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作者： Zicong Xia Yang Liu College of Mathematics and Computer Science Zhejiang Normal University

This paper focuses on the penalty method, which has the potential to turn constrained problems into unconstrained ones, for distributed optimization with inequality and equality constraints. A selection principle to choose penalty gains is proposed based on value-distance ratio. Furthermore, a distributed method is developed to solve the obtained problems while its convergence is rigorously proved. Finally, an example in a numerical way demonstrates the validity of this paper.

关键词： Distributed optimization penalty methods Distributed method Non-smooth analysis

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penalty Algorithms in Hilbert Spaces

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Acta Mathematica Sinica,English Series 2007年第2期23卷 229-236页

作者： Jean Pierre DUSSAULT Hai SHEN André BANDRAUK Département d'Informatique Faculté des Sciences.Université de Sherbrooke Viador inc Laboratoire de Chimie Théorique Faculté des Sciences Université de Sherbrooke

We analyze the classical penalty algorithm for nonlinear programming in Hilbert spaces and obtain global convergence results, as well as asymptotic superlinear convergence order. These convergence results generalize similar results obtained for finite-dimensional problems. Moreover, the nature of the algorithms allows us to solve the unconstrained subproblems in finite-dimensional spaces.

关键词： penalty methods infinite dimensional optimization

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Deleting Outliers in Robust Regression with Mixed Integer Programming

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Acta Mathematicae Applicatae Sinica 2005年第2期21卷 323-334页

作者： Georgios Zioutas Antonios Avramidis Division of Computational methods and Computer Programming General Department Faculty of Technology Aristotle University of Thessaloniki 541 24 Thessaloniki GreeceDivision of Computational Methods and Computer Programming General Department Faculty of Technology Aristotle University of Thessaloniki 541 24 Thessaloniki Greece

In robust regression we often have to decide how many are the unusualobservations, which should be removed from the sample in order to obtain better fitting for the restof the observations. Generally, we use the basic principle of LTS, which is to fit the majority ofthe data, identifying as outliers those points that cause the biggest damage to the robust ***, in the LTS regression method the choice of default values for high break down-point affectsseriously the efficiency of the estimator. In the proposed approach we introduce penalty cost fordiscarding an outlier, consequently, the best fit for the majority of the data is obtained bydiscarding only catastrophic observations. This penalty cost is based on robust design weights andhigh break down-point residual scale taken from the LTS estimator. The robust estimation is obtainedby solving a convex quadratic mixed integer programming problem, where in the objective functionthe sum of the squared residuals and penalties for discarding observations is minimized. Theproposed mathematical programming formula is suitable for small-sample data. Moreover, we conduct asimulation study to compare other robust estimators with our approach in terms of their efficiencyand robustness.

关键词： Robust regression quadratic mixed integer programming least trimmedsquares deleting outliers penalty methods

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On the 4D Variational Data Assimilation with Constraint Conditions

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Advances in Atmospheric Sciences 2001年第6期18卷 1131-1145页

作者：朱克云 Chengdu University of Information Technology Chengdu

An investigation is carried out on the problem involved in 4D variational data assimilation (VDA) with constraint conditions based on a finite-element shallow-water equation model. In the investigation, the adjoint technology, penalty method and augmented Lagrangian method are used in constraint optimization field to minimize the defined constraint objective functions. The results of the numerical experiments show that the optimal solutions are obtained if the functions reach the minima. VDA with constraint conditions controlling the growth of gravity oscillations is efficient to eliminate perturbation and produces optimal initial field. It seems that this method can also be applied to the problem in numerical weather prediction. Key words Variational data assimilation - Constraint conditions - penalty methods - finite-element model This research is supported by National Natural Science Foundation of China (Grant No. 49575269) and by National Key Basic Research on the Formation Mechanism and Prediction Theory of Severe Synoptic Disasters (Grant No. G1998040910).

关键词： Variational data assimilation Constraint conditions penalty methods finite-element model

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The Algebraic Immersed Interface and Boundary Method for Elliptic Equations with Jump Conditions

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Open Journal of Fluid Dynamics 2020年第3期10卷 239-269页

作者： Arthur Sarthou Stéphane Vincent Philippe Angot Jean-Paul Caltagirone Buf Compagnie Paris France Laboratoire MSME Université Gustave Eiffel Champs-sur-Marne France Institut de Mathématiques de Marseille Centrale Marseille Marseille France Institut Polytechnique de Bordeaux Université de Bordeaux Talence France

A new simple fictitious domain method, the algebraic immersed interface and boundary (AIIB) method, is presented for elliptic equations with immersed interface conditions. This method allows jump conditions on immersed interfaces to be discretized with a good accuracy on a compact stencil. Auxiliary unknowns are created at existing grid locations to increase the degrees of freedom of the initial problem. These auxiliary unknowns allow imposing various constraints to the system on interfaces of complex shapes. For instance, the method is able to deal with immersed interfaces for elliptic equations with jump conditions on the solution or discontinuous coefficients with a second order of spatial accuracy. As the AIIB method acts on an algebraic level and only changes the problem matrix, no particular attention to the initial discretization is required. The method can be easily implemented in any structured grid code and can deal with immersed boundary problems too. Several validation problems are presented to demonstrate the interest and accuracy of the method.

关键词： Fictitious Domain Immersed Interface Method Immersed Boundary Method penalty methods Finite Volumes Elliptic Equations Jump Embedded Conditions

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