Measures as Lagrange multipliers in multistage stochastic programming

A duality theory is developed for multistage convex stochastic programming problems whose decision (or recourse) functions can be approximated by continuous functions satisfying the same constraints. Necessary and sufficient conditions for optimality are obtained in terms of the existence of multipliers in the class of regular Borel measures on the underlying probability space, these being decomposable, of course, into absolutely continuous and singular components with respect to the given probability measure. This provides an alternative to the approach where the multipliers are elements of the dual of $\text{L}$^∞ with an analogous decomposition. However, besides the existence of strictly feasible solutions, special regularity conditions are required, such as the “laminarity” of the probability measure, a property introduced in an earlier paper. These are crucial in ensuring that the minimum in the optimization problem can indeed be approached by continuous functions.     

Cores of partitioning games

A generalization of assignment games, called partitioning games, is introduced. Given a finite set N of players, there is an a priori given set π of coalitions of N and only coalitions in π play an essential role. Necessary and sufficient conditions for the nonemptiness of the cores of all games with essential coalitions π are developed. These conditions appear extremely restrictive. However when N is ‘large’, there are relatively few ‘types’ of players, and members of π are ‘small’ and defined in terms of numbers of players of each type contained in subsets, then approximate cores are nonempty.     

Second Order Sufficient Conditions for Optimal Control Problems with Non-unique Minimizers: An Abstract Framework

Standard second order sufficient conditions in optimal control theory provide not only the information that an extremum is a weak local minimizer, but also tell us that the extremum is locally unique. It follows that such conditions will never cover problems in which the extremum is continuously embedded in a family of constant cost extrema. Such problems arise in periodic control, when the cost is invariant under time translations, in shape optimization, where the cost is invariant under Euclidean transformations (translations and rotations of the extremal shape), and other areas where the domain of the optimization problem does not really comprise elements in a linear space, but rather an equivalence class of such elements. We supply a set of sufficient conditions for minimizers that are not locally unique, tailored to problems of this nature. The sufficient conditions are in the spirit of earlier conditions for ‘non-isolated’ minima, in the context of general infinite dimensional nonlinear programming problems provided by Bonnans, Ioffe and Shapiro, and require coercivity of the second variation in directions orthogonal to the constant cost set. The emphasis in this paper is on the derivation of directly verifiable sufficient conditions for a narrower class of infinite dimensional optimization problems of special interest. The role of the conditions in providing easy-to-use tests of local optimality of a non-isolated minimum, obtained by numerical methods, is illustrated by an example in optimal control.     

Combinatorial results on the fitting problems of the multivariate gamma distribution introduced by Prékopa and Szántai

Generalizations of the results of an earlier paper of the second author, related to the problem of fitting a multivarite gamma distribution to empirical data, are discussed in the paper. The multivariate gamma distribution under consideration is the one that was introduced in the paper of Prékopa and Szántai (in Water Resources Research, 14:19?C24, 1978), some earlier results on the fitting problem were given in the paper of Szántai (in Alkalmazott Matematikai Lapok 10:35?C60, 1984). In the present paper it is proved that the necessary conditions given earlier are not sufficient and some further new, mostly computational results are provided, too. Using the more efficient computation tools we are able now to give the sufficient conditions for dimensions 5 and 6 as well. For higher dimensions we have only necessary conditions and the invention of a suitable necessary and sufficient condition remains an open problem when n is greater than 6. The miscellaneousness of the necessary and sufficient conditions obtained in our new project for n=6 indicates that finding necessary and sufficient conditions in general should be a very hard problem.     

The Logical vs. the Ontological Understanding of Conditions

According to the truth-functional analysis of conditions, to be ‘necessary for’ and ‘sufficient for’ are converse relations. From this, it follows that to be ‘necessary and sufficient for’ is a symmetric relation, that is, that if P is a necessary and sufficient condition for Q, then Q is a necessary and sufficient condition for P. This view is contrary to common sense. In this paper, I point out that it is also contrary to a widely accepted ontological view of conditions, according to which if P is a necessary and sufficient condition for Q, then Q is in no sense a condition for P; it is a mere consequence of P.     

On uniqueness and time regularity of flows of power‐law like non‐Newtonian fluids

Large class of non‐Newtonian fluids can be characterized by index p, which gives the growth of the constitutively determined part of the Cauchy stress tensor. In this paper, the uniqueness and the time regularity of flows of these fluids in an open bounded three‐dimensional domain is established for subcritical ps, i.e. for p>11/5. Our method works for ‘all’ physically relevant boundary conditions, the Cauchy stress need not be potential and it may depend explicitly on spatial and time variable. As a simple consequence of time regularity, pressure can be introduced as an integrable function even for Dirichlet boundary conditions. Moreover, these results allow us to define a dynamical system corresponding to the problem and to establish the existence of an exponential attractor. Copyright © 2010 John Wiley & Sons, Ltd.     

Doob's Decomposition Theorem for Fuzzy (Super) Submartingales

Abstract

The notions of fuzzy random variables and fuzzy (super) submartingales are introduced. In this paper we provide the necessary and sufficient conditions of Doob's decomposition for fuzzy (super) submartingales. Finally, we discuss the decomposition of fuzzy (super) submartingales on R, and an example is given which explains that not every fuzzy (super) submartingale has Doob's decomposition.     

Semistability of switched dynamical systems,Part I: Linear system theory

This paper develops semistability and uniform semistability analysis results for switched linear systems. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system’s initial conditions. Since solutions to switched systems are a function of the system’s initial conditions as well as the switching signals, uniformity here refers to the convergence rate of the multiple solutions as the switching signal evolves over a given switching set. The main results of the paper involve sufficient conditions for semistability and uniform semistability using multiple Lyapunov functions and sufficient regularity assumptions on the class of switching signals considered.     

Weak convergence for generalized semi-Markov processes

Generalized semi-Markov schemes were introduced by Matthes in 1962 under the designation ‘Bedienungsschemata’ (service schemes). They include a large variety of familiar stochastic models. It is shown in this paper that under appropriate regularity conditions the associated stochastic process describing the state at timet,t≥0, and the stationary distribution are continuous functions of the life-times of the active components. The supplementary-variable Markov process is shown to be the limit process of a sequence of discrete-state-process obtained through approximating the life-time distributions by mixtures of Erlang distributions and measuring ages and residual life-times in phases. This approach supplements the phase method.     

Asymptotic structural characteristics of fuzzy measure and their applications

In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary.     

Study of the heat equation in a symmetric conical type domain of

This paper is devoted to the analysis of the N‐space dimensional heat equation, subject to Cauchy–Dirichlet boundary conditions. The problem is set in a symmetric conical type domain. More precisely, we look for sufficient conditions on the lateral boundary of the domain, as weak as possible in order to obtain the maximal regularity of the solution in an anisotropic Hilbertian Sobolev space. For this purpose, the domain decomposition method is employed. Copyright © 2013 John Wiley & Sons, Ltd.     

Exceptional Families of Elements for Optimization Problems in Reflexive Banach Spaces with Applications

In this paper, we propose a new notion of ‘exceptional family of elements’ for convex optimization problems. By employing the notion of ‘exceptional family of elements’, we establish some existence results for convex optimization problem in reflexive Banach spaces. We show that the nonexistence of an exceptional family of elements is a sufficient and necessary condition for the solvability of the optimization problem. Furthermore, we establish several equivalent conditions for the solvability of convex optimization problems. As applications, the notion of ‘exceptional family of elements’ for convex optimization problems is applied to the constrained optimization problem and convex quadratic programming problem and some existence results for solutions of these problems are obtained.     

A unified theory for coherent systems in reliability I. A generalization for some classic theory: Monotone binary coherent systems

Binary coherent system theory has played an important part in reliability. Its extension to (‘degradable’ or ‘multistate’ or) multinary systems has recently been considered in various papers, through various definitions. This paper lays the foundations of a unified theory for coherent systems by first giving unified arguments to apply and to investigate further binary and multinary systems. Monotone binary systems are introduced and examined by generalizing classic deterministic and probabilistic results. Applications of monotone coherence to the multinary case are proposed in a companion paper with a unified viewpoint on multinary coherent systems. As an indication, monotone constraints are defined with a partition of the component set and some total orderings imposed on the elements of the concerned partition. The discrete partition retrieves the classic theory of (free) binary coherent systems; some constraints defined from component levels lead to multinary coherent systems; some other constraints apply to systems submitted to some ‘common stresses’, e.g. the organizing system of a monotone coherent decomposition.     

Adaptive stability in combinatorial optimization problems

We consider a general approach to the construction of necessary, sufficient, and necessary and sufficient conditions that allow to ‘adapt’ a known optimal solution of an abstract combinatorial problem with a certain structure to a change in the initial data set for a fixed cost function ‘easily’ from the combinatorial point of view. We call this approach adaptive stability. Apparently, it is the first time that the approach is described for an abstract problem in a rigorous mathematical formalization.     

Sufficient conditions for error bounds and linear regularity in Banach spaces

In this paper, we study error bounds for lower semicontinuous functions defined on Banach space and linear regularity for finitely many closed subset in Banach spaces. By using Clarke＇s subd- ifferentials and Ekeland variational principle, we establish several sufficient conditions ensuring error bounds and linear regularity in Banach spaces.     

Limit properties in a family of quasi-arithmetic means

It is known that the Power Means tend to the maximum of their arguments when the exponents tend to \({+\infty}\). We give certain necessary and sufficient conditions for a 1-parameter family of quasi-arithmetic means generated by functions satisfying certain smoothness conditions to have an analogous property. Our results are deeply connected with operators introduced by Mikusiński and Páles in the late 1940s and late 1980s, respectively. The main result is a generalization of the author’s earlier results.     

Isometric embeddings into Heisenberg groups

We study isometric embeddings of a Euclidean space or a Heisenberg group into a higher dimensional Heisenberg group, where both the source and target space are equipped with an arbitrary left-invariant homogeneous distance that is not necessarily sub-Riemannian. We show that if all infinite geodesics in the target are straight lines, then such an embedding must be a homogeneous homomorphism. We discuss a necessary and certain sufficient conditions for the target space to have this ‘geodesic linearity property’, and we provide various examples.     

Blow-up and global solutions for quasilinear parabolic equations with Neumann boundary conditions

Using the upper and lower solution techniques and Hopf's maximum principle, the sufficient conditions for the existence of blow-up positive solution and global positive solution are obtained for a class of quasilinear parabolic equations subject to Neumann boundary conditions. An upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also specified.     

Symmetry in de Branges almost Pontryagin spaces

In many examples of de Branges spaces symmetry appears naturally. Presence of symmetry gives rise to a decomposition of the space into two parts, the ‘even’ and the ‘odd’ part, which themselves can be regarded as de Branges spaces. The converse question is to decide whether a given space is the ‘even’ part or the ‘odd’ part of some symmetric space, and, if yes, to describe the totality of all such symmetric spaces. We consider this question in an indefinite (almost Pontryagin space) setting, and give a complete answer. Interestingly, it turns out that the answers for the ‘even’ and ‘odd’ cases read quite differently; the latter is significantly more complex.     

Some results on regularity in vector optimization

In this paper we suggest an approach to regularity in, vector optimization which extends the one given in [9]; some necessary or sufficient regularity conditions are given for a wide class of nondifferentiable vector optimization problems which embraces the convex ones.