact, a selection theorem for the efficient points set of images of a continuous mapping with respect to a convex closed pointed cone. As a corollary we prove existence of a continuous selection of the support points of a closed convex bounded set depending continuously (in the Hausdorff sense) on a parameter (existence of such support points is garanteed by B.

Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity

1996

In Gfiteaux or bornologically differentiable spaces there are two natural generalizations of the concept of a Fr6chet subderivative. In this paper we study the viscosity subderivative (which is the more robust of the two) and establish refined fuzzy sum rules for it in a smooth Banach space. These rules are applied to obtain comparison results for viscosity solutions of Hamilton-Jacobi equations in smooth spaces. A unified treatment of metric regularity in smooth spaces completes the paper. This illustrates the flexibility of viscosity subderivatives as a tool for analysis.

Variational methods in convex analysis

2006

We use variational methods to provide a concise development of a number of basic results in convex and functional analysis. This illuminates the parallels between convex analysis and smooth subdifferential theory.

The viscosity subdifferential of the sum of two functions in Banach spaces. I. First order case

Journal of Convex Analysis, 1996

We present a formula for the viscosity subdifferential of the sum of two uniformly continuous functions on smooth Banach spaces. This formula is deduced from a new variational principle with constraints. We obtain as a consequence a weak form of Preiss’ theorem for uniformly continuous functions. We use these results to give simple proofs of some uniqueness results of viscosity solutions of Hamilton-Jacobi equations and we show how singlevaluedness of the associated Hamilton-Jacobi operators is related to the geometry of Banach spaces.

On Optimization via-Generalized Weak Subdifferentials

2018

In this paper, we study -generalized weak subdifferential for functions defined on a real topological vector space. Some necessary and sufficient conditions for having nonempty -generalized weak subdifferential of a function are presented. The positively homogenous of the -generalized weak subdifferential operator is proved. A necessary and sufficient conditions for achieving a global minimum of a -generalized weak subdifferentiable function is stated. A link between sudifferential and Fréchet differential with -generalized weak subdifferential is established. Finally a result about the equality of the fuzzy sum rule inclusion is investigated.