Lipschitz Continuous Research Articles

On locally Lipschitz convex functions defined on subsets with empty interior of locally convex spaces

The generic Fréchet and/or Gateaux differentiability of a continuous convex function on an open convex subset of a Banach space is well studied and has important theoretical implications. An example of Rainwater [Yet more on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(3):773–778] shows that similar results are not true when the openness of the domain is not ensured, but such properties can be obtained if the respective convex function is assumed to be locally Lipschitz, as shown by Verona [More on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(1):137–140]; Rainwater [Yet more on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(3):773–778]; Noll [Generic Gâteaux-differentiability of convex functions on small sets. J Math Anal Appl. 1990;147(2):531–544] and others. It is our aim to extend such results for convex functions defined on (not necessarily open) convex subsets of locally convex spaces. In the same framework we extend Gale's duality theorem (1967), as well as a result of H. Ergin & T. Sarver on the unique dual representation of a convex function (2010). Moreover, a detailed study of the Rainwater example is done.

Stochastic theta methods for random periodic solution of stochastic differential equations under non-globally Lipschitz conditions

Stochastic theta methods for random periodic solution of stochastic differential equations under non-globally Lipschitz conditions

Fractional Newton‐type integral inequalities by means of various function classes

The authors of the paper present a method to examine some Newton‐type inequalities for various function classes using Riemann‐Liouville fractional integrals. Namely, some fractional Newton‐type inequalities are established by using convex functions. In addition, several fractional Newton‐type inequalities are proved by using bounded functions by fractional integrals. Moreover, we construct some fractional Newton‐type inequalities for Lipschitzian functions. Furthermore, several Newton‐type inequalities are acquired by fractional integrals of bounded variation. Finally, we provide our results by using special cases of obtained theorems and examples.

Numerical solution of fuzzy stochastic Volterra integral equations with constant delay

This paper considers the nonlinear fuzzy stochastic Volterra integral equations with constant delay, which are general and include many fuzzy stochastic integral and differential equations discussed in literature. Since Doob's martingale inequality is no longer applicable to such equations, a new maximum inequality is obtained. Combining with the Picard approximation method, the existence and uniqueness of solutions to nonlinear fuzzy stochastic Volterra integral equations with constant delay are given. Moreover we prove that the solution behaves stably in the case of small changes of initial values, kernels and nonlinearities. We further develop a Euler-Maruyama (EM) scheme and prove the strong convergence of the scheme. It is shown that the strong convergence order of the EM method is 0.5 under Lipschitz condition. Moreover, the strong superconvergence order is 1 if further, the kernel h(t,s) of the stochastic term satisfies h(t,t)=0. Numerical examples demonstrated that the numerical results are consistent with the theoretical research conclusions. Furthermore, the application model of the fuzzy stochastic Volterra integral equation with constant delay in population dynamics is considered, and the exact solution of the numerical example is given in explicit form.

Inertial Halpern-type methods for variational inequality with application to medical image recovery

In this paper, we propose inertial Halpern-type algorithms involving a quasi-monotone operator for approximating solutions of variational inequality problems which are fixed points of quasi-nonexpansive mappings in reflexive Banach spaces. We use Bregman distance functions to enhance the efficiency of our algorithms and obtain strong convergence results, even in cases where the Lipschitz constant of the operator involved is unknown a priori. Furthermore, we illustrate the practical applicability of our methods through numerical experiments. Notably, our algorithms excel when compared to recent techniques in the literature. Of particular significance is their successful application in restoring computed tomography medical images that have been affected by motion blur and random noise. Our algorithms consistently outperform established state-of-the-art methods in all conducted experiments, showcasing their competitiveness and potential to advance variational inequality problem-solving, especially in the field of medical image recovery.

Presenting the Sierpinski Gasket in Various Categories of Metric Spaces

This paper studies presentations of the Sierpinski gasket as a final coalgebra for a functor on three categories of metric spaces with additional designated points. The three categories which we study differ on their morphisms: one uses short (non-expanding) maps, the second uses continuous maps, and the third uses Lipschitz maps. The functor in all cases is very similar to what we find in the standard presentation of the gasket as an attractor. It was previously known that the Sierpinski gasket is bilipschitz equivalent (though not isomorhpic) to the final coalgebra of this functor in the category with short maps, and that final coalgebra is obtained by taking the completion of the initial algebra. In this paper, we prove that the Sierpiniski gasket itself is the final coalgebra in the category with continuous maps, though it does not occur as the completion of the initial algebra. In the Lipschitz setting, we show that the final coalgebra for this functor does not exist.

Initialization-free Lie-bracket Extremum Seeking

Stability results for extremum seeking control in Rn have predominantly been restricted to local or, at best, semi-global practical stability. Extending semi-global stability results of extremum-seeking systems to unbounded sets of initial conditions often demands a stringent global Lipschitz condition on the cost function, which is rarely satisfied by practical applications. In this paper, we address this challenge by leveraging tools from higher-order averaging theory. In particular, we establish a novel second-order averaging result with global (practical) stability implications. By leveraging this result, we characterize sufficient conditions on cost functions under which uniform global (i.e., under any initialization) practical asymptotic stability can be established for a class of extremum-seeking systems acting on static maps. Our sufficient conditions include the case when the gradient of the cost function, rather than the cost function itself, satisfies a global Lipschitz condition, which covers quadratic cost functions. Our results are also applicable to vector fields that are not necessarily Lipschitz continuous at the origin, opening the door to non-smooth Lie-bracket ES dynamics. We illustrate all the results via different analytical and/or numerical examples.

Extragradient method with feasible inexact projection to variational inequality problem

The variational inequality problem in finite-dimensional Euclidean space is addressed in this paper, and two inexact variants of the extragradient method are proposed to solve it. Instead of computing exact projections on the constraint set, as in previous versions extragradient method, the proposed methods compute feasible inexact projections on the constraint set using a relative error criterion. The first version of the proposed method provided is a counterpart to the classic form of the extragradient method with constant steps. In order to establish its convergence we need to assume that the operator is pseudo-monotone and Lipschitz continuous, as in the standard approach. For the second version, instead of a fixed step size, the method presented finds a suitable step size in each iteration by performing a line search. Like the classical extragradient method, the proposed method does just two projections into the feasible set in each iteration. A full convergence analysis is provided, with no Lipschitz continuity assumption of the operator defining the variational inequality problem.

Strong Cosmic Censorship with bounded curvature

Abstract In this paper we propose a weaker version of Penrose’s much heeded Strong Cosmic Censorship (SCC) conjecture, asserting inextendability of maximal Cauchy developments by manifolds with Lipschitz continuous Lorentzian metrics and Riemann curvature bounded in Lp. Lipschitz continuity is the threshold regularity for causal structures, and curvature bounds rule out infinite tidal accelerations, arguing for physical significance of this weaker SCC conjecture. The main result of this paper, under the assumption that no extensions exist with higher connection regularity W1,p loc, proves in the affirmative this SCC conjecture with bounded curvature for p sufficiently large, (p > 4 with uniform bounds, p > 2 without uniform bounds).

Error bounds for one-dimensional constrained Langevin approximations for nearly density-dependent Markov chains

AbstractContinuous-time Markov chains are frequently used to model the stochastic dynamics of (bio)chemical reaction networks. However, except in very special cases, they cannot be analyzed exactly. Additionally, simulation can be computationally intensive. An approach to address these challenges is to consider a more tractable diffusion approximation. Leite and Williams (Ann. Appl. Prob.29, 2019) proposed a reflected diffusion as an approximation for (bio)chemical reaction networks, which they called the constrained Langevin approximation (CLA) as it extends the usual Langevin approximation beyond the first time some chemical species becomes zero in number. Further explanation and examples of the CLA can be found in Anderson et al. (SIAM Multiscale Modeling Simul.17, 2019).In this paper, we extend the approximation of Leite and Williams to (nearly) density-dependent Markov chains, as a first step to obtaining error estimates for the CLA when the diffusion state space is one-dimensional, and we provide a bound for the error in a strong approximation. We discuss some applications for chemical reaction networks and epidemic models, and illustrate these with examples. Our method of proof is designed to generalize to higher dimensions, provided there is a Lipschitz Skorokhod map defining the reflected diffusion process. The existence of such a Lipschitz map is an open problem in dimensions more than one.

Infinitesimal Lipschitz conditions on a family of analytic varieties: genericity and necessity

Infinitesimal Lipschitz conditions on a family of analytic varieties: genericity and necessity

T-controllability of higher-order fractional stochastic delay system via integral contractor

The existence, uniqueness and trajectory (T)-controllability results for mild solutions to the fractional neutral stochastic integro-delay differential system are presented in this article. To demonstrate the results, the concept of bounded integral contractors is combined with the stochastic result and sequencing technique. In contrast to previous publications, we do not need to specify the induced inverse of the controllability operator to prove the controllability results, and the relevant non-linear function does not have to meet the Lipschitz condition. Furthermore, T-controllability results for neutral integro-delay differential systems with non-local conditions have been established. Finally, an application to demonstrate the acquired results is discussed.

Novel hybrid integrated Pix2Pix and WGAN model with Gradient Penalty for binary images denoising

This paper introduces a novel approach to image denoising that leverages the advantages of Generative Adversarial Networks (GANs). Specifically, we propose a model that combines elements of the Pix2Pix model and the Wasserstein GAN (WGAN) with Gradient Penalty (WGAN-GP). This hybrid framework seeks to capitalize on the denoising capabilities of conditional GANs, as demonstrated in the Pix2Pix model, while mitigating the need for an exhaustive search for optimal hyperparameters that could potentially ruin the stability of the learning process. In the proposed method, the GAN’s generator is employed to produce denoised images, harnessing the power of a conditional GAN for noise reduction. Simultaneously, the implementation of the Lipschitz continuity constraint during updates, as featured in WGAN-GP, aids in reducing susceptibility to mode collapse. This innovative design allows the proposed model to benefit from the strong points of both Pix2Pix and WGAN-GP, generating superior denoising results while ensuring training stability. Drawing on previous work on image-to-image translation and GAN stabilization techniques, the proposed research highlights the potential of GANs as a general-purpose solution for denoising. The paper details the development and testing of this model, showcasing its effectiveness through numerical experiments. The dataset was created by adding synthetic noise to clean images. Numerical results based on real-world dataset validation underscore the efficacy of this approach in image-denoising tasks, exhibiting significant enhancements over traditional techniques. Notably, the proposed model demonstrates strong generalization capabilities, performing effectively even when trained with synthetic noise.

A necessary conditions for optimal singular control of McKean-Vlasov stochastic differential equations driven by spatial parameters local martingale

In this paper, we focus on stochastic singular control problems involving McKean-Vlasov stochastic differential equations driven by a spatially parameterized continuous local martingale. The drift coefficient in these equations depends on the state of the solution process and its law. The control variable consists of two components: an absolutely continuous control and a singular one. Firstly, under Lipschitz conditions, we establish the existence and uniqueness of its strong solution. Next, we derive the necessary conditions for optimal singular control under the assumption that the control domain is convex. These optimality conditions differ from the classical ones in the sense that here the adjoint equation is a McKean-Vlasov backward stochastic differential equation driven by a continuous local martingale with spatial parameters. The proof of our result is based on the derivatives of the local martingale with respect to spatial parameters and the derivative of the drift coefficient with respect to the probability measure.

Indirect adaptive observer control (I-AOC) design for truck–trailer model based on T–S fuzzy system with unknown nonlinear function

Tracking is a crucial problem for nonlinear systems as it ensures stability and enables the system to accurately follow a desired reference signal. Using Takagi–Sugeno (T–S) fuzzy models, this paper addresses the problem of fuzzy observer and control design for a class of nonlinear systems. The Takagi–Sugeno (T–S) fuzzy models can represent nonlinear systems because it is a universal approximation. Firstly, the T–S fuzzy modeling is applied to get the dynamics of an observational system in order to estimate the unmeasurable states of an unknown nonlinear system. There are various kinds of nonlinear systems that can be modeled using T–S fuzzy systems by combining the input state variables linearly. Secondly, the T–S fuzzy systems can handle unknown states as well as parameters known to the indirect adaptive fuzzy observer. A simple feedback method is used to implement the proposed controller. As a result, the feedback linearization method allows for solving the singularity problem without using any additional algorithms. A fuzzy model representation of the observation system comprises parameters and a feedback gain. The Lyapunov function and Lipschitz conditions are used in constructing the adaptive law. This method is then illustrated by an illustrative example to prove its effectiveness with different kinds of nonlinear functions. A well-designed controller is effective and its performance index minimizes network utilization—this factor is particularly significant when applied to wireless communication systems.

A Stancu type extension of the Cheney-Sharma Chlodovsky operators

In this paper we introduce a Stancu type extension of the Cheney-Sharma Chlodovsky operators based on the ideas presented by Cătinaș and Buda, Bostanci and Bașcanbaz-Tunca, respectively Söylemez and Tașdelen. For this new operators we study some approximation and convexity properties and the preservation of the Lipschitz constant and order. Finally, we study approximation properties of the new operators with the help of Korovkin type theorems.

Chaos synchronization of two coupled map lattice systems using safe reinforcement learning

When synchronizing two continuous typical chaotic systems, the state variables of the response system are usually bounded and satisfy the Lipschitz condition. This permits a universal synchronization method. In the synchronization of two high-dimensional discrete chaotic systems such as the coupled map lattice systems, the state variables of the response system often diverge, which makes it difficult to seek a universal synchronization method. To overcome this difficulty, bounded and hard constraints, which correspond to the concept of safety level III in reinforcement learning terms, on the state variables must be imposed, when the discrete response system is perturbed. We propose a universal method for synchronizing two discrete systems based on safe reinforcement learning (RL). In this method, the RL agent’s policy is used to reach the goal of synchronization and a safety layer added directly on top of the policy is used to satisfy safety level III. The safety layer consists of a one-step predictor for the perturbed response system and an action correction formulation. The one-step predictor, based on a next generation reservoir computing, is used to identify whether the next state of the perturbed system is within the chaotic domain, and if not, the action correction formula is activated to modify the corresponding perturbing force component to zero. In this way, the state of the perturbed system will remain in the bounded chaotic domain. We demonstrate that the proposed method succeeds in synchronization without divergence through a numerical example with two coupled map lattice systems, and the average synchronization time is 12.66 iteration steps over 1000 different initial conditions. The method works even when parameter mismatches and disturbances exist in the system. We compare the performance in both cases with and without the safety layer and find that successful synchronization is only possible in the former case, emphasizing the significance of the safety layer. The performance of the algorithm with optimal hyper-parameters obtained by Bayesian optimization is robust and stable.

Strong convergence of the tamed Euler-Maruyama method for stochastic singular initial value problems with non-globally Lipschitz continuous coefficients

In our previous works [1] and [2], we delved into numerical methods for solving stochastic singular initial value problems (SSIVPs) that involve coefficients satisfying the global Lipschitz condition. The paper addresses the limitations of our previous work by introducing an explicit method, called the tamed Euler-Maruyama method, for numerically solving SSIVPs with non-globally Lipschitz continuous coefficients, which is both easy-to-implement and exceptionally efficient. We prove the existence and uniqueness theorem and the boundedness of the moments of the solution to SSIVPs under the non-globally Lipschitz condition. Moreover, we provide a sharp analysis of the strong convergence of the proposed method, along with the uniform boundedness of numerical solutions. We also apply our results to the stochastic singular Ginzburg-Landau system and provide numerical simulations to illustrate our theoretical findings.

Random coordinate descent: A simple alternative for optimizing parameterized quantum circuits

Variational quantum algorithms rely on the optimization of parameterized quantum circuits in noisy settings. The commonly used back-propagation procedure in classical machine learning is not directly applicable in this setting due to the collapse of quantum states after measurements. Thus, gradient estimations constitute a significant overhead in a gradient-based optimization of such quantum circuits. This paper introduces a random coordinate descent algorithm as a practical and easy-to-implement alternative to the full gradient descent algorithm. This algorithm only requires one partial derivative at each iteration. Motivated by the behavior of measurement noise in the practical optimization of parameterized quantum circuits, this paper presents an optimization problem setting that is amenable to analysis. Under this setting, the random coordinate descent algorithm exhibits the same level of stochastic stability as the full gradient approach, making it as resilient to noise. The complexity of the random coordinate descent method is generally no worse than that of the gradient descent and can be much better for various quantum optimization problems with anisotropic Lipschitz constants. Theoretical analysis and extensive numerical experiments validate our findings. Published by the American Physical Society 2024

Convex integrals of molecules in Lipschitz-free spaces

We introduce convex integrals of molecules in Lipschitz-free spaces F(M) as a continuous counterpart of convex series considered elsewhere, based on the de Leeuw representation. Using optimal transport theory, we show that these elements are determined by cyclical monotonicity of their supports, and that under certain finiteness conditions they agree with elements of F(M) that are induced by Radon measures on M, or that can be decomposed into positive and negative parts. We also show that convex integrals differ in general from convex series of molecules. Finally, we present some standalone results regarding extensions of Lipschitz functions which, combined with the above, yield applications to the extremal structure of F(M). In particular, we show that all elements of F(M) are convex series of molecules when M is uniformly discrete and identify all extreme points of the unit ball of F(M) in that case.