Weak Continuity Condition Research Articles

A Relaxed Tseng’s Method-based Algorithm for Solving Split Variational Inequalities with Multiple Output Sets

In this paper, the split variational inequality problem with multiple output sets is considered in a more general framework of reflexive Banach spaces. We introduce a relaxed Tseng extragradient method for approximating the solution of this problem when the cost operators are pseudomonotone and non-Lipschitz. Moreover, our proposed algorithm employs the inertial technique and a new self-adaptive step size to guarantee high rate of convergence. Strong convergence of the algorithm is proved in this case without the need for the sequential weak continuity condition. Finally, some numerical experiments are given to show the applicability of our proposed algorithm.

Fixed point theorems for $ b $-generalized contractive mappings with weak continuity conditions

<abstract><p>The purpose of this paper was to introduce several $ b $-generalized contractive mappings in the framework of cone $ b $-metric spaces over Banach algebras. The obtained contractions generalized and extended the counterparts in metric spaces, cone metric spaces, and $ b $-metric spaces. Moreover, via weakening the completeness of the spaces, we gave some fixed point theorems for asymptotically regular mappings without considering the orbital continuity and $ k $-continuity of the mappings. Those who need a specification is our results do not rely on the continuity of $ b $-metric and the normality of cones. In addition, some nontrivial examples were presented to illustrate the superiority of our fixed point theorems.</p></abstract>

A fixed point theorem for non-negative functions

<p>In this paper, we are concerned with the study of the existence and uniqueness of fixed points for the class of functions $ f: C\to C $ satisfying the inequality</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \ell\left(\alpha f(t)+(1-\alpha)f(s)\right)\leq \sigma \ell(\alpha t+(1-\alpha)s) $\end{document} </tex-math></disp-formula></p><p>for every $ t, s\in C $ with $ f(t)\neq f(s) $, where $ C $ is a closed subset of $ [0, \infty) $, $ \alpha, \sigma\in (0, 1) $ are constants, and $ \ell: [0, \infty)\to [0, \infty) $ is a function satisfying the condition $ \inf_{t > 0} \frac{\ell(t)}{t^\rho} > 0 $ for some constant $ \rho > 0 $. Namely, under a weak continuity condition imposed on $ f $, we show that $ f $ possesses a unique fixed point, and for every $ t_0\in C $, the Picard sequence defined by $ t_{n+1} = f(t_n) $, $ n\geq 0 $, converges to this fixed point. Next, we study the special cases when $ C $ is a closed interval and $ \ell $ is a convex or concave function. Namely, making use of the Hermite-Hadamard inequalities, we obtain several new fixed point theorems. To the best of our knowledge, the considered class of functions was never previously investigated in the literature.</p>

An Inertial Iterative Algorithm for Approximating Common Solutions to Split Equalities of Some Nonlinear Optimization Problems

AbstractIn this paper, we introduce a new inertial Tseng’s extragradient method with self-adaptive step sizes for approximating a common solution of split equalities of equilibrium problem (EP), non-Lipschitz pseudomonotone variational inequality problem (VIP) and fixed point problem (FPP) of nonexpansive semigroups in real Hilbert spaces. We prove that the sequence generated by our proposed method converges strongly to a common solution of the EP, pseudomonotone VIP and FPP of nonexpansive semigroups without any linesearch procedure nor the sequential weak continuity condition often assumed by authors when solving non-Lipschitz VIPs. Finally, we provide some numerical experiments for the proposed method in comparison with related methods in the literature. Our result improves, extends and generalizes several of the existing results in this direction.

An inertial subgradient extragradient method with Armijo type step size for pseudomonotone variational inequalities with non-Lipschitz operators in Banach spaces

In this paper, we study the pseudomonotone variational inequality problems with non-Lipschitz operators. We propose an inertial subgradient extragradient method with Halpern technique and Armijo type step size for approximating the solution of the problem in the framework of 2-uniformly convex real Banach spaces. We prove that the sequence generated by our proposed method converges strongly to the solution of the problem under some mild conditions and without the weak sequential continuity condition often assumed by authors in solving pseudomonotone variational inequality problems. Finally, we provide some numerical experiments for the proposed method in comparison with other existing methods in the literature. Our result extends and improves several of the existing results in the current literature in this direction.

Ball Convergence of Modified Homeier-Like's Method in Banach Spaces under Weak Continuity Condition

The aim of this study is to analyze the local convergence of the multi-step Homeier's-like method for solving nonlinear equations in Banach space. Furthermore, we avoid hypotheses on high order derivatives which limit the applicability of the method. Instead, we only use hypotheses on the first derivative. Thus the applicability of the method has been extended by preserving the order of convergence. The convergence of the solution is proved under the weak hypotheses i.e. $\omega$-continuity condition. Some numerical instances where earlier results cannot be applied to solve equations but our results can be applied are provided to validate the theoretical contribution.

Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations

We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms both biased and unbiased simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory.

Efficient Simulation of Generalized SABR and Stochastic Local Volatility Models based on Markov Chain Approximations

We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms both biased and unbiased simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory.

Efficient Simulation of Stochastic Differential Equations Based on Markov Chain Approximations With Applications

We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms \textit{both biased and unbiased} simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory. Note: this is an earlier version of the work Efficient Simulation of Generalized SABR and Stochastic Local Volatility Models based on Markov Chain Approximations.

Asymptotic Optimality of Finite Model Approximations for Partially Observed Markov Decision Processes With Discounted Cost

We consider finite model approximations of discrete-time partially observed Markov decision processes (POMDPs) under the discounted cost criterion. After converting the original partially observed stochastic control problem to a fully observed one on the belief space, the finite models are obtained through the uniform quantization of the state and action spaces of the belief space Markov decision process (MDP). Under mild assumptions on the components of the original model, it is established that the policies obtained from these finite models are nearly optimal for the belief space MDP, and so, for the original partially observed problem. The assumptions essentially require that the belief space MDP satisfies a mild weak continuity condition. We provide an example and introduce explicit approximation procedures for the quantization of the set of probability measures on the state space of POMDP (i.e., belief space).

Free vibration of a micro-scale composite laminated Reddy plate using a finite element method based on the new modified couple stress theory

In this study, a non-confirming element which is a combination of the BCIZ and refined triangular non-conforming element is developed for the free vibration analysis of micro-scale angle-ply laminated Reddy plate based on the new modified couple stress theory. Strain and curvature components of the micro-scale composite laminated Reddy plates based on the new modified couple stress theory use the first- and the second-order derivatives of the transverse displacement. Therefore, the functions are required to satisfy both C0 and C1 to be continuous. There are two types of shape functions for the transverse displacement in the present element. One satisfies the C0 continuity condition exactly while the other satisfies the C1 weak continuity condition. Numerical examples are given in which the results for a simply supported cross-ply micro-scale composite laminated plate obtained by the present element coincide well with the solutions of the analytical Fourier series solutions. We also investigated the effects of ply orientation, boundary conditions and length-to-thickness ratio on the natural frequencies of a micro-scale composite laminated plate using the finite element method (FEM).

Constrained discounted Markov decision processes with Borel state spaces

We study discrete-time discounted constrained Markov decision processes (CMDPs) with Borel state and action spaces. These CMDPs satisfy either weak (W) continuity conditions, that is, the transition probability is weakly continuous and the reward function is upper semicontinuous in state–action pairs, or setwise (S) continuity conditions, that is, the transition probability is setwise continuous and the reward function is upper semicontinuous in actions. Our main goal is to study models with unbounded reward functions, which are often encountered in applications, e.g., in consumption/investment problems. We provide some general assumptions under which the optimization problems in CMDPs are solvable in the class of randomized stationary policies and in the class of chattering policies introduced in this paper. If the initial distribution and transition probabilities are atomless, then using a general “purification result” of Feinberg and Piunovskiy we show the existence of a deterministic (stationary) optimal policy. Our main results are illustrated by examples.

Connectedness of random set attractors

We examine the question whether random set attractors for continuous-time random dynamical systems on a connected state space are connected. In the deterministic case, these attractors are known to be connected. In the probabilistic setup, however, connectedness has only been shown under stronger connectedness assumptions on the state space. Under a weak continuity condition on the random dynamical system we prove connectedness of the pullback attractor on a connected space. Additionally, we provide an example of a weak random set attractor of a random dynamical system with even more restrictive continuity assumptions on an even path-connected space which even attracts all bounded sets and which is not connected. On the way to proving connectedness of a pullback attractor we prove a lemma which may be of independent interest and which holds without the assumption that the state space is connected. It states that even though pullback convergence to the attractor allows for exceptional nullsets which may depend on the compact set, these nullsets can be chosen independently of the compact set (which is clear for \sigma -compact spaces but not at all clear for spaces which are not \sigma -compact).

On Wold’s approach to representation of preferences

This paper presents easily verifiable sufficient conditions on sequence spaces that guarantee representation of preference orders. Our approach involves identifying a suitable subset of the set of alternatives, such that (a) the preference order is representable on this subset, and (b) the subset has the property that for each alternative, there is some element in this subset which is indifferent to it. We follow Wold in choosing this subset to be the diagonal. Our first result uses a weak monotonicity condition (on the diagonal), and a substitution condition, and may be identified as the essence of Wold’s contribution. In the second result, we show that one can obtain a Wold-type representation result when weak monotonicity is replaced by a weak continuity condition. We use the countable order-dense characterization of representability in the proofs of both results, thereby integrating the contributions of Wold (1943) and Debreu (1954). Through a series of examples we show that our representation results are robust; they cannot be improved upon by dropping any of our conditions. An example is also presented to show that existence of degenerate indifference classes is compatible with the representation of monotone preferences. Our study thereby indicates that while the presence of substitution possibilities can be useful in representing preferences, they are not necessary for such results to hold.

Existence and computation of Berge equilibrium and of two refinements

This paper presents existence conditions as well as computation methods for Berge equilibrium and two refinements: Berge–Vaisman equilibrium and Berge–Nash equilibrium. Each equilibrium concept is interpreted and illustrated on the basis of relevant examples and general existence conditions satisfying weak continuity and quasi-concavity conditions are provided.

An oversampling technique for the multiscale finite volume method to simulate electromagnetic responses in the frequency domain

In order to reduce the computational cost of the simulation of electromagnetic responses in geophysical settings that involve highly heterogeneous media, we develop a multiscale finite volume method with oversampling for the quasi-static Maxwell’s equations in the frequency domain. We assume a coarse mesh nested within a fine mesh that accurately discretizes the problem. For each coarse cell, we independently solve a local version of the original Maxwell’s system subject to linear boundary conditions on an extended domain, which includes the coarse cell and a neighborhood of fine cells around it. The local Maxwell’s system is solved using the fine mesh contained in the extended domain and the mimetic finite volume method. Next, these local solutions (basis functions) together with a weak-continuity condition are used to construct a coarse-mesh version of the global problem. The basis functions can be used to obtain the fine-mesh details from the solution of the coarse-mesh problem. Our approach leads to a significant reduction in the size of the final system of equations and the computational time, while accurately approximating the behavior of the fine-mesh solutions. We demonstrate the performance of our method using two 3D synthetic models: one with a mineral deposit in a geologically complex medium and one with random isotropic heterogeneous media. Both models are discretized using an adaptive mesh refinement technique.

A refined three-node triangular element based on the HW variational theorem for multilayered composite plates

For the three-node triangular elements, the displacement parameters are linear variation within one element, so that the second and the third derivatives of displacement parameters are close to zero. Therefore, the three-node triangular element will encounter difficulty to accurately predict transverse shear stresses by integrating the three-dimensional equilibrium equation through the thickness direction. Thus, the few three-node triangular plate elements can accurately yield the transverse shear stresses of multilayered composite plates. A higher-order zig-zag model accounting for the zero transverse shear strain conditions at the top and the bottom surfaces is firstly presented in this work. Based on the proposed model, a refined three-node triangular plate element satisfying the requirement of C1 weak-continuity conditions in the inter-element is developed to accurately produce the distributions of displacements and stresses through the thickness of multilayered composite plates. To obtain the improved transverse shear stresses, the simplified three-dimensional equilibrium equation has been a priori used. It is significant that the higher-order derivatives of displacement parameters have been taken out from the expression of transverse shear stresses by employing the three-field Hu-Washizu (HW) variational principle, which is convenient for the finite element implementation. Performance of the present element is tested by comparing with three-dimensional elasticity solutions as well as the reference results evaluated using other models. A detailed study is conducted to highlight the influences of boundary condition, number of layers and aspect ratio on the static response of multilayered composite plates, and numerical results can show the accuracy and range of applicability for the proposed element.

Near optimality of quantized policies in stochastic control under weak continuity conditions

This paper studies the approximation of optimal control policies by quantized (discretized) policies for a very general class of Markov decision processes (MDPs). The problem is motivated by applications in networked control systems, computational methods for MDPs, and learning algorithms for MDPs. We consider the finite-action approximation of stationary policies for a discrete-time Markov decision process with discounted and average costs under a weak continuity assumption on the transition probability, which is a significant relaxation of conditions required in earlier literature. The discretization is constructive, and quantized policies are shown to approximate optimal deterministic stationary policies with arbitrary precision. The results are applied to the fully observed reduction of a partially observed Markov decision process, where weak continuity is a much more reasonable assumption than more stringent conditions such as strong continuity or continuity in total variation.

Abstract Cauchy problem for weakly continuous operators

We discuss the solvability of the abstract Cauchy problem u′(t)=Au(t) for t>0 and u(0)=u0∈D, where D is a weakly closed subset of a Banach space Y continuously embedded in the underlying Banach space X and A is a nonlinear operator from D into X satisfying a local weak continuity condition described by a family of functionals. We introduce a new type of weak compactness on the level sets of functionals and give a necessary and sufficient condition for the existence of a weakly continuously differentiable solution satisfying a growth condition. We apply the abstract result obtained to the Cauchy problem for a nonlinear Schrödinger equation and a mixed problem for a logarithmic wave equation.

Average Optimality for Continuous-Time Markov Decision Processes Under Weak Continuity Conditions

This paper considers the average optimality for a continuous-time Markov decision process in Borel state and action spaces, and with an arbitrarily unbounded nonnegative cost rate. The existence of a deterministic stationary optimal policy is proved under the conditions that allow the following; the controlled process can be explosive, the transition rates are weakly continuous, and the multifunction defining the admissible action spaces can be neither compact-valued nor upper semicontinuous.