Lectures on Stochastic Programming-Model
这是一本关于随机规划比较全面的书！比较难，不太容易啃，但是读了之后收获很大。这是高清版的！To Julia, Benjamin, Daniel, Nalan, and Yael;to Tsonka Konstatin and Marekand to the memory of feliks, Maria, and dentcho2009/8/20pagContentsList of notationserace1 Stochastic Programming ModelsIntroduction1.2 Invento1.2.1The news vendor problem1.2.2Constraints12.3Multistage modelsMultiproduct assembl1.3.1Two-Stage Model1.3.2Chance Constrained ModeMultistage modelPortfolio selection131.4.1Static model14.2Multistage Portfolio selection14.3Decision rule211.5 Supply Chain Network Design22Exercises2 Two-Stage Problems272.1 Linear Two-Stage Problems2.1.1Basic pi272.1.2The Expected Recourse Cost for Discrete Distributions 302.1.3The Expected Recourse Cost for General Distributions.. 322.1.4Optimality Conditions垂Polyhedral Two-Stage Problems422.2.1General Properties422.2.2Expected recourse CostOptimality conditions2.3 General Two-Stage Problems82.3.1Problem Formulation, Interchangeability482.3.2Convex Two-Stage Problems2.4 Nonanticipativity2009/8/20page villContents2.4.1Scenario formulation2.4.2Dualization of Nonanticipativity Constraints2.4.3Nonanticipativity duality for general Distributions2.4.4Value of perfect infExercises3 Multistage problems3. 1 Problem Formulation633.1.1The general setting3.1The Linear case653.1.3Scenario trees3.1.4Algebraic Formulation of nonanticipativity constraints 7lDuality....763.2.1Convex multistage problems·763.2.2Optimality Conditions3.2.3Dualization of Feasibility Constraints3.2.4Dualization of nonanticipativity ConstraintsExercises4 Optimization models with Probabilistic Constraints874.1 Introduction874.2 Convexity in Probabilistic Optimization4.2Generalized Concavity of Functions and measures4.2.2Convexity of probabilistically constrained sets1064.2.3Connectedness of Probabilistically Constrained Sets... 113Separable probabilistic Constraints.1144.3Continuity and Differentiability Properties ofDistribution functions4.3.2p-Efficient Points.1154.3.3Optimality Conditions and Duality Theory1224 Optimization Problems with Nonseparable Probabilistic Constraints.. 1324.4Differentiability of Probability Functions and OptimalityConditions13344.2Approximations of Nonseparable ProbabilisticConstraints134.5 Semi-infinite Probabilistic Problems144E1505 Statistical Inference155Statistical Properties of Sample Average Approximation Estimators.. 1555.1.1Consistency of SAA estimators1575.1.2Asymptotics of the saa Optimal value1635.1.3Second order asStochastic Programs5.2 Stoch1745.2.1Consistency of solutions of the SAA GeneralizedEquatio1752009/8/20pContents5.2.2Atotics of saa generalized equations estimators 1775.3 Monte Carlo Sampling Methods180Exponential Rates of Convergence and Sample sizeEstimates in the Case of a finite Feasible se1815.3.2Sample size estimates in the General Case1855.3.3Finite Exponential Convergence1915.4 Quasi-Monte Carlo Methods1935.Variance-Reduction Techniques198Latin hmpling1985.5.2Linear Control random variables method200ng and likelihood ratio methods 205.6 Validation analysis5.6.1Estimation of the optimality g2025.6.2Statistical Testing of Optimality Conditions2075.7Constrained Probler5.7.1Monte Carlo Sampling Approach2105.7.2Validation of an Optimal solution5.8 SAA Method Applied to Multistage Stochastic Programmin205.8.1Statistical Properties of Multistage SAA Estimators22l5.8.2Complexity estimates of Multistage Programs2265.9 Stochastic Approximation Method2305.9Classical Approach5.9.2Robust sA approach..23359.3Mirror Descent sa method235.9.4Accuracy Certificates for Mirror Descent Sa Solutions.. 244Exercis6 Risk Averse Optimi2536.1 Introductio6.2 Mean-Risk models.2546.2.1Main ideas of mean -Risk analysis546.2.2Semideviation6.2.3Weighted Mean Deviations from Quantiles.2566.2.4Average value-at-Risk2576.3 Coherent risk measures2616.3.1Differentiability Properties of Risk Measures2656.3.2Examples of risk Measures..2696.3.3Law invariant risk measures and Stochastic orders2796.3.4Relation to Ambiguous Chance Constraints2856.4 Optimization of risk measures.2886.4.1Dualization of Nonanticipativity Constraints2916.4.2Examples...2956.5 Statistical Properties of Risk measures6.5.IAverage value-at-Ris6.52Absolute semideviation risk measure301Von mises statistical functionals3046.6The problem of moments306中2009/8/20page xContents6.7 Multistage Risk Averse Optimization3086.7.1Scenario tree formulation3086.7.2Conditional risk mappings3156.7.3Risk Averse multistage Stochastic Programming318Exercises3287 Background material3337.1 Optimization and Convex Analysis..334Directional Differentiability3347.1.2Elements of Convex Analysis3367.1.3Optimization and duality3397.1.4Optimality Conditions.............3467.1.5Perturbation analysis3517.1.6Epiconvergence3572 Probability3597.2.1Probability spaces and random variables7.2.2Conditional Probability and Conditional Expectation... 36372.3Measurable multifunctions and random functions3657.2.4Expectation Functions.3687.2.5Uniform Laws of Large Numbers...,,3747.2.6Law of Large Numbers for Random Sets andSubdifferentials3797.2.7Delta method7.2.8Exponential Bounds of the Large Deviations Theory3877.2.9Uniform Exponential Bounds7.3 Elements of Functional analysis3997.3Conjugate duality and differentiability.......... 4017.3.2Lattice structure4034058 Bibliographical remarks407Biibliography415Index4312009/8/20pageList of Notationsequal by definition, 333IR", n-dimensional space, 333A, transpose of matrix(vector)A, 3336I, domain of the conjugate of risk mea-C(X) space of continuous functions, 165sure p, 262CK, polar of cone C, 337Cn, the space of nonempty compact sub-C(v,R"), space of continuously differ-sets of r 379entiable mappings,176set of probability density functions,I Fr influence function. 3042L, orthogonal of (linear) space L, 41Sz, set of contact points, 3990(1), generic constant, 188b(k; a, N), cdf of binomial distribution,Op(), term, 382214S, the set of &-optimal solutions of theo, distance generating function, 236true problem, 18g(x), right-hand-side derivative, 297Va(a), Lebesgue measure of set A C RdCl(A), topological closure of set A, 334195conv(C), convex hull of set C, 337W,(U), space of Lipschitz continuousCorr(X, Y), correlation of X and Y 200functions. 166. 353CoV(X, Y, covariance of X and y, 180[a]+=max{a,0},2ga, weighted mean deviation, 256IA(, indicator function of set A, 334Sc(, support function of set C, 337n(n.f. p). space. 399A(x), set ofdist(x, A), distance from point x to set Ae multipliers vectors334348dom f, domain of function f, 333N(μ,∑), nonmal distribution,16Nc, normal cone to set C, 337dom 9, domain of multifunction 9, 365IR, set of extended real numbers. 333o(z), cdf of standard normal distribution,epif, epigraph of function f, 333IIx, metric projection onto set X, 231epiconvergence, 377convergence in distribution, 163SN, the set of optimal solutions of the0(x,h)d order tangent set 348SAA problem. 156AVOR. Average value-at-Risk. 258Sa, the set of 8-optimal solutions of thef, set of probability measures, 306SAA problem. 181ID(A, B), deviation of set A from set Bn,N, optimal value of the Saa problem,334156IDIZ], dispersion measure of random vari-N(x), sample average function, 155able 7. 2541A(, characteristic function of set A, 334吧, expectation,361int(C), interior of set C, 336TH(A, B), Hausdorff distance between setsLa」, integer part of a∈R,219A and B. 334Isc f, lower semicontinuous hull of funcN, set of positive integers, 359tion f, 3332009/8/20pageList of notationsRc, radial cone to set C, 337C, tangent cone to set C, 337V-f(r), Hessian matrix of second orderpartial derivatives, 179a. subdifferential. 338a, Clarke generalized gradient, 336as, epsilon subdifferential, 380pos w, positive hull of matrix W, 29Pr(A), probability of event A, 360ri relative interior. 337upper semideviation, 255Le, lower semideviation, 255@R. Value-at-Risk. 25Var[X], variance of X, 149, optimal value of the true problem, 1565=(51,……,5), history of the process,{a,b},186r, conjugate of function/, 338f(x, d), generalized directional deriva-g(x, h), directional derivative, 334O,(, term, 382p-efficient point, 116lid, independently identically distributed,1562009/8/20page xlllPrefaceThe main topic of this book is optimization problems involving uncertain parametersfor which stochastic models are available. Although many ways have been proposed tomodel uncertain quantities stochastic models have proved their flexibility and usefulnessin diverse areas of science. This is mainly due to solid mathematical foundations andtheoretical richness of the theory of probabilitystochastic processes, and to soundstatistical techniques of using real dataOptimization problems involving stochastic models occur in almost all areas of scienceand engineering, from telecommunication and medicine to finance This stimulates interestin rigorous ways of formulating, analyzing, and solving such problems. Due to the presenceof random parameters in the model, the theory combines concepts of the optimization theory,the theory of probability and statistics, and functional analysis. Moreover, in recent years thetheory and methods of stochastic programming have undergone major advances. all thesefactors motivated us to present in an accessible and rigorous form contemporary models andideas of stochastic programming. We hope that the book will encourage other researchersto apply stochastic programming models and to undertake further studies of this fascinatinand rapidly developing areaWe do not try to provide a comprehensive presentation of all aspects of stochasticprogramming, but we rather concentrate on theoretical foundations and recent advances inselected areas. The book is organized into seven chapters The first chapter addresses modeling issues. The basic concepts, such as recourse actions, chance(probabilistic)constraintsand the nonanticipativity principle, are introduced in the context of specific models. Thediscussion is aimed at providing motivation for the theoretical developments in the book,rather than practical recommendationsChapters 2 and 3 present detailed development of the theory of two-stage and multistage stochastic programming problems. We analyze properties of the models and developoptimality conditions and duality theory in a rather general setting. Our analysis coversgeneral distributions of uncertain parameters and provides special results for discrete distributions, which are relevant for numerical methods. Due to specific properties of two- andmultistage stochastic programming problems, we were able to derive many of these resultswithout resorting to methods of functional analvsisThe basic assumption in the modeling and technical developments is that the proba-bility distribution of the random data is not influenced by our actions(decisions). In someapplications, this assumption could be unjustified. However, dependence of probability dis-tribution on decisions typically destroys the convex structure of the optimization problemsconsidered, and our analysis exploits convexity in a significant way
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Gabor小波+PCA+LDA特征提取方法的人脸表情识别
基于Gabor小波+PCA+LDA特征提取方法的人脸表情识别 论文机电技术2011年12月(k)(×()4eXp|-0.5(0(4)-2×3.1415926)×U第四步:初始模型确定下来以后,采用则将该待识别人脸表情判别为第讠类表情Baum- Welch算法对参数进行重估计,选代调整模人脸表情实验及分析型参数以达最优化。这样就完成了训练建模工作,获得了在粗分本文在JAFE( Japanese Female Facial Expre-类层次下代表7种人睑表情的IMM模型sion)表情数据库中进行实验。 JAFFE人脸表情库a1=(A,B,)(=1,2,…,7)10个人,共213幅图像组成,每人都包含愤怒、23HMM人脸表情分类厌恶、恐惧、髙兴、悲伤、惊讶和中性7种表情,对待识别的人脸表情图像,按本节22中模型每人每种表情2~4幅,每和表情共有27幅本文实验与 Mihcac等的测试方法相似,把数训练步骤的第一步获得观察向量O,然后计算观据库中的210图像分成3个部分,每个部分包含察向量O与已训练好的HMM模型A1=(A,B,z)10个人,且每个人包含7幅不同表情图像,每次使的似然概率P(O|λ)(=,2,…,7)。通过用其中的一个部分训练各个表情的HMM模型,剩Forward-Backward算法来计算似然概率,如果下的两个部分用来测试。循环操作3次,然后对这P(O|A1),i∈(1.2,…,)在P(OA)(=1,2,…,7)3次取平均值,得出表情识别率。本方法获取的表中最人,即情识别率见表1P(Ol, )=max p(ola)表1人脸表情识别结果愤怒厌恶恐惧高兴中性悲伤惊讶识别率/愤怒厌恶恐惧5591高兴中性悲伤0201100095.00020惊讶05693.33整体识别率92.14从表1中可以看出,本方法可以获得较好的表高的问题,提出了一种新的人脸表情识别方法。情识别率,但是,从实验中也可发现,不管是有本方法通过更加有效的 Gabor小波+PCA+LDA表没有减少分类类别数,实验中始终存在误识别。情特征提取方法,然后结合IMM模型建立7个对从实验数据斥中发现,悴中杲些人的种表情变化应基本表情+中性表情的模型,进行识别分类,并不明显,即使人眼也无法确定其表情类别,以并且取得了92.14%的整体识别率。同时,在实验致算法在识别时出现错误。中发现,由于人脸实验数据库中某些人的7种表情4结论变化并不明显,即使人眼也无法确定其表情类别,以致算法在识别时出现错误。本文针对目前人脸衣情的整体识别卒普遍不(下转第35页)21994-2015ChinaacAdemicJOurnalElcctronicPublishingHousc.Allrightsrcscrved.http://www.cnki.nct第6期林彩邴等:磁流变阻尼器减振系统的分数阶微分方程硏究35[2] DROZDOVAD Mechanicsofviscoelasticsolids[M]. New York: John Wiley &Sons Ltd, 1998: 21-65B3 LAKES RS. Viscoelastic solids[M]. London: CRC Press, 1998: 63-1104 Stanway R, Sproston, J L, Stevens N (i. Non-liner modeling of an electrorheological vibration damper.J. Electrostatics,1987,20[S]周强,瞿伟廉.磁流变阻尼器的两种力学模型和试验验证[.地震工程与工程振动,2002,22(40.[6] Gamoto DR, Filisko F E. Dynamic mechanical studies of electrorheological materials moderate frequencies. J. Rheology,991,35(3).[刀]汪建晓,孟光.磁流变阻尼器用于振动搾制的理论及实验研究J.振动与冲击,2001,20(2)[8 Wen Y K. Method of random vibration of hysteretic systems[J. Journal of Engineering Mechanics Division, ASCE102(EM2),19769SpencerJr. BF, Dyke SJ, Sain M K, Carlson, J.D. Phenomenological model of a magnetorheological damper. JEngrg. MechASCE.1997,123[10]薛定宇陈阳泉高等应用数学问题的 MATLAB解[M清华大学出版社,2010[l]l云,谭半.磁流变阻尼控制理论与技术[M科学出版社2007[1】]王振滨曹广义分数微积分的两种系统建模方汏系统仿真学报,2004,6(4:810-81上h与比比比比匙比比知5印(上接第20页)参考文献[1] ZHAO Quan-you, PAN Bao-chang, PAN Jian-jia, et al. Facial expression recognition based on fusion of Gabor and LBPfeaturesLA] In: Proceedings of the 2008 International Conference on Wavelet Analysis and Pattern Recognition[C), HongKong,IEEE,2008(8):362-367[2] Tang Fang-qi, Deng Ben-zai. Facial Expression Recognition using AAM and Local Facial Features [A]. In: Proceedings ofthe Third International Conference on Natural Computation[C], Haikou, China, IEEE, 2007(8): 632-635[3]陈培俊基于静态图像的人脸表情识别研咒[M成都:西南交通大学,2007[4] Lades M, Vorbruggen JC, Buhmann J. Distortion invariant object recognition in the dynamic linkarchitecture[J].IEEE Transactions on Computers, IEEE, 1993, 42(3): 300-311[5]彭辉张长水荣钢等基于KL变换的人脸自动识别方法清华大学学报(自然科学版),2007,37(6):6770[6]Duda RO, Hart PE, Stork DG Pattern Classification(second edition). New York: Wiley, 20017]李云霞,李治柱,吴亚栋基于HMM的关键词识别系统计算机工程,2004,30(7):130-132谨致作者敬请本刊作者允诺:稿件中没有侵犯他人著作权或其它杈利的内容并且文责自负:投寄给本刊的槁件(论文、图表、照片等)自发表之日起,其专有出版权和网络传播权即授于本刊,并许可本刊在本刊网站或本刊授权的网站上传播及屮国核心期刊(遴选)数据库、屮国学术期刊综合评价数据库、屮文科技期刊数据库、屮国台湾华艺数据库全文收录。对上述合作若有异议者,烦请来时向本刊申明,未作申明者,本刊将视为同意,谢谢合作。并致诚挚敬意。《机电技术》编辑韶21994-2015ChinaacAdemicJOurnalElcctronicPublishingHousc.Allrightsrcscrved.http://www.cnki.nct
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