We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit funct...
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We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.
The global boundness and existence are presented for the kind of the Rosseland equation with a general growth condition. A linearized map in a closed convex set is defined. The image set is precompact, and thus a fixe...
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The global boundness and existence are presented for the kind of the Rosseland equation with a general growth condition. A linearized map in a closed convex set is defined. The image set is precompact, and thus a fixed point exists. A multi-scale expansion method is used to obtain the homogenized equation. This equation satisfies a similar growth condition.
The augmented Lagrangian method is a classical method for solving constrained ***,the augmented Lagrangian method attracts much attention due to its applications to sparse optimization in compressive sensing and low r...
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The augmented Lagrangian method is a classical method for solving constrained ***,the augmented Lagrangian method attracts much attention due to its applications to sparse optimization in compressive sensing and low rank matrix optimization ***,most Lagrangian methods use first order information to update the Lagrange multipliers,which lead to only linear *** this paper,we study an update technique based on second order information and prove that superlinear convergence can be *** properties of the update formula are given and some implementation issues regarding the new update are also discussed.
Examines a nonoverlapping domain decomposition method based on the natural boundary reduction. Development of the D-N alternating algorithm; Studies the convergence of the D-N method for exterior spherical domain; Dis...
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Examines a nonoverlapping domain decomposition method based on the natural boundary reduction. Development of the D-N alternating algorithm; Studies the convergence of the D-N method for exterior spherical domain; Discussion of the discrete form of the D-N alternating algorithm.
Background: computational cardiovascular flow modeling plays a crucial role in understanding blood flow dynamics. While 3D models provide acute details, they are computationally expensive, especially with fluid–struc...
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Background: computational cardiovascular flow modeling plays a crucial role in understanding blood flow dynamics. While 3D models provide acute details, they are computationally expensive, especially with fluid–structure interaction (FSI) simulations. 1D models offer a computationally efficient alternative, by simplifying the 3D Navier–Stokes equations through axisymmetric flow assumption and cross-sectional averaging. However, traditional 1D models based on finite element methods (FEM) often lack accuracy compared to 3D averaged solutions. Methods: This study introduces a novel physics-constrained machine learning technique that enhances the accuracy of 1D cardiovascular flow models while maintaining computational efficiency. Our approach, utilizing a physics-constrained coupled neural differential equation (PCNDE) framework, demonstrates superior performance compared to conventional FEM-based 1D models across a wide range of inlet boundary condition waveforms and stenosis blockage ratios. A key innovation lies in the spatial formulation of the momentum conservation equation, departing from the traditional temporal approach and capitalizing on the inherent temporal periodicity of blood flow. Results: This spatial neural differential equation formulation switches space and time and overcomes issues related to coupling stability and smoothness, while simplifying boundary condition implementation. The model accurately captures flow rate, area, and pressure variations for unseen waveforms and geometries, having 3–5 times smaller error than 1D FEM, and less than 1.2% relative error compared to 3D averaged training data. We evaluate the model's robustness to input noise and explore the loss landscapes associated with the inclusion of different physics terms. Conclusion: This advanced 1D modeling technique offers promising potential for rapid cardiovascular simulations, achieving computational efficiency and accuracy. By combining the strengths of physics-based and data-d
Hamilton-Jacobiequation appears frequently in applications, e.g., in differential games and control theory, and is closely related to hyperbolic conservation laws[3, 4, 12]. This is helpful in the design of difference...
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Hamilton-Jacobiequation appears frequently in applications, e.g., in differential games and control theory, and is closely related to hyperbolic conservation laws[3, 4, 12]. This is helpful in the design of difference approximations for Hamilton-Jacobi equation and hyperbolic conservation laws. In this paper we present the relaxing system for HamiltonJacobiequations in arbitrary space dimensions, and high resolution relaxing schemes for Hamilton-Jacobi equation, based on using the local relaxation approximation. The schemes are numerically tested on a variety of 1D and 2D problems, including a problem related to optimal control problem. High-order accuracy in smooth regions, good resolution of discontinuities, and convergence to viscosity solutions are observed.
An extended semi-definite programming, the SDP with an additional quadratic term in the objective function, is studied. Our generalization is similar to the generalization from linear programming to quadratic programm...
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An extended semi-definite programming, the SDP with an additional quadratic term in the objective function, is studied. Our generalization is similar to the generalization from linear programming to quadratic programming. Optimal conditions for this new class of problems are discussed and a potential reduction algorithm for solving QSDP problems is presented. The convergence properties of this algorithm are also given.
作者：
Peng, JMAssistant Professor
State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific Engineering Computing Academia Sinica Beijing China
The implicit Lagrangian has attracted much attention recently because of its utility in reformulating complementarity and variational inequality problems as unconstrained minimization problems, II was first proposed b...
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The implicit Lagrangian has attracted much attention recently because of its utility in reformulating complementarity and variational inequality problems as unconstrained minimization problems, II was first proposed by Mangasarian and Solodov as a merit function for the nonlinear complementarity problem (Ref. 1). Three open problems were also raised in the same paper, This paper addresses, among other issues, one of these problems by giving the properties of the implicit Lagrangian and establishing its convexity under appropriate assumptions.
This paper proposes an accelerated consensus-based distributed iterative algorithm for resource allocation and scheduling. The proposed gradient-tracking algorithm introduces an auxiliary variable to add momentum towa...
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