Proof of covering minimality by generalizing the notion of independence

A method is proposed for obtaining lower bounds for the length of the shortest cover and complexity of the minimal cover based on the notion of independent family of sets. For the problem of minimization of Boolean functions, we provide the functions and construct coverings by faces of the set of unit vertices for which the suggested lower bounds can be achieved in the case of five or more variables. The lower bounds, based on independent sets, are unreachable and cannot be used as sufficient conditions for minimality of such functions.     

Sufficient sets in weighted Fréchet spaces of entire functions

Under study are sufficient sets in Fréchet spaces of entire functions with uniform weighted estimates. We obtain general results on the a priori overflow of these sets and introduce the concept of their minimality. We also establish necessary and sufficient conditions for a sequence of points on the complex plane to be a minimal sufficient set for a weighted Fréchet space. Applications are given to the problem of representation of holomorphic functions in a convex domain with certain growth near the boundary by exponential series.     

On complexity measures of complexes of faces in the unit cube

Under study is the problem of proving the minimality of complexes of faces in the unit cube. Basing on the ordinal properties of a complexity measure functional and the structural properties of Boolean functions, we formulated some sufficient conditions that can be used to prove that a complex of faces is minimal. This allowed us to expand the set of complexes of faces that were proved to be minimal with respect to the complexity measures with certain properties. The strict inclusion is proved for the sets of complexes of faces: kernel, minimal for an arbitrary complexity measure, and minimal for every complexity measure that is invariant under replacement of faces with isomorphic faces.     

The Basis Properties of Some Systems of Exponential Functions,Cosines, and Sines

We consider some systems of exponential functions, cosines, and sines with complex-valued coefficients and establish a necessary and sufficient condition for completeness and minimality of these systems in Lebesgue spaces.     

指数函数系的不完备性和极小性

Necessary and sufficient conditions are obtained for the incompleteness and the minimality of the exponential system E（Λ,M） = {z l e λ n z ： l = 0,1,...,m n-1;n = 1,2,...} in the Banach space E 2 [σ] consisting of some analytic functions in a half strip.If the incompleteness holds,each function in the closure of the linear span of exponential system E（Λ,M） can be extended to an analytic function represented by a Taylor-Dirichlet series.Moreover,by the conformal mapping ζ = φ（z） = e z ,the similar results hold for the incompleteness and the minimality of the power function system F （Λ,M） = {（log ζ） l ζ λ n ： l = 0,1,...,m n-1;n = 1,2,...} in the Banach space F 2 [σ] consisting of some analytic functions in a sector.     

First and second order optimality conditions in vector optimization problems with nontransitive preference relation

We present first and second order conditions, both necessary and sufficient, for ?-minimizers of vector-valued mappings over feasible sets with respect to a nontransitive preference relation ?. Using an analytical representation of a preference relation ? in terms of a suitable family of sublinear functions, we reduce the vector optimization problem under study to a scalar inequality, from which, using the tools of variational analysis, we derive minimality conditions for the initial vector optimization problem.     

On Completeness and Minimality of Random Exponential System in a Weighted Banach Space of Functions Continuous on the Real Line

In this paper, the completeness and minimality properties of some random exponential system in a weighted Banach space of complex functions continuous on the real line for convex nonnegative weight are studied. The results may be viewed as a probabilistic version of Malliavin's classical results.     

Stability and independence of the shifts of finitely many refinable functions

Typical constructions of wavelets depend on the stability of the shifts of an underlying refinable function. Unfortunately, several desirable properties are not available with compactly supported orthogonal wavelets, e.g., symmetry and piecewise polynomial structure. Presently, multiwavelets seem to offer a satisfactory alternative. The study of multiwavelets involves the consideration of the properties of several (simultaneously) refinable functions. In Section 2 of this article, we characterize stability and linear independence of the shifts of a finite refinable function set in terms of the refinement mask. Several illustrative examples are provided. The characterizations given in Section 2 actually require that the refinable functions be minimal in some sense. This notion of minimality is made clear in Section 3, where we provide sufficient conditions on the mask to ensure minimality. The conditions are shown to be necessary also under further assumptions on the refinement mask. An example is provided illustrating how the software package MAPLE can be used to investigate at least the case of two simultaneously refinable functions.     

复指数函数系的极小性

A sufficient condition is obtained for the minimality of the complex exponential system E（A, M） = {z^le^λnz： l = 0, 1,,.., mn - 1; n = 1, 2,...} in the Banaeh space La^p consisting of all functions f such that f^-a ∈ LP（N）. Moreover, if the incompleteness holds, each function in the closure of the linear span of exponential system E（A, M） can be extended to an analytic function represented by a Taylor-Dirichlet series.     

Some Dual Conditions for Global Weak Sharp Minimality of Nonconvex Functions

Weak sharp minimality is a notion emerged in optimization whose utility is largely recognized in the convergence analysis of algorithms for solving extremum problems as well as in the study of the perturbation behavior of such problems. In this article, some dual constructions of nonsmooth analysis, mainly related to quasidifferential calculus and its recent developments, are employed in formulating sufficient conditions for global weak sharp minimality. They extend to nonconvex functions a condition, which is known to be valid in the convex case. A feature distinguishing the results here proposed is that they avoid to assume the Asplund property on the underlying space.     

Stability of semi-infinite vector optimization problems under functional perturbations

This paper is devoted to the study of continuity properties of Pareto solution maps for parametric semi-infinite vector optimization problems (PSVO). We establish new necessary conditions for lower and upper semicontinuity of Pareto solution maps under functional perturbations of both objective functions and constraint sets. We also show that the necessary condition becomes sufficient for the lower and upper semicontinuous properties in the special case where the constraint set mapping is lower semicontinuous at the reference point. Examples are given to illustrate the obtained results.     

A Minty variational principle for set optimization

Extremal problems are studied involving an objective function with values in (order) complete lattices of sets generated by so-called set relations. Contrary to the popular paradigm in vector optimization, the solution concept for such problems, introduced by F. Heyde and A. Löhne, comprises the attainment of the infimum as well as a minimality property. The main result is a Minty type variational inequality for set optimization problems which provides a sufficient optimality condition under lower semicontinuity assumptions and a necessary condition under appropriate generalized convexity assumptions. The variational inequality is based on a new Dini directional derivative for set-valued functions which is defined in terms of a “lattice difference quotient.” A residual operation in a lattice of sets replaces the inverse addition in linear spaces. Relationships to families of scalar problems are pointed out and used for proofs. The appearance of improper scalarizations poses a major difficulty which is dealt with by extending known scalar results such as Diewert's theorem to improper functions.     

Reducing Exhausters

The notions of upper and lower exhausters were introduced by Demyanov (Optimization 45:13–29, 1999). Upper and lower exhausters can be employed to study a very wide range of positively homogeneous functions, for example, various directional derivatives of nonsmooth functions. Exhausters are not uniquely defined; hence, the problem of minimality arises naturally. This paper describes some techniques for reducing exhausters, both in size and amount of sets. We define also a modified convertor which provides much more flexibility in converting upper exhausters to lower ones and vice versa, and allows us to obtain much smaller sets.     

Kahane–Khintchine inequalities and functional central limit theorem for stationary random fields

We establish new Kahane–Khintchine inequalities in Orlicz spaces induced by exponential Young functions for stationary real random fields which are bounded or satisfy some finite exponential moment condition. Next, we give sufficient conditions for partial sum processes indexed by classes of sets satisfying some metric entropy condition to converge in distribution to a set-indexed Brownian motion. Moreover, the class of random fields that we study includes φ-mixing and martingale difference random fields.     

The weak Rayleigh hypothesis and certain properties of systems of metaharmonic functions

We consider a two-dimensional problem of scattering on periodic boundaries. We study properties, such as minimality and property of being a basis, of systems of functions arising during application of the generalized separation-of-variables method. A rigorous formulation of the weak Rayleigh hypothesis, both in the narrow and the wide sense, is given and necessary and sufficient conditions for their validity are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 195, pp. 19–28, 1991.     

On necessary and sufficient optimality conditions for multicriterial optimization problems

In this paper we derive first order necessary and sufficient optimality conditions for nonsmooth optimization problems with multiple criteria. These conditions are given for different optimality notions (i.e. weak, Pareto- and proper minimality) and for different types of derivatives of nonsmooth objective functions (locally Lipschitz continuous and quasidifferentiable) mappings. The conditions are given, if possible, in terms of a derivative and a subdifferential of those mappings.     

Exponential stability of mild solutions of stochastic partial differential equations with delays

A semiiinear stochastic partial differential equation with variable delays is considered. Sufficient conditions for the exponential stability in the p-th mean of mild solutions are obtained. Also, pathwise exponential stability is proved. Since the technique ofLyapunov functions is not suitable for delayed equations, the results have been proved by using the properties of the stochastic convolution. As the sufficient conditions obtained are also valid for the case without delays, one can ensure exponential stability of mild solution in some cases where the sufficient conditions in Ichikawa [11] do not give any answer. The results are illustrated with some examples     

Reduction and Minimality of Coexhausters

V.F. Demyanov introduced exhausters for the study of nonsmooth functions. These are families of convex compact sets that enable one to represent the main part of the increment of a considered function in a neighborhood of the studied point as MaxMin or MinMax of linear functions. Optimality conditions were described in terms of these objects. This provided a way for constructing new algorithms for solving nondifferentiable optimization problems. Exhausters are defined not uniquely. It is obvious that the smaller an exhauster, the less are the computational expenses when working with it. Thus, the problem of reduction of an available family arises. For the first time, this problem was considered by V.A. Roshchina. She proposed conditions for minimality and described some methods of reduction in the case when these conditions are not satisfied. However, it turned out that the exhauster mapping is not continuous in the Hausdorff metrics, which leads to the problems with convergence of numerical methods. To overcome this difficulty, Demyanov proposed the notion of coexhausters. These objects enable one to represent the main part of the increment of the considered function in a neighborhood of the studied point in the form of MaxMin or MinMax of affine functions. One can define a class of functions with the continuous coexhauster mapping. Optimality conditions can be stated in terms of these objects too. But coexhausters are also defined not uniquely. The problem of reduction of coexhausters is considered in this paper for the first time. Definitions of minimality proposed by Roshchina are used. In contrast to ideas proposed in the works of Roshchina, the minimality conditions and the technique of reduction developed in this paper have a clear and transparent geometric interpretation.     

Generalized weak subdifferentials

In this article, generalized weak subgradient (gw-subgradient) and generalized weak subdifferential (gw-subdifferential) are defined for nonconvex functions with values in an ordered vector space. Convexity and closedness of the gw-subdifferential are stated and proved. By using the gw-subdifferential, it is shown that the epigraph of nonconvex functions can be supported by a cone instead of an affine subspace. A generalized lower (locally) Lipschitz function is also defined. By using this definition, some existence conditions of the gw-subdifferentiability of any function are stated and some properties of gw-subdifferentials of any function are examined. Finally, by using gw-subdifferential, a global minimality condition is obtained for nonconvex functions.     

Strict Efficiency in Vector Optimization

The notion of strict minimum of order m for real optimization problems is extended to vector optimization. Its properties and characterization are studied in the case of finite-dimensional spaces (multiobjective problems). Also the notion of super-strict efficiency is introduced for multiobjective problems, and it is proved that, in the scalar case, all of them coincide. Necessary conditions for strict minimality and for super-strict minimality of order m are provided for multiobjective problems with an arbitrary feasible set. When the objective function is Fréchet differentiable, necessary and sufficient conditions are established for the case m = 1, resulting in the situation that the strict efficiency and super-strict efficiency notions coincide.