Showing papers on "Monotone polygon published in 2020"

A Forward-Backward Splitting Method for Monotone Inclusions Without Cocoercivity

Abstract: In this work, we propose a simple modification of the forward-backward splitting method for finding a zero in the sum of two monotone operators. Our method converges under the same assumptions as T...

Distributed GNE Seeking Under Partial-Decision Information Over Networks via a Doubly-Augmented Operator Splitting Approach

Abstract: We consider distributed computation of generalized Nash equilibrium (GNE) over networks, in games with shared coupling constraints. Existing methods require that each player has full access to opponents’ decisions. In this paper, we assume that players have only partial-decision information, and can communicate with their neighbors over an arbitrary undirected graph. We recast the problem as that of finding a zero of a sum of monotone operators through primal-dual analysis. To distribute the problem, we doubly augment variables, so that each player has local decision estimates and local copies of Lagrangian multipliers. We introduce a single-layer algorithm, fully distributed with respect to both primal and dual variables. We show its convergence to a variational GNE with fixed step sizes, by reformulating it as a forward–backward iteration for a pair of doubly-augmented monotone operators.

Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability

Abstract: This survey article addresses the class of continuous-time systems where a system modeled by ordinary differential equations is coupled with a static or time-varying set-valued operator in the feed...

A Strong Convergence Theorem for Solving Pseudo-monotone Variational Inequalities Using Projection Methods

Abstract: Several iterative methods have been proposed in the literature for solving the variational inequalities in Hilbert or Banach spaces, where the underlying operator A is monotone and Lipschitz continuous. However, there are very few methods known for solving the variational inequalities, when the Lipschitz continuity of A is dispensed with. In this article, we introduce a projection-type algorithm for finding a common solution of the variational inequalities and fixed point problem in a reflexive Banach space, where A is pseudo-monotone and not necessarily Lipschitz continuous. Also, we present an application of our result to approximating solution of pseudo-monotone equilibrium problem in a reflexive Banach space. Finally, we present some numerical examples to illustrate the performance of our method as well as comparing it with related method in the literature.

A comment on a controversial issue: A generalized fractional derivative cannot have a regular kernel

Abstract: The problem whether a given pair of functions can be used as the kernels of a generalized fractional derivative and the associated generalized fractional integral is reduced to the problem of existence of a solution to the Sonine equation. It is shown for some selected classes of functions that a necessary condition for a function to be the kernel of a fractional derivative is an integrable singularity at 0. It is shown that locally integrable completely monotone functions satisfy the Sonine equation if and only if they are singular at 0.

Monotone operator equilibrium networks.

Abstract: Implicit-depth models such as Deep Equilibrium Networks have recently been shown to match or exceed the performance of traditional deep networks while being much more memory efficient. However, these models suffer from unstable convergence to a solution and lack guarantees that a solution exists. On the other hand, Neural ODEs, another class of implicit-depth models, do guarantee existence of a unique solution but perform poorly compared with traditional networks. In this paper, we develop a new class of implicit-depth model based on the theory of monotone operators, the Monotone Operator Equilibrium Network (MON). We show the close connection between finding the equilibrium point of an implicit network and solving a form of monotone operator splitting problem, which admits efficient solvers with guaranteed, stable convergence. We then develop a parameterization of the network which ensures that all operators remain monotone, which guarantees the existence of a unique equilibrium point. Finally, we show how to instantiate several versions of these models, and implement the resulting iterative solvers, for structured linear operators such as multi-scale convolutions. The resulting models vastly outperform the Neural ODE-based models while also being more computationally efficient. Code is available at this http URL.

A Novel Inertial Projection and Contraction Method for Solving Pseudomonotone Variational Inequality Problems

Abstract: In this paper, we introduce a new algorithm which combines the inertial contraction projection method and the Mann-type method (Mann in Proc. Am. Math. Soc. 4:506–510, 1953) for solving monotone variational inequality problems in real Hilbert spaces. The strong convergence of our proposed algorithm is proved under some standard assumptions imposed on cost operators. Finally, we give some numerical experiments to illustrate the proposed algorithm and the main result.

On generalized fractional integral inequalities for the monotone weighted Chebyshev functionals

Abstract: In this paper, we establish the generalized Riemann–Liouville (RL) fractional integrals in the sense of another increasing, positive, monotone, and measurable function Ψ. We determine certain new double-weighted type fractional integral inequalities by utilizing the said integrals. We also give some of the new particular inequalities of the main result. Note that we can form various types of new inequalities of fractional integrals by employing conditions on the function Ψ given in the paper. We present some corollaries as particular cases of the main results.

On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients

Abstract: We develop a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the $L^{p}$-distance between the solution process of an SDE and an arbitrary Ito process, which we view as a perturbation of the solution process of the SDE, by the $L^{q}$-distances of the differences of the local characteristics for suitable $p,q>0$. As one application of the developed perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with nonglobally monotone coefficients. As another application of the developed perturbation theory, we prove strong convergence rates for spatial spectral Galerkin approximations of solutions of semilinear SPDEs with nonglobally monotone nonlinearities including Cahn–Hilliard–Cook-type equations and stochastic Burgers equations. Further applications of the developed perturbation theory include regularity analyses of solutions of SDEs with respect to their initial values as well as small-noise analyses for ordinary and partial differential equations.

Strong approximation of monotone stochastic partial differential equations driven by white noise

Abstract: We establish an optimal strong convergence rate of a fully discrete numerical scheme for second order parabolic stochastic partial differential equations with monotone drifts, including the stochastic Allen-Cahn equation, driven by an additive space-time white noise Our first step is to transform the original stochastic equation into an equivalent random equation whose solution possesses more regularity than the original one Then we use the backward Euler in time and spectral Galerkin in space to fully discretize this random equation By the monotone assumption, in combination with the factorization method and stochastic calculus in martingale-type 2 Banach spaces, we derive a uniform maximum norm estimation and a Holder-type regularity for both stochastic and random equations Finally, the strong convergence rate of the proposed fully discrete scheme under the $l_t^\infty L^2_\omega L^2_x \cap l_t^q L^q_\omega L^q_x$-norm is obtained Several numerical experiments are carried out to verify the theoretical result

An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions

Abstract: In this paper, we propose a new inertial extrapolation method for solving the generalized split feasibility problems over the solution set of monotone variational inclusion problems in real Hilbert...

Halpern Iteration for Near-Optimal and Parameter-Free Monotone Inclusion and Strong Solutions to Variational Inequalities

Abstract: We leverage the connections between nonexpansive maps, monotone Lipschitz operators, and proximal mappings to obtain near-optimal (i.e., optimal up to poly-log factors in terms of iteration complexity) and parameter-free methods for solving monotone inclusion problems. These results immediately translate into near-optimal guarantees for approximating strong solutions to variational inequality problems, approximating convex-concave min-max optimization problems, and minimizing the norm of the gradient in min-max optimization problems. Our analysis is based on a novel and simple potential-based proof of convergence of Halpern iteration, a classical iteration for finding fixed points of nonexpansive maps. Additionally, we provide a series of algorithmic reductions that highlight connections between different problem classes and lead to lower bounds that certify near-optimality of the studied methods.

The one-way communication complexity of submodular maximization with applications to streaming and robustness

Abstract: We consider the classical problem of maximizing a monotone submodular function subject to a cardinality constraint, which, due to its numerous applications, has recently been studied in various computational models. We consider a clean multi-player model that lies between the offline and streaming model, and study it under the aspect of one-way communication complexity. Our model captures the streaming setting (by considering a large number of players), and, in addition, two player approximation results for it translate into the robust setting. We present tight one-way communication complexity results for our model, which, due to the above-mentioned connections, have multiple implications in the data stream and robust setting. Even for just two players, a prior information-theoretic hardness result implies that no approximation factor above 1/2 can be achieved in our model, if only queries to feasible sets, i.e., sets respecting the cardinality constraint, are allowed. We show that the possibility of querying infeasible sets can actually be exploited to beat this bound, by presenting a tight 2/3-approximation taking exponential time, and an efficient 0.514-approximation. To the best of our knowledge, this is the first example where querying a submodular function on infeasible sets leads to provably better results. Through the above-mentioned link to the robust setting, both of these algorithms improve on the current state-of-the-art for robust submodular maximization, showing that approximation factors beyond 1/2 are possible. Moreover, exploiting the link of our model to streaming, we settle the approximability for streaming algorithms by presenting a tight 1/2+e hardness result, based on the construction of a new family of coverage functions. This improves on a prior 1−1/e+e hardness and matches, up to an arbitrarily small margin, the best known approximation algorithm.

An efficient projection-type method for monotone variational inequalities in Hilbert spaces

Abstract: We consider the monotone variational inequality problem in a Hilbert space and describe a projection-type method with inertial terms under the following properties: (a) The method generates a strongly convergent iteration sequence; (b) The method requires, at each iteration, only one projection onto the feasible set and two evaluations of the operator; (c) The method is designed for variational inequality for which the underline operator is monotone and uniformly continuous; (d) The method includes an inertial term. The latter is also shown to speed up the convergence in our numerical results. A comparison with some related methods is given and indicates that the new method is promising.

Gradientless Descent: High-Dimensional Zeroth-Order Optimization

Abstract: Zeroth-order optimization is the process of minimizing an objective $f(x)$, given oracle access to evaluations at adaptively chosen inputs $x$. In this paper, we present two simple yet powerful GradientLess Descent (GLD) algorithms that do not rely on an underlying gradient estimate and are numerically stable. We analyze our algorithm from a novel geometric perspective and we show that for {\it any monotone transform} of a smooth and strongly convex objective with latent dimension $k \ge n$, we present a novel analysis that shows convergence within an $\epsilon$-ball of the optimum in $O(kQ\log(n)\log(R/\epsilon))$ evaluations, where the input dimension is $n$, $R$ is the diameter of the input space and $Q$ is the condition number. Our rates are the first of its kind to be both 1) poly-logarithmically dependent on dimensionality and 2) invariant under monotone transformations. We further leverage our geometric perspective to show that our analysis is optimal. Both monotone invariance and its ability to utilize a low latent dimensionality are key to the empirical success of our algorithms, as demonstrated on synthetic and MuJoCo benchmarks.

A strong convergence theorem for Tseng’s extragradient method for solving variational inequality problems

Abstract: In this paper, we introduce a new algorithm for solving variational inequality problems with monotone and Lipschitz-continuous mappings in real Hilbert spaces. Our algorithm requires only to compute one projection onto the feasible set per iteration. We prove under certain mild assumptions, a strong convergence theorem for the proposed algorithm to a solution of a variational inequality problem. Finally, we give some numerical experiments illustrating the performance of the proposed algorithm for variational inequality problems.

Fractional generalized Hadamard and Fejér-Hadamard inequalities for m -convex functions

Abstract: The objective of this paper is to present the fractional Hadamard and Fejer-Hadamard inequalities in generalized forms. By employing a generalized fractional integral operator containing extended generalized Mittag-Leffler function involving a monotone increasing function, we generalize the well known fractional Hadamard and Fejer-Hadamard inequalities for m-convex functions. Also we study the error bounds of these generalized inequalities. In connection with some published results from presented inequalities are obtained.

A note on the spectral gradient projection method for nonlinear monotone equations with applications

Abstract: In this work, we provide a note on the spectral gradient projection method for solving nonlinear equations. Motivated by recent extensions of the spectral gradient method for solving nonlinear monotone equations with convex constraints, in this paper, we note that choosing the search direction as a convex combination of two different positive spectral coefficients multiplied with the residual vector is more efficient and robust compared with the standard choice of spectral gradient coefficients combined with the projection strategy of Solodov and Svaiter (A globally convergent inexact newton method for systems of monotone equations. In: Reformulation: Nonsmooth. Piecewise Smooth, Semismooth and Smoothing Methods, pp 355–369. Springer, 1998). Under suitable assumptions, the convergence of the proposed method is established. Preliminary numerical experiments show that the method is promising. In this paper, the proposed method was used to recover sparse signal and restore blurred image arising from compressive sensing.

A Modified Hestenes-Stiefel-Type Derivative-Free Method for Large-Scale Nonlinear Monotone Equations

Abstract: The goal of this paper is to extend the modified Hestenes-Stiefel method to solve large-scale nonlinear monotone equations. The method is presented by combining the hyperplane projection method (Solodov, M.V.; Svaiter, B.F. A globally convergent inexact Newton method for systems of monotone equations, in: M. Fukushima, L. Qi (Eds.)Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers. 1998, 355-369) and the modified Hestenes-Stiefel method in Dai and Wen (Dai, Z.; Wen, F. Global convergence of a modified Hestenes-Stiefel nonlinear conjugate gradient method with Armijo line search. Numer Algor. 2012, 59, 79-93). In addition, we propose a new line search for the derivative-free method. Global convergence of the proposed method is established if the system of nonlinear equations are Lipschitz continuous and monotone. Preliminary numerical results are given to test the effectiveness of the proposed method.

Nonlinear inviscid damping for a class of monotone shear flows in finite channel

Abstract: We prove the nonlinear inviscid damping for a class of monotone shear flows in $T\times [0,1]$ for initial perturbation in Gevrey-$1/s$($s>2$) class with compact support. The main idea of the proof is to use the wave operator of a slightly modified Rayleigh operator in a well chosen coordinate system.

The Inertial Sub-Gradient Extra-Gradient Method for a Class of Pseudo-Monotone Equilibrium Problems

Abstract: In this article, we focus on improving the sub-gradient extra-gradient method to find a solution to the problems of pseudo-monotone equilibrium in a real Hilbert space. The weak convergence of our method is well-established based on the standard assumptions on a bifunction. We also present the application of our results that enable to solve numerically the pseudo-monotone and monotone variational inequality problems, in addition to the particular presumptions required by the operator. We have used various numerical examples to support our well-proved convergence results, and we can show that the proposed method involves a considerable influence over-running time and the total number of iterations.

Modified extragradient algorithms for solving monotone variational inequalities and fixed point problems

Abstract: In this paper, we investigate the monotone variational inequalities and fixed point problems in Hilbert spaces. Two modified extragradient algorithms are presented for finding a common element of t...

Weak and strong convergence theorems for solving pseudo-monotone variational inequalities with non-Lipschitz mappings

Abstract: The aim of this paper is to study a classical pseudo-monotone and non-Lipschitz continuous variational inequality problem in real Hilbert spaces. Weak and strong convergence theorems are presented under mild conditions. Our methods generalize and extend some related results in the literature and the main advantages of proposed algorithms there is no use of Lipschitz condition of the variational inequality associated mapping. Numerical illustrations in finite and infinite dimensional spaces illustrate the behaviors of the proposed schemes.

Integral inequalities for a fractional operator of a function with respect to another function with nonsingular kernel

Abstract: At first, we construct a connection between the Atangana–Baleanu and the Riemann–Liouville fractional integrals of a function with respect to a monotone function with nonsingular kernel. By examining this relationship and the iterated form of Prabhakar fractional model, we are able to find some new Hermite–Hadamard inequalities and related results on integral inequalities for the two models of fractional calculus which are defined using monotone functions with nonsingular kernels.

A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space

Abstract: In this paper, we introduce a viscosity-type proximal point algorithm which comprises of a finite sum of resolvents of monotone operators, and a generalized asymptotically nonexpansive mapping. We prove that the algorithm converges strongly to a common zero of a finite family of monotone operators, which is also a fixed point of a generalized asymptotically nonexpansive mapping in an Hadamard space. Furthermore, we give two numerical examples of our algorithm in finite dimensional spaces of real numbers and one numerical example in a non-Hilbert space setting, in order to show the applicability of our results.

New strong convergence theorem of the inertial projection and contraction method for variational inequality problems

Abstract: In this paper, we introduce a new algorithm which combines the inertial projection and contraction method and the viscosity method for solving monotone variational inequality problems in real Hilbert spaces and prove a strong convergence theorem of our proposed algorithm under the standard assumptions imposed on cost operators. Finally, we give some numerical experiments to illustrate the proposed algorithm.

Mixed Monotonicity for Reachability and Safety in Dynamical Systems

Abstract: A dynamical system is mixed monotone if its vector field or update-map is decomposable into an increasing component and a decreasing component. In this tutorial paper, we study both continuous-time and discrete-time mixed monotonicity and consider systems subject to an input that accommodates, e.g., unknown parameters, an unknown disturbance input, or an exogenous control input. We first define mixed monotonicity with respect to a decomposition function, and we recall sufficient conditions for mixed monotonicity based on sign properties of the state and input Jacobian matrices for the system dynamics. The decomposition function allows for constructing an embedding system that lifts the dynamics to another dynamical system with twice as many states but where the dynamics are monotone with respect to a particular southeast order. This enables applying the powerful theory of monotone systems to the embedding system in order to conclude properties of the original system. In particular, a single trajectory of the embedding system provides hyperrectangular over-approximations of reachable sets for the original dynamics. In this way, mixed monotonicity enables efficient reachable set approximation for applications such as optimization-based control and abstraction-based formal methods in control systems.