A novel adaptive method for operator inclusions

2021 ◽  
pp. 33-35
Author(s):  
Vladimir V. Semenov ◽  
Serhii Denysov ◽  
Yana Vedel
Keyword(s):  
Banach Space ◽  
Lipschitz Constant ◽  
Maximal Monotone ◽  
Continuous Operator ◽  
Adaptive Method ◽  
Step Size ◽  
Splitting Algorithm ◽  
Operator Inclusion ◽  
Lipschitz Continuous ◽  
Lipschitz Continuous Operator

A novel splitting algorithm for solving operator inclusion with the sum of the maximal monotone operator and the monotone Lipschitz continuous operator in the Banach space is proposed and studied. The proposed algorithm is an adaptive variant of the forward-reflected-backward algorithm, where the rule used to update the step size does not require knowledge of the Lipschitz constant of the operator. For operator inclusions in 2-uniformly convex and uniformly smooth Banach space, the theorem on the weak convergence of the method is proved.

scholarly journals Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems

2020 ◽  
Vol 53 (1) ◽  
pp. 208-224 ◽  
Author(s):  
Timilehin Opeyemi Alakoya ◽  
Lateef Olakunle Jolaoso ◽  
Oluwatosin Temitope Mewomo
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Extragradient Method ◽  
Lipschitz Constant ◽  
Monotone Mapping ◽  
Variational Inequality Problems ◽  
Step Size ◽  
Lipschitz Continuous ◽  
Convergence Results ◽  
Monotone Variational Inequality

AbstractIn this work, we introduce two new inertial-type algorithms for solving variational inequality problems (VIPs) with monotone and Lipschitz continuous mappings in real Hilbert spaces. The first algorithm requires the computation of only one projection onto the feasible set per iteration while the second algorithm needs the computation of only one projection onto a half-space, and prior knowledge of the Lipschitz constant of the monotone mapping is not required in proving the strong convergence theorems for the two algorithms. Under some mild assumptions, we prove strong convergence results for the proposed algorithms to a solution of a VIP. Finally, we provide some numerical experiments to illustrate the efficiency and advantages of the proposed algorithms.

CONVERGENCE OF THE EXTRAPOLATION METHOD FROM THE PAST AND THE OPERATOR EXTRAPOLATION METHOD

2021 ◽  
Vol 3 ◽  
pp. 58-72
Author(s):  
Vladimir Semenov ◽  
◽  
Sergei Denisov ◽  
Dmitry Siryk ◽  
Oleg Kharkov ◽  
...  
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Low Cost ◽  
Lipschitz Constant ◽  
Extrapolation Method ◽  
Bregman Projection ◽  
Step Size ◽  
Lipschitz Continuous ◽  
Extrapolation Algorithm ◽  
Weakly Continuous

One of the popular areas of modern applied nonlinear analysis is the study of variational inequalities. Many important problems of operations research and mathematical physics can be written in the form of variational inequalities. With the advent of generating adversarial neural networks, interest in algorithms for solving variational inequalities arose in the ML-community. This paper is devoted to the study of three new algorithms with Bregman projection for solving variational inequalities in Hilbert space. The first algorithm is the result of a modification of the two-stage Bregman method by low-cost adjusting the step size that without the prior knowledge of the Lipschitz constant of operator. The second algorithm, which we call the operator extrapolation algorithm, is obtained by replacing the Euclidean metric in the Malitsky–Tam method with the Bregman divergence. An attractive feature of the algorithm is only one computation at the iterative step of the Bregman projection onto the feasible set. The third algorithm is an adaptive version of the second, where the used rule for updating the step size does not require knowledge of Lipschitz constants and the calculation of operator values at additional points. For variational inequalities with pseudo-monotone, Lipschitz-continuous, and sequentially weakly continuous operators acting in a Hilbert space, convergence theorems are proved.

scholarly journals An accelerated differential equation system for generalized equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Qiyuan Wei ◽  
Liwei Zhang
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Maximal Monotone ◽  
Equation System ◽  
Generalized Equation ◽  
Generalized Equations ◽  
Differential Equation System ◽  
Step Size ◽  
Yosida Regularization ◽  
Numerical Discretization

<p style='text-indent:20px;'>An accelerated differential equation system with Yosida regularization and its numerical discretized scheme, for solving solutions to a generalized equation, are investigated. Given a maximal monotone operator <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> on a Hilbert space, this paper will study the asymptotic behavior of the solution trajectories of the differential equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \dot{x}(t)+T_{\lambda(t)}(x(t)-\alpha(t)T_{\lambda(t)}(x(t))) = 0,\quad t\geq t_0\geq 0, \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>to the solution set <inline-formula><tex-math id="M2">\begin{document}$ T^{-1}(0) $\end{document}</tex-math></inline-formula> of a generalized equation <inline-formula><tex-math id="M3">\begin{document}$ 0 \in T(x) $\end{document}</tex-math></inline-formula>. With smart choices of parameters <inline-formula><tex-math id="M4">\begin{document}$ \lambda(t) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \alpha(t) $\end{document}</tex-math></inline-formula>, we prove the weak convergence of the trajectory to some point of <inline-formula><tex-math id="M6">\begin{document}$ T^{-1}(0) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M7">\begin{document}$ \|\dot{x}(t)\|\leq {\rm O}(1/t) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M8">\begin{document}$ t\rightarrow +\infty $\end{document}</tex-math></inline-formula>. Interestingly, under the upper Lipshitzian condition, strong convergence and faster convergence can be obtained. For numerical discretization of the system, the uniform convergence of the Euler approximate trajectory <inline-formula><tex-math id="M9">\begin{document}$ x^{h}(t) \rightarrow x(t) $\end{document}</tex-math></inline-formula> on interval <inline-formula><tex-math id="M10">\begin{document}$ [0,+\infty) $\end{document}</tex-math></inline-formula> is demonstrated when the step size <inline-formula><tex-math id="M11">\begin{document}$ h \rightarrow 0 $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Convergence theorems for generalized projections and maximal monotone operators in Banach spaces

2003 ◽  
Vol 2003 (10) ◽  
pp. 621-629 ◽  
Author(s):  
Takanori Ibaraki ◽  
Yasunori Kimura ◽  
Wataru Takahashi
Keyword(s):  
Banach Space ◽  
Pointwise Convergence ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Convex Banach Space ◽  
Limit Set ◽  
Strong And Weak Convergence ◽  
Strictly Convex Banach Space ◽  
Generalized Projections

We study a sequence of generalized projections in a reflexive, smooth, and strictly convex Banach space. Our result shows that Mosco convergence of their ranges implies their pointwise convergence to the generalized projection onto the limit set. Moreover, using this result, we obtain strong and weak convergence of resolvents for a sequence of maximal monotone operators.

Convergence of derivative free iterative methods

2019 ◽  
Vol 28 (1) ◽  
pp. 19-26
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
SANTHOSH GEORGE ◽  
Keyword(s):  
Banach Space ◽  
Iterative Methods ◽  
Local Convergence ◽  
Practical Interest ◽  
Systems Of Equations ◽  
Hölder Continuous ◽  
Numerical Examples ◽  
Lipschitz Continuous ◽  
Derivative Free ◽  
Special Cases

We present the local as well as the semi-local convergence of some iterative methods free of derivatives for Banach space valued operators. These methods contain the secant and the Kurchatov method as special cases. The convergence is based on weak hypotheses specializing to Lipschitz continuous or Holder continuous hypotheses. The results are of theoretical and practical interest. In particular the method is compared favorably ¨ to other methods using concrete numerical examples to solve systems of equations containing a nondifferentiable term.

scholarly journals Relaxed Inertial Tseng’s Type Method for Solving the Inclusion Problem with Application to Image Restoration

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 818 ◽  
Author(s):  
Jamilu Abubakar ◽  
Poom Kumam ◽  
Abdulkarim Hassan Ibrahim ◽  
Anantachai Padcharoen
Keyword(s):  
Lipschitz Constant ◽  
Monotone Mapping ◽  
Splitting Method ◽  
Split Feasibility Problem ◽  
Monotone Operators ◽  
Feasibility Problem ◽  
Inclusion Problem ◽  
Variable Step Size ◽  
Step Size ◽  
Type Method

The relaxed inertial Tseng-type method for solving the inclusion problem involving a maximally monotone mapping and a monotone mapping is proposed in this article. The study modifies the Tseng forward-backward forward splitting method by using both the relaxation parameter, as well as the inertial extrapolation step. The proposed method follows from time explicit discretization of a dynamical system. A weak convergence of the iterates generated by the method involving monotone operators is given. Moreover, the iterative scheme uses a variable step size, which does not depend on the Lipschitz constant of the underlying operator given by a simple updating rule. Furthermore, the proposed algorithm is modified and used to derive a scheme for solving a split feasibility problem. The proposed schemes are used in solving the image deblurring problem to illustrate the applicability of the proposed methods in comparison with the existing state-of-the-art methods.

Piecewise continuous solutions of nonlinear pseudoparabolic equations in two space dimensions*

1992 ◽  
Vol 121 (3-4) ◽  
pp. 203-217 ◽  
Author(s):  
Dao-Qing Dai ◽  
Wei Lin
Keyword(s):  
Lipschitz Constant ◽  
Complex Function ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Lipschitz Continuous ◽  
Two Space Dimensions ◽  
Image Position ◽  
Initial Boundary Value ◽  
Pseudoparabolic Equations ◽  
Nonlinear Pseudoparabolic Equation

SynopsisAn initial boundary value problem of Riemann type is solved for the nonlinear pseudoparabolic equation with two space variablesThe complex functionHis measurable on ℂ ×I × ℂ5, withIbeing an interval of the real line ℝ, Lipschitz continuous with respect to the last five variables, with the Lipschitz constant for the last variable being strictly less than one (ellipticity condition). No smallness assumption is needed in the argument.

scholarly journals Strong and Weak Convergence of Modified Mann Iteration for New Resolvents of Maximal Monotone Operators in Banach Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-20 ◽  
Author(s):  
Somyot Plubtieng ◽  
Wanna Sriprad
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Convergence Rate ◽  
Minimization Problem ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Convex Minimization Problem ◽  
Mann Iteration ◽  
Strong And Weak Convergence

We prove strong and weak convergence theorems for a new resolvent of maximal monotone operators in a Banach space and give an estimate of the convergence rate of the algorithm. Finally, we apply our convergence theorem to the convex minimization problem. The result present in this paper extend and improve the corresponding result of Ibaraki and Takahashi (2007), and Kim and Xu (2005).

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