The adaptivedouglas-rachford splitting algorithm iteratively applies the operator T = \kappa nQAQB+ (1 - \kappa n)Id to solve the inclusion problem zer(A + B). By taking a Yosida approximation standpoint, we express ...
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The adaptivedouglas-rachford splitting algorithm iteratively applies the operator T = \kappa nQAQB+ (1 - \kappa n)Id to solve the inclusion problem zer(A + B). By taking a Yosida approximation standpoint, we express in canonical form QAQB = (Id - (\gamma + \lambda ) \gamma A) \circ (Id - (\gamma + \lambda ) \lambda B). We extend the domain of indices \gamma , \lambda to the entire real line, so that the adaptivealgorithm is able to encompass a forward-backward splitting algorithm into one unified framework. Convergence results for both primal and dual problems are proved for different combinations of (strongly and weakly) monotone and comonotone operators. Under the ``monotone + comonotone"" assumption, we obtain a better rate bound for linear convergence than the classical douglas-rachfordalgorithm.
The douglas-rachford splitting method is a classical and powerful method that is widely used in engineering fields for finding a zero of the sum of two operators. In this paper, we begin by proposing an abstract secon...
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The douglas-rachford splitting method is a classical and powerful method that is widely used in engineering fields for finding a zero of the sum of two operators. In this paper, we begin by proposing an abstract second-order dynamic system involving a generalized cocoercive operator to find a zero of the operator in a real Hilbert space. Then we develop a second-order adaptivedouglas-rachford dynamic system for finding a zero of the sum of two operators, one of which is strongly monotone while the other one is weakly monotone. With proper tuning of the parameters such that the adaptivedouglas-rachford operator is quasi-nonexpansive, we demonstrate that the trajectory of the proposed adaptive system converges weakly to a fixed point of the adaptive operator. When the strong monotonicity strictly outweighs the weak one, we further derive the strong convergence of shadow trajectories to the solution of the original problem. Finally, two simulation examples are reported to corroborate the effectiveness of the proposed adaptive system.
We study a conical extension of averaged nonexpansive operators and the role it plays in convergence analysis of fixed point algorithms. Various properties of conically averaged operators are systematically investigat...
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We study a conical extension of averaged nonexpansive operators and the role it plays in convergence analysis of fixed point algorithms. Various properties of conically averaged operators are systematically investigated, in particular, the stability under relaxations, convex combinations and compositions. We derive conical averagedness properties of resolvents of generalized monotone operators. These properties are then utilized in order to analyze the convergence of the proximal point algorithm, the forward-backward algorithm, and the adaptive douglas-rachford algorithm. Our study unifies, improves and casts new light on recent studies of these topics.
Demiclosedness principles are powerful tools in the study of convergence of iterative methods. For instance, a multi-operator demiclosedness principle for firmly nonexpansive mappings is useful in obtaining simple and...
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Demiclosedness principles are powerful tools in the study of convergence of iterative methods. For instance, a multi-operator demiclosedness principle for firmly nonexpansive mappings is useful in obtaining simple and transparent arguments for the weak convergence of the shadow sequence generated by the douglas-rachfordalgorithm. We provide extensions of this principle, which are compatible with the framework of more general families of mappings such as cocoercive and conically averaged mappings. As an application, we derive the weak convergence of the shadow sequence generated by the adaptive douglas-rachford algorithm.
Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps invol...
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Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem.
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