Showing papers on "Monotone polygon published in 2023"

Existence of Pulses for Monotone Reaction-Diffusion Systems

Abstract: We consider a monotone reaction-diffusion system of the form w 1 − w1 + f1(w2) = 0, w 2 − w2 + f2(w1) = 0, and address the question of the existence of pulses, that is of positive decaying at infinity solutions. We prove that pulses exist if and only if the wave speed of the associated travelling-wave problem is positive. The proofs are based on the Leray-Schauder method which uses topological degree for elliptic problems in unbounded domains and a priori estimates of solutions in weighted spaces.

On Split Monotone Variational Inclusion Problem with Multiple Output Sets with Fixed Point Constraints

Abstract: Abstract In this paper, we introduce and study the concept of split monotone variational inclusion problem with multiple output sets (SMVIPMOS). We propose a new iterative scheme, which employs the viscosity approximation technique for approximating the solution of the SMVIPMOS with fixed point constraints of a nonexpansive mapping in real Hilbert spaces. The proposed method utilises the inertial technique for accelerating the speed of convergence and a self-adaptive step size so that its implementation does not require prior knowledge of the operator norm. Under mild conditions, we obtain a strong convergence result for the proposed algorithm and obtain a consequent result, which complements several existing results in the literature. Moreover, we apply our result to study the notions of split variational inequality problem with multiple output sets with fixed point constraints and split convex minimisation problem with multiple output sets with fixed point constraints in Hilbert spaces. Finally, we present some numerical experiments to demonstrate the implementability of our proposed method.

Modified Dai-Yuan iterative scheme for nonlinear systems and its application

Abstract: By exploiting the idea employed in the spectral Dai-Yuan method by Xue et al. [IEICE Trans. Inf. Syst. 101 (12)2984-2990 (2018)] and the approach applied in the modified Hager-Zhang scheme for nonsmooth optimization [PLos ONE 11(10): e0164289 (2016)], we develop a Dai-Yuan type iterative scheme for convex constrained nonlinear monotone system. The scheme's algorithm is obtained by combining its search direction with the projection method [Kluwer Academic Publishers, pp. 355-369(1998)]. One of the new scheme's attribute is that it is derivative-free, which makes it ideal for solving non-smooth problems. Furthermore, we demonstrate the method's application in image de-blurring problems by comparing its performance with a recent effective method. By employing mild assumptions, global convergence of the scheme is determined and results of some numerical experiments show the method to be favorable compared to some recent iterative methods.

Projection method with inertial step for nonlinear equations: Application to signal recovery

Abstract: <p style='text-indent:20px;'>In this paper, using the concept of inertial extrapolation, we introduce a globally convergent inertial extrapolation method for solving nonlinear equations with convex constraints for which the underlying mapping is monotone and Lipschitz continuous. The method can be viewed as a combination of the efficient three-term derivative-free method of Gao and He [Calcolo. 55(4), 1-17, 2018] with the inertial extrapolation step. Moreover, the algorithm is designed such that at every iteration, the method is free from derivative evaluations. Under standard assumptions, we establish the global convergence results for the proposed method. Numerical implementations illustrate the performance and advantage of this new method. Moreover, we also extend this method to solve the LASSO problems to decode a sparse signal in compressive sensing. Performance comparisons illustrate the effectiveness and competitiveness of our algorithm.</p>

Training Generative Adversarial Networks via Stochastic Nash Games

Abstract: Generative adversarial networks (GANs) are a class of generative models with two antagonistic neural networks: a generator and a discriminator. These two neural networks compete against each other through an adversarial process that can be modeled as a stochastic Nash equilibrium problem. Since the associated training process is challenging, it is fundamental to design reliable algorithms to compute an equilibrium. In this article, we propose a stochastic relaxed forward-backward (SRFB) algorithm for GANs, and we show convergence to an exact solution when an increasing number of data is available. We also show convergence of an averaged variant of the SRFB algorithm to a neighborhood of the solution when only a few samples are available. In both cases, convergence is guaranteed when the pseudogradient mapping of the game is monotone. This assumption is among the weakest known in the literature. Moreover, we apply our algorithm to the image generation problem.

An Overall Evaluation on Benefits of Competitive Influence Diffusion

Abstract: Influence maximization (IM) is a representative and classic problem that has been studied extensively before. The most important application derived from the IM problem is viral marketing. Take us as a promoter, we want to get benefits from the influence diffusion in a given social network, where each influenced (activated) user is associated with a benefit. However, there is often competing information initiated by our rivals that diffuses in the same social network at the same time. Consider such a scenario, a user is influenced by both our information and our rivals’ information. Here, the benefit from this user should be weakened to a certain degree. How to quantify the degree of weakening? Based on that, we propose an overall evaluation on benefits of influence (OEBI) problem. We prove the objective function of the OEBI problem is not monotone, not submodular, and not supermodular. Fortunately, we can decompose this objective function into the difference of two submodular functions and adopt a modular-modular procedure to approximate it with a data-dependent approximation guarantee. Because of the difficulty to compute the exact objective value, we design a group of unbiased estimators by exploiting the idea of reverse influence sampling, which can improve time efficiency significantly without losing its approximation ratio. Finally, numerical experiments on real datasets verified the effectiveness of our approaches regardless of performance and efficiency.

Growing Echo State Network With an Inverse-Free Weight Update Strategy

Abstract: An echo state network (ESN) draws widespread attention and is applied in many scenarios. As the most typical approach for solving the ESN, the matrix inverse operation of high computational complexity is involved. However, in the modern big data era, addressing the heavy computational burden problem is necessary. In order to reduce the computational load, an inverse-free ESN (IFESN) is proposed for the first time in this article. Besides, an incremental IFESN is constructed to attain the network topology with theoretical proof on the training error's monotone decline property. Simulations and experiments are conducted on several numerical and real-world time-series benchmarks, and corresponding results indicate that the proposed model is superior to some existing models and possesses excellent practical application potential. The source code is publicly available at https://github.com/LongJin-lab/the-supplementary-file-for-CYB-E-2021-04-0944.

Monotone decreasing LKF method for secure consensus of second-order mass with delay and switching topology

Abstract: This letter investigates secure global asymptotic consensus (SGAC) with non-weighted L2 gain of second-order multi-agent systems (MASs) under deception attacks by designing time-delayed state feedback control with switching topology. Compared with the existing methods in which the Lyapunov–Krasovskii functional (LKF) has to jump high at switching instants, the most important breakthrough is that a discretized LKF is designed such that it is monotone decreasing at switching instants, which not only can obtain low conservative results but also is convenient to study consensus with non-weighted L2 gain without any additional inequality transformation. The Results are universal since they can be applied with no difficulty to any other switched time-delay systems. A numerical simulation illustrates the less conservative SGAC.

Algorithms for Approximating Solutions of Split Variational Inclusion and Fixed-Point Problems

Abstract: In this paper, the split fixed point and variational inclusion problem is considered. With the help of fixed point technique, Tseng-type splitting method and self-adaptive rule, an iterative algorithm is proposed for solving this split problem in which the involved operators S and T are demicontractive operators and g is plain monotone. Strong convergence theorem is proved under some mild conditions.

Semi-Decentralized Generalized Nash Equilibrium Seeking in Monotone Aggregative Games

Abstract: We address the generalized Nash equilibrium seeking problem for a population of agents playing aggregative games with affine coupling constraints. We focus on semi-decentralized communication architectures, where there is a central coordinator able to gather and broadcast signals of aggregative nature to the agents. By exploiting the framework of monotone operator theory and operator splitting, we first critically review the most relevant available algorithms and then design two novel schemes: (i) a single-layer, fixed-step algorithm with convergence guarantee for general (non cocoercive, non-strictly) monotone aggregative games and (ii) a single-layer proximal-type algorithm for a class of monotone aggregative games with linearly coupled cost functions. We also design novel accelerated variants of the algorithms via (alternating) inertial and over-relaxation steps. Finally, we show via numerical simulations that the proposed algorithms outperform those in the literature in terms of convergence speed.

Modified accelerated Bregman projection methods for solving quasi-monotone variational inequalities

Abstract: In this paper, we introduce three new inertial-like Bregman projection methods with a nonmonotone adaptive step-size for solving quasi-monotone variational inequalities in real Hilbert spaces. Under some suitable conditions, the weak convergence of these methods is proved without the prior knowledge of the Lipschitz constant of the operator and the strong convergence of some proposed methods under a strong quasi-monotonicity assumption of the mapping is also provided. Finally, several numerical experiments and applications in image restoration problems are provided to illustrate the performance of the proposed methods.

Group fairness in non-monotone submodular maximization

Abstract: Maximizing a submodular function has a wide range of applications in machine learning and data mining. One such application is data summarization whose goal is to select a small set of representative and diverse data items from a large dataset. However, data items might have sensitive attributes such as race or gender, in this setting, it is important to design fairness-aware algorithms to mitigate potential algorithmic bias that may cause over- or under- representation of particular groups. Motivated by that, we propose and study the classic non-monotone submodular maximization problem subject to novel group fairness constraints. Our goal is to select a set of items that maximizes a non-monotone submodular function, while ensuring that the number of selected items from each group is proportionate to its size, to the extent specified by the decision maker. We develop the first constant-factor approximation algorithms for this problem. We also extend the basic model to incorporate an additional global size constraint on the total number of selected items.

Decay in full von Kármán beam with temperature and microtemperatures effects

Abstract: In this article we derive the equations that constitute the mathematical model of the full von K\'{a}rm\'{a}n beam with temperature and microtemperatures effects. The nonlinear governing equations are derived by using Hamilton principle in the framework of Euler–Bernoulli beam theory. Under quite general assumptions on nonlinear damping function acting on the transversal component and based on nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions to the derived problem. Then using the multiplier method, we show that solutions decay exponentially. Finally we consider the case of zero thermal conductivity and we show that the dissipation given only by the microtemperatures is strong enough to produce exponential stability.

On a coupled system under coupled integral boundary conditions involving non-singular differential operator

Abstract: In this work, a coupled system under coupled integral boundary conditions with Caputo-Fabrizio derivative (CFD) is considered. We intend to derive some necessary and sufficient results for the existence of at least one solution. In addition, we extend our analysis further to develop a monotone iterative scheme coupled with the upper and lower solution method to compute extremal solutions. Therefore, in this regard, Perov's fixed point theorem is applied to study the existing criteria for the solution. Also, results related to at least one solution are derived by using Schauder's fixed point theorem. Finally, we use a monotone iterative procedure together with upper and lower solution methods to study extremal solutions. Graphical presentations of upper and lower solutions are provided for some examples to illustrate our results.

Tight Remainder-Form Decomposition Functions With Applications to Constrained Reachability and Guaranteed State Estimation

Abstract: This paper proposes a tractable family of remainder-form mixed-monotone decomposition functions that are useful for over-approximating the image set of nonlinear mappings in reachability and estimation problems. Our approach applies to a new class of nonsmooth, discontinuous nonlinear systems that we call either-sided locally Lipschitz semicontinuous (ELLS) systems, which we show to be a strict superset of locally Lipschitz continuous (LLC) systems, thus expanding the set of systems that are formally known to be mixed-monotone. In addition, we derive lower and upper bounds for the over-approximation error and show that the lower bound is achieved with our proposed approach, i.e., our approach constructs the tightest, tractable remainder-form mixed-monotone decomposition function. Moreover, we introduce a set inversion algorithm that along with the proposed decomposition functions, can be used for constrained reachability analysis and guaranteed state estimation for continuous- and discrete-time systems with bounded noise.

Sums of squares I: Scalar functions

Abstract: This paper is devoted to proving the strong averaging principle for slow–fast stochastic partial differential equations with locally monotone coefficients, where the slow component is a stochastic partial differential equations with locally monotone coefficients and the fast component is a stochastic partial differential equations with strongly monotone coefficients. The result is applicable to a large class of examples, such as the stochastic porous medium equation, the stochastic p-Laplace equation, the stochastic Burgers type equation and the stochastic 2D Navier–Stokes equation, which are the nonlinear stochastic partial differential equations. The main techniques are based on time discretization and the variational approach to stochastic partial differential equations.

Strong Averaging Principle for Slow–Fast Stochastic Partial Differential Equations with Locally Monotone Coefficients

Abstract: This paper is devoted to proving the strong averaging principle for slow–fast stochastic partial differential equations with locally monotone coefficients, where the slow component is a stochastic partial differential equations with locally monotone coefficients and the fast component is a stochastic partial differential equations with strongly monotone coefficients. The result is applicable to a large class of examples, such as the stochastic porous medium equation, the stochastic p-Laplace equation, the stochastic Burgers type equation and the stochastic 2D Navier–Stokes equation, which are the nonlinear stochastic partial differential equations. The main techniques are based on time discretization and the variational approach to stochastic partial differential equations.

Monotone one-port circuits

Abstract: Maximal monotonicity is explored as a generalization of the linear theory of passivity, aiming at an algorithmic input/output analysis of physical models. The theory is developed for maximal monotone one-port circuits, formed by the series and parallel interconnection of basic elements. These circuits generalize passive LTI transfer functions. Periodic input signals are shown to be mapped to periodic output signals, and these input-output behaviors can be efficiently computed using a maximal monotone splitting algorithm, which decomposes the computation according to the circuit topology. A new splitting algorithm is presented, which applies to any monotone one-port circuit defined as a port interconnection of monotone elements.

Regularity properties of monotone measure-preserving maps

Abstract: Abstract In this note, we extend the regularity theory for monotone measure-preserving maps, also known as optimal transports for the quadratic cost optimal transport problem, to the case when the support of the target measure is an arbitrary convex domain and, on the low-regularity end, between domains carrying certain invariant measures.

Existence for Nonlinear Fourth-Order Two-Point Boundary Value Problems

Abstract: The present paper is devoted to the solvability of various two-point boundary value problems for the equation y(4)=f(t,y,y′,y″,y‴), where the nonlinearity f may be defined on a bounded set and is needed to be continuous on a suitable subset of its domain. The established existence results guarantee not just a solution to the considered boundary value problems but also guarantee the existence of monotone solutions with suitable signs and curvature. The obtained results rely on a basic existence theorem, which is a variant of a theorem due to A. Granas, R. Guenther and J. Lee. The a priori bounds necessary for the application of the basic theorem are provided by the barrier strip technique. The existence results are illustrated with examples.

Iterative roots of two-dimensional mappings

Abstract: Abstract As a weak version of embedding flow, the problem of iterative roots is studied extensively in one dimension, especially in monotone case. There are few results in high dimensions because the constructive method dealing with monotone mappings is unavailable. In this paper, by introducing a kind of partial order, we define the monotonicity for two-dimensional mappings and then present some results on the existence of iterative roots for linear mappings, triangle-type mappings, and co-triangle-type mappings, respectively. Our theorems show that even the property of monotonicity for iterative roots of monotone mappings, which is a trivial result in one dimension, does not hold anymore in high dimensions. At the end of this paper, the problem of iterative roots for two well-known planar mappings, that is, Hénon mappings and coupled logistic mappings, are also discussed.

Fixed-Time Nash Equilibrium Seeking in Time-Varying Networks

Abstract: In this article, we introduce first-order and zeroth-order Nash equilibrium seeking dynamics with fixed-time and practical fixed-time convergence certificates for noncooperative games having finitely many players. The first-order algorithms achieve exact convergence to the Nash equilibrium of the game in a finite time that can be additionally upper bounded by a constant that is independent of the initial conditions of the actions of the players. Moreover, these fixed-time bounds can be prescribed a priori by the system designer under an appropriate tuning of the parameters of the algorithms. When players have access only to measurements of their cost functions, we consider a class of distributed multitime scale zeroth-order model-free adaptive dynamics that achieve semiglobal practical fixed-time stability, qualitatively preserving the fixed-time bounds of the first-order dynamics as the time scale separation increases. Moreover, by leveraging the property of fixed-time input-to-state stability, further results are obtained for mixed games where some of the players implement different seeking dynamics. Fast and slow switching communication graphs are also incorporated using tools from hybrid systems. We consider potential games as well as general nonpotential strongly monotone games. Numerical examples illustrate our results.

A Linearly Convergent Distributed Nash Equilibrium Seeking Algorithm for Aggregative Games

Abstract: This article considers distributed Nash equilibrium (NE) seeking of strongly monotone aggregative games over a multiagent network. Each player can only observe its own strategy while can exchange information with its neighbors via a communication graph. To solve the problem, we propose a distributed algorithm with multiple rounds of communication, where the players need constant rounds of communication with their neighbors at each iteration. We then prove that our algorithm converges to the (unique) NE with a linear convergence rate. We further study a single-round communication version of our algorithm, which can also achieve linear convergence rate with an additional condition related to the structure of the graph and the properties of the aggregative game. Finally, we provide numerical simulations to verify our results.

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Abstract: We study the Dirichlet problem for the pseudo-parabolic equation ut−diva(x,t)|∇u|p(x,t)−2∇u−Δut=b(x,t)|u|q(x,t)−2uin the cylinder QT=Ω×(0,T), where Ω⊂Rd is a sufficiently smooth domain. The positive coefficients a, b and the exponents p≥2, q>2 are given Lipschitz-continuous functions. The functions a, p are monotone decreasing, and b, q are monotone increasing in t. It is shown that there exists a positive constant M=M(|Ω|,sup(x,t)∈QTp(x,t),sup(x,t)∈QTq(x,t)), such if the initial energy is negative, E(0)=∫Ωa(x,0)p(x,0)|∇u0(x)|p(x,0)−b(x,0)q(x,0)|u0(x)|q(x,0)dx<−M,then the problem admits a local in time solution with negative energy E(t). If p and q are independent of t, then M=0. For the solutions from this class, sufficient conditions for the finite time blow-up are derived.