Class Of Generalized Functions Research Articles

Sufficient optimality conditions and duality for nonsmooth multiobjective optimization problems via higher order strong convexity

In this paper, we define some new generalizations of strongly convex functions of order m for locally Lipschitz functions using Clarke subdifferential. Suitable examples illustrating the non emptiness of the newly defined classes of functions and their relationships with classical notions of pseudoconvexity and quasiconvexity are provided. These generalizations are then employed to establish sufficient optimality conditions for a nonsmooth multiobjective optimization problem involving support functions of compact convex sets. Furthermore, we formulate a mixed type dual model for the primal problem and establish weak and strong duality theorems using the notion of strict efficiency of order m. The results presented in this paper extend and unify several known results from the literature to a more general class of functions as well as optimization problems.

Non-Stationary Additive Utility and Time Consistency

Within a continuous time life cycle model of consumption and savings, I study the properties of the most general class of additive intertemporal utility functionals. They are not necessarily stationary, and do not necessarily multiplicatively separate a discount factor from utility. I prove rigorously that time consistency holds if and only if the per-period felicity function is multiplicatively separable in t, the date of decision and in s, the date of consumption, or equivalently, if the Fisherian instantaneous subjective discount rate does not depend on t. The model allows to explain anomalies in intertemporal choice and various empirical regularities, even when the agents are time consistent. On the other hand, the model allows to characterize mathematically the effective consumption profile of naive, time-inconsistent agents.

Dimension Reduction in Statistical Estimation of Partially Observed Multiscale Processes

We consider partially observed multiscale diffusion models that are specified up to an unknown vector parameter. We establish for a very general class of test functions that the filter of the original model converges to a filter of reduced dimension. Then, this result is used to justify statistical estimation for the unknown parameters of interest based on the model of reduced dimension but using the original available data. This allows us to learn the unknown parameters of interest while working in lower dimensions, as opposed to working with the original high dimensional system. Simulation studies support and illustrate the theoretical results.

The Fekete-Szegö problem for a general class of bi-univalent functions satisfying subordinate conditions

In this work, we obtain the Fekete-Szego inequalities for the class $P_{Sigma }left( lambda ,phi right) $ of bi-univalent functions. The results presented in this paper improve the recent work of Prema and Keerthi [11].

The gradient flow of a generalized Fisher information functional with respect to modified Wasserstein distances

This article is concerned with the existence of nonnegative weak solutions to a particular fourth-order partial differential equation: it is a formal gradient flow with respect to a generalized Wasserstein transportation distance with nonlinear mobility. The corresponding free energy functional is referred to as generalized Fisher information functional since it is obtained by autodissipation of another energy functional which generates the heat flow as its gradient flow with respect to the aforementioned distance. Our main results are twofold: For mobility functions satisfying a certain regularity condition, we show the existence of weak solutions by construction with the well-known minimizing movement scheme for gradient flows. Furthermore, we extend these results to a more general class of mobility functions: a weak solution can be obtained by approximation with weak solutions of the problem with regularized mobility.

Classical integrability for three-point functions: cognate structure at weak and strong couplings

Abstract In this paper, we develop a new method of computing three-point functions in the SU(2) sector of the $$ \mathcal{N}=4 $$ N = 4 super Yang-Mills theory in the semi-classical regime at weak coupling, which closely parallels the strong coupling analysis. The structure threading two disparate regimes is the so-called monodromy relation, an identity connecting the three-point functions with and without the insertion of the monodromy matrix. We shall show that this relation can be put to use directly for the semi-classical regime, where the dynamics is governed by the classical Landau-Lifshitz sigma model. Specifically, it reduces the problem to a set of functional equations, which can be solved once the analyticity in the spectral parameter space is specified. To determine the analyticity, we develop a new universal logic applicable at both weak and strong couplings. As a result, compact semi-classical formulas are obtained for a general class of three-point functions at weak coupling including the ones whose semi-classical behaviors were not known before. In addition, the new analyticity argument applied to the strong coupling analysis leads to a modification of the integration contour, producing the results consistent with the recent hexagon bootstrap approach. This modification also makes the Frolov-Tseytlin limit perfectly agree with the weak coupling form.

Tail probability estimates for additive functionals

In this paper, based on techniques of Malliavin calculus, we obtain an explicit bound for tail probabilities of a general class of additive functionals. Applications to fractional Brownian motion and Cox–Ingersoll–Ross process are given to illustrate the theory.

Multistability of complex-valued neural networks with time-varying delays

In this paper, the multistability problem is studied for an n-dimensional delayed complex-valued neural networks with two general classes of activation functions. After splitting the state space to multiple subsets, based on the fixed point theorem, it is shown that such complex-valued neural networks can have 9n equilibria, each of which is located in one of the subsets. Furthermore, some sufficient conditions are derived for the local exponential stability of some equilibria by employing the property of activation functions and inequality technique. As an application of these results, some criteria are obtained for checking the coexistence and exponential stability of multiple equilibria of real-valued neural networks. Two examples are performed to illustrate and validate the theoretical findings.

Global saddle points of nonlinear augmented Lagrangian functions

We notice that the results for the existence of global (local) saddle points of augmented Lagrangian functions in the literature were only sufficient conditions of some special types of augmented Lagrangian. In this paper, we introduce a general class of nonlinear augmented Lagrangian functions for constrained optimization problem. In two different cases, we present sufficient and necessary conditions for the existence of global saddle points. Moreover, as corollaries of the two results above, we not only obtain sufficient and necessary conditions for the existence of global saddle points of some special types of augmented Lagrangian functions mentioned in the literature, but also give some weaker sufficient conditions than the ones in the literature. Compared with our recent work (Wang et al. in Math Oper Res 38:740---760, 2013), the nonlinear augmented Lagrangian functions in this paper are more general and the results in this paper are original. We show that some examples (such as improved barrier augmented Lagrangian) satisfy the assumptions of this paper, but not available in Wang et al. (2013).

On two functional equations connected with distributivity of fuzzy implications

The distributivity law for a fuzzy implication \(I\colon [0,1]^2 \to [0,1]\) with respect to a fuzzy disjunction \(S\colon [0,1]^2 \to [0,1]\) states that the functional equation \( I(x,S(y,z))=S(I(x,y),I(x,z)) \) is satisfied for all pairs \((x,y)\) from the unit square. To compare some results obtained while solving this equation in various classes of fuzzy implications, Wanda Niemyska has reduced the problem to the study of the following two functional equations: \( h(\min(xg(y),1)) = \min(h(x)+ h(xy),1)\), \(x \in (0,1)\), \(y \in (0,1]\), and \( h(xg(y)) = h(x)+ h(xy)\), \(x,y \in (0, \infty)\), in the class of increasing bijections \(h\colon [0,1] \to [0,1]\) with an increasing function \(g\colon (0,1] \to [1, \infty)\) and in the class of monotonic bijections \(h\colon (0, \infty) \to (0, \infty)\) with a function \(g\colon (0, \infty) \to (0, \infty)\), respectively. A description of solutions in more general classes of functions (including nonmeasurable ones) is presented.

Stationary Increments Harmonizable Stable Fields: Upper Estimates on Path Behaviour

Studying sample path behaviour of stochastic fields/processes is a classical research topic in probability theory and related areas such as fractal geometry. To this end, many methods have been developed for a long time in Gaussian frames. They often rely on some underlying “nice” Hilbertian structure and can also require finiteness of moments of high order. Therefore, they can hardly be transposed to frames of heavy-tailed stable probability distributions. However, in the case of some linear non-anticipative moving average stable fields/processes, such as the linear fractional stable sheet and the linear multi-fractional stable motion, rather new wavelet strategies have already proved to be successful in order to obtain sharp moduli of continuity and other results on sample path behaviour. The main goal of our article is to show that, despite the difficulties inherent in the frequency domain, such kind of a wavelet methodology can be generalized and improved, so that it also becomes fruitful in a general harmonizable stable setting with stationary increments. Let us point out that there are large differences between this harmonizable setting and the moving average stable one. The real-valued harmonizable stable stochastic field X on which we focus is defined on $$\mathbb {R}^d$$ through an arbitrary spectral density belonging to a general and wide class of functions. First, we introduce a wavelet-type random series representation of X and express it as the finite sum $$X=\sum _\eta X^\eta $$ , where the fields $$X^\eta $$ are called the $$\eta $$ -frequency parts, since they extend the usual low-frequency and high-frequency parts. Moreover, we show the continuity of the sample paths of the $$X^\eta $$ ’s and X; also, we discuss the existence and continuity of their partial derivatives of an arbitrary order. Thereafter, we obtain several almost sure upper estimates related to: (a) the anisotropic behaviour of generalized directional increments of the $$X^\eta $$ ’s and X, on an arbitrary fixed compact cube of $$\mathbb {R}^d$$ ; (b) the behaviour at infinity of the $$X^\eta $$ ’s, of X, and of their partial derivatives, when they exist. We mention that all the results on sample paths obtained in the article are valid on the same event of probability 1; furthermore, this event is “universal”, in the sense that it does not depend, in any way, on the spectral density associated with X.

Logistic Modeling for Optimal Resource Allocation in Network Assisted Power Control

Controlling the distribution of resources is imperative in any wireless system. Much research has considered the efficiency of the allocation in terms of bits per Joule of radiated energy. The most common model used in analysis of this utility has been the logistic curve. Many game theoretical concepts have been used to propose algorithms that help such systems reach an optimal operating point. This paper investigates network assisted power control in general and, unlike other work that has previously been done, proves that for a very general class of functions it is Pareto Optimal. Furthermore, a methodology to modeling wireless systems with various modulation and channel conditions as logistic functions is presented and their trade-offs investigated.

Reconstruction of noisy signals by minimization of non-convex functionals

Non-convex functionals have shown sharper results in signal reconstruction as compared to convex ones, although the existence of a minimum has not been established in general. This paper addresses the study of a general class of either convex or non-convex functionals for denoising signals which combines two general terms for fitting and smoothing purposes, respectively. The first one measures how close a signal is to the original noisy signal. The second term aims at removing noise while preserving some expected characteristics in the true signal such as edges and fine details. A theoretical proof of the existence of a minimum for functionals of this class is presented. The main merit of this result is to show the existence of minimizer for a large family of non-convex functionals.

Random Conical Tessellations

We consider tessellations of the Euclidean $$(d-1)$$ -sphere by $$(d-2)$$ -dimensional great subspheres or, equivalently, tessellations of Euclidean d-space by hyperplanes through the origin; these we call conical tessellations. For random polyhedral cones defined as typical cones in a conical tessellation by random hyperplanes, and for random cones which are dual to these in distribution, we study expectations for a general class of geometric functionals. They include combinatorial quantities, such as face numbers, as well as, for example, conical intrinsic volumes. For isotropic conical tessellations (those generated by random hyperplanes with spherically symmetric distribution), we determine the complete covariance structure of the random vector whose components are the k-face contents of the induced spherical random polytopes. This result can be considered as a spherical counterpart of a classical result due to Roger Miles.

Catastrophes and ex post shadow prices—How the value of the last fish in a lake is infinity and why we should not care (much)

Catastrophic risk is currently a hotly debated topic. This paper contributes to this debate by showing two results. First, it is shown that for a certain class of optimal control problems, the derivative of the value function with respect to the initial state may approach infinity as the state variable goes to zero, even when the first-order partial derivatives of the integrand and transition functions are finite. In the process, it is shown that standard phase diagrams used in optimal control theory contain more information than generally recognized and that the value function itself may be easily illustrated in these diagrams. Second, we show that even if the value function has an infinite derivative at some point, it is not correct to avoid this point in finite time at almost any cost. The results are illustrated in a simple linear-quadratic fisheries model and proven for a more general class of growth functions.

Error bounds for parametric polynomial systems with applications to higher-order stability analysis and convergence rates

The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be viewed, in particular, as solution sets to problems of generalized semi-infinite programming with polynomial data. Exploiting the imposed polynomial structure together with powerful tools of variational analysis and semialgebraic geometry, we establish a far-going extension of the \L ojasiewicz gradient inequality to the general nonsmooth class of supremum marginal functions as well as higher-order (H\"older type) local error bounds results with explicitly calculated exponents. The obtained results are applied to higher-order quantitative stability analysis for various classes of optimization problems including generalized semi-infinite programming with polynomial data, optimization of real polynomials under polynomial matrix inequality constraints, and polynomial second-order cone programming. Other applications provide explicit convergence rate estimates for the cyclic projection algorithm to find common points of convex sets described by matrix polynomial inequalities and for the asymptotic convergence of trajectories of subgradient dynamical systems in semialgebraic settings.

Gradual Adjustment and Equilibrium Uniqueness Under Noisy Monitoring

We study the implications of flexible adjustment in strategic interactions using a class of finite-horizon models in continuous time. Players take costly actions to affect the evolution of state variables that are commonly observable and perturbed by Brownian noise. The values of these state variables influence players' terminal payoffs at the deadline, as well as their flow payoffs. In contrast to the static case, the equilibrium is unique under a general class of terminal payoff functions. Our characterization of the equilibrium builds on recent developments in the theory of backward stochastic differential equations (BSDEs). We use this tool to analyze applications, including team production, hold-up problems, and dynamic contests. In a team production model, the unique equilibrium selects an efficient outcome when frictions vanish.

Supply and Demand Functions in Inventory Models

The central issue in supply chain management is to match supply with demand, and the heart of a planning model is the modeling of supply and demand functions. To allow for analytical tractability, the existing literature often assumes almost surely linear supply and demand functions, which greatly limits the applicability of the models. The goal of this paper is to provide a unified approach to analyze general random supply and demand functions. By transforming the problem into one defined on a higher dimension, we show that many of the seemingly highly nonlinear supply and demand functions (in the almost sure sense) are linear in the stochastic sense. With this new notion of linearity, called stochastic linearity in mid-point, our ability to analyze supply chain problems is much enhanced. We are able to prove the concavity of the profit function in the transformed supply and demand decisions for a general class of supply and demand functions that include, but are not restricted to, the ones studied in the existing literature. Thus, many of the challenging problems now become tractable. Moreover, we characterize a set of easy-to-verify conditions for the stochastic linearity in mid-point that allows us to test functions with known forms. For functions with unknown forms, we develop a nonparametric approach to allow for empirical estimation and verification of the needed stochastic properties directly from the data.

Bounded extremum seeking with discontinuous dithers

The analysis of discontinuous extremum seeking (ES) controllers, e.g. those applicable to digital systems, has historically been more complicated than that of continuous controllers. We establish a simple and general extension of a recently developed bounded form of ES to a general class of oscillatory functions, including functions discontinuous with respect to time, such as triangle or square waves with dead time. We establish our main results by combining a novel idea for oscillatory control with an extension of functional analytic techniques originally utilized by Kurzweil, Jarnik, Sussmann, and Liu in the late 80s and early 90s and recently studied by Dürr et al. We demonstrate the value of the result with an application to inverter switching control.

Random geometric graphs with general connection functions

In the original (1961) Gilbert model of random geometric graphs, nodes are placed according to a Poisson point process, and links formed between those within a fixed range. Motivated by wireless ad hoc networks "soft" or "probabilistic" connection models have recently been introduced, involving a "connection function" H(r) that gives the probability that two nodes at distance r are linked (directly connect). In many applications (not only wireless networks), it is desirable that the graph is connected; that is, every node is linked to every other node in a multihop fashion. Here the connection probability of a dense network in a convex domain in two or three dimensions is expressed in terms of contributions from boundary components for a very general class of connection functions. It turns out that only a few quantities such as moments of the connection function appear. Good agreement is found with special cases from previous studies and with numerical simulations.