Generalized Monotonicity of Subdifferentials and Generalized Convexity

Characterizations of convexity and quasiconvexity of lower semicontinuous functions on a Banach space X are presented in terms of the contingent and Fréchet subdifferentials. They rely on a general mean-value theorem for such subdifferentials, which is valid in a class of spaces which contains the class of Asplund spaces.     

Compact Subdifferentials in Banach Cones

We construct, in general terms, the theory of first-order compact subdifferentials for mappings acting in Banach cones. The basic properties of K-subdifferentials up to the mean-value theorem and its nontrivial corollaries are studied. An application to variational functionals with nonsmooth integrand is considered.     

Mean-Value Theorem with Small Subdifferentials

We prove a mean-value theorem for lower semicontinuous functions on a large class of Banach spaces which contains the class of Asplund spaces, in particular reflexive Banach spaces and Banach spaces with a separable dual. It involves the lower subdifferential (or contingent subdifferential) and the Fréchet subdifferentials, which are among the smallest subdifferentials known to date. It follows that the estimates which it provides require weak assumptions and are accurate. When the function is locally Lipschitzian, we get a simple statement which refines the Lebourg mean-value theorem.     

Proximal analysis in reflexive smooth Banach spaces

In the present paper, we introduce and study a new proximal normal cone in reflexive Banach spaces in terms of a generalized projection operator. Two new variants of generalized proximal subdifferentials are also introduced in reflexive smooth Banach spaces. The density theorem for both proximal subdifferentials has been proved in p-uniformly convex and q-uniformly smooth Banach spaces. Various important properties and applications of our concepts are also proved.     

Relationships Between Convexificators and Greenberg–Pierskalla Subdifferentials for Quasiconvex Functions

In this paper, the relationship between convexificators and Greenberg–Pierskalla-based (GP-based) subdifferentials for quasiconvex functions is proved. The established results lead to a mean value theorem, a chain rule, and the closedness property for GP-based subdifferentials. Furthermore, the connection between Clarke generalized gradient and Mordukhovich subdifferential with GP-based subdifferentials is highlighted.     

Error Bounds and Implicit Multifunction Theorem in Smooth Banach Spaces and Applications to Optimization

In this paper we provide an error bound estimate and an implicit multifunction theorem in terms of smooth subdifferentials and abstract subdifferentials. Then, we derive a subdifferential calculus and Fritz–John type necessary optimality conditions for constrained minimization problems.     

Application of Michael's theorem and its converse to sublinear operators

A theorem of Michael on continuous selectors and its converse are used in this article to study subdifferentials of continuous sublinear operators with values in a cone of lower semicontinuous functions. It is proved that such operators are subdifferentiable (i.e., have nonempty subdifferentials) if their domains are separable Banach spaces. Sublinear operators that are not subdifferentiable are found.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 67–75, July, 1992.     

ON CONE D.C. OPTIMIZATION AND CONJUGATE DUALITY

This paper derives first order necessary and sufficient conditions for unconstrained cone d.c. programming problems where the underlined space is partially ordered with respect to a cone. These conditions are given in terms of directional derivatives and subdifferentials of the component functions. Moreover, conjugate duality for cone d.c. optimization is discussed and weak duality theorem is proved in a more general partially ordered linear topological vector space (generalizing the results in [11]).     

A divergence theorem for Finsler metrics

By means of the general form of Stokes' theorem on manifolds a divergence theorem is derived for hypersurfaces which bound a compact region of ann-dimensional Finsler spaceF _n . In general the integrand of then-fold volume integral will depend on the covariant derivatives of an arbitrary vector field which defines the element of support; certain conditions under which this dependence may be circumvented are discussed. The scalar curvature ofF _n is expressed in terms of the divergence of a certain vector field: forn=2 this formula reduces to a particularly simple form, and its substitution into the aforementioned divergence theorem gives rise to a formula which represents a generalization of the classical Gauss-Bonnet Theorem.

     

Weak sharp efficiency in multiobjective optimization

By using the generalized Fermat rule, the Mordukhovich subdifferential for maximum functions, the fuzzy sum rule for Fréchet subdifferentials and the sum rule for Mordukhovich subdifferentials, we establish a necessary optimality condition for the local weak sharp efficient solution of a constrained multiobjective optimization problem. Moreover, by employing the approximate projection theorem, and some appropriate convexity and affineness conditions, we also obtain some sufficient optimality conditions respectively for the local and global weak sharp efficient solutions of such a multiobjective optimization problem.     

Weyl’s theorem and thin spectra

Using the notion of thin sets we prove a theorem of Weyl type for the Wolf essential spectrum ofT ∈β (H). *Further we show that Weyl’s theorem holds for a restriction convexoid operator and consequently modify some results of Berberian. Finally we show that Weyl’s theorem holds for a paranormal operator and that a polynomially compact paranormal operator is a compact perturbation of a diagnoal normal operator. A structure theorem for polynomially compact paranormal operators is also given.     

Vector variational-like inequalities and vector optimization problems in Asplund spaces

In this paper, the Minty vector variational-like inequality, the Stampacchia vector variational-like inequality, and the weak formulations of these two inequalities defined by means of Mordukhovich limiting subdifferentials are introduced and studied in Asplund spaces. Some relations between the vector variational-like inequalities and vector optimization problems are established by using the properties of Mordukhovich limiting subdifferentials. An existence theorem of solutions for the weak Minty vector variational-like inequality is also given.     

Multifunctions on abstract measurable spaces and application to stochastic decision theory

Summary The main results are some very general theorems about measurable multifunctions on abstract measurable spaces with compact values in a separable metric space. It is shown that measurability is equivalent to the existence of a pointwise dense countable family of measurable selectors, and that the intersection of two compact-valued measurable multifunctions is measurable. These results are used to obtain a Filippov type implicit function theorem, and a general theorem concerning the measurability of y(t)=min f({t} × Γ(t)) when f is a real valued function and Γ a compact valued multifunction. An application to stochastic decision theory is given generalizing a result of Benes. The research in this paper was partially supported by University of Kansas General Research Fund Grants 3918-5038 and 3199-5038. Entrata in Redazione il 20 dicembre 1972.     

The Bartle Bilinear Integration and Carleman Operators

Some characterizations of integrable functions in the bilinear sense of Bartle with respect to the injective tensor product are obtained. As a consequence it is shown that the kernels of Carleman compact operators coincide with these Bartle integrable functions. This result is applied to prove that every Carleman L-weak-compact operator is compact. An example showing the different behavior of the integrability with respect to the projective tensor product is given. A general Fubini theorem in this setting is shown.     

Ergodic theory of differentiable dynamical systems

Iff is a C^{1 + ɛ} diffeomorphism of a compact manifold M, we prove the existence of stable manifolds, almost everywhere with respect to everyf-invariant probability measure on M. These stable manifolds are smooth but do not in general constitute a continuous family. The proof of this stable manifold theorem (and similar results) is through the study of random matrix products (multiplicative ergodic theorem) and perturbation of such products. Dedicated to the memory of Rufus Bowen     

Nonsmooth multiobjective optimization using limiting subdifferentials

In this study, using the properties of limiting subdifferentials in nonsmooth analysis and regarding a separation theorem, some weak Pareto-optimality (necessary and sufficient) conditions for nonsmooth multiobjective optimization problems are proved.     

On global subdifferentials with applications in nonsmooth optimization

The notions of global subdifferentials associated with the global directional derivatives are introduced in the following paper. Most common used properties, a set of calculus rules along with a mean value theorem are presented as well. In addition, a diversity of comparisons with well-known subdifferentials such as Fréchet, Dini, Clarke, Michel–Penot, and Mordukhovich subdifferential and convexificator notion are provided. Furthermore, the lower global subdifferential is in fact proved to be an abstract subdifferential. Therefore, the lower global subdifferential satisfies standard properties for subdifferential operators. Finally, two applications in nonconvex nonsmooth optimization are given: necessary and sufficient optimality conditions for a point to be local minima with and without constraints, and a revisited characterization for nonsmooth quasiconvex functions.

     

On a theorem of Wiener

 Wiener has shown that an integrable function on the circle T which is square integrable near the identity and has nonnegative Fourier transform, is square integrable on all of T. In the last 30 years this has been extended by the work of various authors step by step. The latest result states that, in a suitable reformulation, Wiener's theorem with ``p-integrable' in place of ``square integrable' holds for all even p and fails for all other p  (1, ∞) in the case of a general locally compact abelian group. We extend this to all IN-groups (locally compact groups with at least one invariant compact neighbourhood) and show that an extension to all locally compact groups is not possible: Wiener's theorem fails for all p < ∞ in the case of the ax + b-group. Received: 12 September 2000 Mathematics Subject Classification (2000): 43A35     

Exact calculus of Fréchet subdifferentials for Hadamard directionally differentiable functions

We continue the study of the calculus of the generalized subdifferentials started in [V.F. Demyanov, V. Roshchina, Exhausters and subdifferentials in nonsmooth analysis, Optimization (2006) (in press)] and [V. Roshchina, Relationships between upper exhausters and the basic subdifferential in Variational Analysis, Journal of Mathematical Analysis and Applications 334 (2007) 261–272] and provide some basic calculus rules for the Fréchet subdifferentials via collections of compact convex sets associated with Hadamard directional derivative. The main result of this paper is the sum rule for the Fréchet subdifferential in the form of an equality, which holds for Hadamard directionally differentiable functions, and is of significant interest from the points of view of both theory and applications.