简介：有双性人目标策略的一个扩充Lagrangian信任区域方法被建议因为解决非线性的平等抑制了优化，它在惩罚类型方法和没有惩罚的之间掉落。在每次重复，试用步被在一个信任区域以内最小化一个二次的近似模型到扩充Lagrangian功能计算。模型是为非强迫的优化的标准信任区域subproblem并且能被许多存在方法高效地因此解决。选择惩罚参数，辅助信任区域subproblem与限制违背有关被介绍。结果，惩罚参数不必monotonically正在增加并且不趋于到无穷。双性人目标策略，与客观功能和限制违背的措施有关，被利用决定试用步是否将被接受。方法的全球集中在温和假设下面被建立。数字实验被做，它在各种各样的困难的状况上说明算法的效率。

简介：Inthispaperwedeveloptwoconformingfiniteelementmethodsforafourthorderbi-waveequationarisingasasimplifiedGinzburg-Landau-typemodelford-wavesuperconductorsinabsenceofappliedmagneticfield.UnlikethebiharmonicoperatorA2,thebi-waveoperator□~2isnotanellipticoperator,sotheenergyspaceforthebi-waveequationismuchlargerthantheenergyspaceforthebiharmonicequation.Thisthenmakesitpossibletoconstructloworderconformingfiniteelementsforthebi-waveequation.However,theexistenceandconstructionofsuchfiniteelementsstronglydependsonthemesh.Inthepaper,wefirstcharacterizemeshconditionswhichallowandnotallowconstructionofloworderconformingfiniteelementsforapproximatingthebi-waveequation.Wethenconstructacubicandaquarticconformingfiniteelement.Itisprovedthatbothelementshavethedesiredapproximationproperties,andgiveoptimalordererrorestimatesintheenergynorm,suboptimal(andoptimalinsomecases)ordererrorestimatesintheH~1andL~2norm.Finally,numericalexperimentsarepresentedtoguagetheefficiencyoftheproposedfiniteelementmethodsandtovalidatethetheoreticalerrorbounds.