双层凸优化问题A Global Optimization Method for Solving Convex

第50卷第23期电力系统保护与控制Vol.50 No.23 2022年12月1日Power System Protection and Control Dec. 1, 2022 DOI: 10.19783/ki.pspc.220202计及源荷不确定影响的不平衡配电网两阶段优化苏向敬1，刘一航1，张知宇2，符 杨1(1.上海电力大学电气工程学院，上海 200090；2.国网浙江省电力有限公司海盐县供电公司，浙江 嘉兴 314300)摘要：在“碳达峰、碳中和”国家战略背景下，分布式电源(distributed generators, DG)的大量接入虽然带来显著的社会和环境效益，但其高间歇性与不确定性也导致峰谷差拉大、电压频繁越限，危害配网安全稳定运行。

作为缓解上述问题的有效手段，储能与DG的协调控制近年来得到广泛关注。

但现有研究普遍基于三相平衡网络模型假设，且受储能自身特性限制对DG不确定性影响考虑不足。

为此，提出一种基于储能和DG协同控制的不平衡配电网两阶段优化策略。

其中，第I阶段基于三相配网模型和源荷功率预测，对储能开展长时尺度(24 h)优化调度，以提升运行经济性。

针对第I阶段存在的电压越限风险，第II阶段基于源荷不确定性建模和DG逆变器无功能力，在短时间尺度(15 min)开展不平衡配网电压分布式鲁棒优化，以提升运行安全性。

对上述两阶段优化问题采用凸化和对偶重构手段进行求解，并基于真实配电网案例进行仿真验证。

结果表明：所提两阶段优化策略能有效计及三相不平衡和源荷不确定影响，并提升配电网运行的经济性与安全性。

关键词：分布式电源；电池储能；不平衡配电网；分布式鲁棒优化；凸优化Two-stage optimization of unbalanced distribution networks considering impacts ofDG and load uncertaintiesSU Xiangjing1, LIU Yihang1, ZHANG Zhiyu2, FU Yang1(1. College of Electrical Engineering, Shanghai University of Electric Power, Shanghai 200090, China;2. Haiyan Power Supply Company, State Grid Zhejiang Electric Power Co., Ltd., Jiaxing 314300, China)Abstract: In the context of a national strategy of ‘emission peak and carbon neutrality’, the penetration of distributed generators (DG) is increasing dramatically, bringing significant social and environmental benefits. However, DG features such as high intermittency and uncertainty lead to problems such as high peak-valley difference and frequent voltage violation, causing negative impacts on the safe and stable operation of distribution networks. As one of the most effective solutions, the coordination of a battery energy storage system (BESS) and DG has been attracting a lot of attention.However, existing studies of BESS and DG coordination unreasonably assume distribution networks are three-phase balanced, and are less capable to deal with uncertainties because of the limitations of BESS characteristics. Thus, a BESS and DG-based two-stage control strategy of unbalanced distribution networks is proposed. Specifically, based on a three-phase network model and predictions of DG and load power, stage I performs BESS control on a long-time scale(24 hours) to improve the operational economy. To deal with the voltage violation risk in stage I, stage II conductsdistributionally robust voltage optimization of the unbalanced distribution network over a short time scale (15 min), based on source-load uncertainty modeling and the reactive capability of a DG inverter, thus improving the operational safety.Convex optimization and dual reconstruction are used to solve the above two-stage optimization problem, and simulations are carried out on a real distribution network. The results show that the proposed two-stage optimization strategy can effectively take into account the impacts of three-phase unbalance and source-load uncertainty, and also improve the economy and safety of distribution network operation.This work is supported by the General Program of National Natural Science Foundation of China (No. 61873159).Key words: distributed generation; battery energy storage system; unbalanced distribution network; distributionally robust optimization; convex optimization基金项目：国家自然科学基金面上项目资助(61873159)；上海绿色能源并网工程技术研究中心项目资助(13DZ2251900)苏向敬，等计及源荷不确定影响的不平衡配电网两阶段优化- 95 -0 引言以风电、光伏为代表的分布式电源(distributed generator, DG)渗透率越来越高，在带来显著社会和环境效益的同时，其高随机性和高波动性导致峰谷差加大、逆向潮流、电压越限和三相不平衡等问题，危害配电网的安全稳定运行[1-6]。

An augmented Lagrangian multiplier method based on a CHKS smoothing function for solving nonlinear bilevel programmingproblemsYan Jiang a ,Xuyong Li a ,⇑,Chongchao Huang b ,Xianing Wu caState Key Laboratory of Urban and Regional Ecology,Research Center for Eco-Environmental Sciences,Chinese Academy of Sciences,Beijing 100085,PR China bSchool of Mathematics and Statistics,Wuhan University,Wuhan 430072,PR China cSinohydro Resources Limited,Beijing 100048,PR Chinaa r t i c l e i n f o Article history:Received 19July 2012Received in revised form 9August 2013Accepted 13August 2013Available online 23August 2013Keywords:Nonlinear bilevel programming Karush–Kuhn–Tucker condition Complementary constraints Smoothing functionAugmented Lagrangian multiplier methoda b s t r a c tBilevel programming techniques deal with decision processes involving two decision makers with a hier-archical structure.In this paper,an augmented Lagrangian multiplier method is proposed to solve non-linear bilevel programming (NBLP)problems.An NBLP problem is first transformed into a single level problem with complementary constraints by replacing the lower level problem with its Karush–Kuhn–Tucker optimality condition,which is sequentially smoothed by a Chen–Harker–Kanzow–Smale (CHKS)smoothing function.An augmented Lagrangian multiplier method is then applied to solve the smoothed nonlinear program to obtain an approximate optimal solution of the NBLP problem.The asymptotic prop-erties of the augmented Lagrangian multiplier method are analyzed and the condition for solution opti-mality is derived.Numerical results showing viability of the approach are reported.Ó2013Elsevier B.V.All rights reserved.1.IntroductionA bilevel programming (BLP)problem is characterized by a nested optimization problem with two levels,namely,an upper and a lower level,in a hierarchy where the constraint region of the upper level problem is implicitly determined by the lower level optimization problem.Such problems arise in game-theoretic and decomposition related applications in engineering design.Re-cently,these problems have received increasing attention and have been applied in various aspects of resource planning,management,and policy making,including water resource management,and financial,land-use,production,and transportation planning [1].After a real-world problem has been modeled as a bilevel deci-sion problem,how to find the optimal solution for this decision problem becomes very important.However,the nested optimiza-tion model described above is quite difficult to solve.It has been proved that solving a linear bilevel program is NP-hard [2,3].In fact,even searching for a locally optimal solution of a linear bilevel program is strongly NP-hard [4].Herminia and Carmen [5]pointed out that the existence of multiple optima for the lower-level problem can result in an inadequate formulation of the bilevel problem.Moreover,the feasible region of the problem is usuallynon-convex,without necessarily being a connected set,and empty,making it difficult to solve BLP problems.Existing methods for solving BLP problems can be classified into the following categories [6]:extreme-point search,transformation,descent and heuristic,interior point,and intelligent computation approaches.Recently,the latter approach has become an impor-tant computational method for solving BLP problems.Genetic algo-rithm [7–9],Neural networks [10],Tabu search [11–13],ant colony optimization [14],and particle swarm optimization [15–17]are typical intelligent algorithms for solving BLP problems.In all these algorithms,the BLP problem is first reduced to a one-level problem with complementary constraints by replacing the lower level prob-lem with its Karush–Kuhn–Tucker (KKT)optimality condition.However,the one-level complementary slackness problem is non-convex and non-differential.The major difficulty in solving this type of problem is that its constraints fail to satisfy the stan-dard Mangasarian–Fromovitz constraint qualification at any feasi-ble point,and that general intelligent algorithms may fail to uncover such local minima or even mere stationary points.To avoid the difficulty of the nonsmooth constraints,a sequence of smooth problems by using smoothing functions have been progressively approximated this nonsmooth problem.To date,there have been many smoothing functions,such as perturbed Fischer–Burmeister (FB),Chen–Harker–Kanzow–Smale (CHKS),Neural networks,uniform,and Picard smoothing functions [18–20].Thereinto,perturbed FB and CHKS smoothing functions0950-7051/$-see front matter Ó2013Elsevier B.V.All rights reserved./10.1016/j.knosys.2013.08.017Corresponding author.Tel.:+861062840210.E-mail address:yanjiang@ (X.Y.Li).have been applied widely to solve BLP problems.Facchinei et al.[21]transformed mathematical program with equilibrium con-straints(MPEC),including bilevel programming problems as a par-ticular case,into an equivalent one-level nonsmooth optimization problem,which was then approximated progressively by applying CHKS smoothing function to a sequence of smooth,regular prob-lems that approximate the nonsmooth problem.Finally,it was solved by standard available software for constrained optimiza-tion.Dempe[22]introduced a smoothing technique by applying perturbed FB function and Lagrangian function method to solve a BLP problem,approximated by a sequence of smooth optimization problems.Wang et al.[23]proposed an approximate programming method based on the simplex method to research the general bile-vel nonlinear programming model by applying perturbed FB smoothing function.Etoa[24]presented a smoothing sequential quadratic programming using perturbed FB functional to compute a solution of a quadratic convex bilevel programming problem.In general,perturbed FB and CHSK smoothing functions are very clo-sely related.These two functions are very similar except for the dif-ference of a constant.In what follows,we choose to focus on the CHKS smoothing function to approximate nonsmooth problem.The purpose of this work is to design an efficient algorithm for solving general nonlinear bilevel programming(NBLP)models.Dif-ferent from the algorithms in[22,23],where Lagrangian optimiza-tion and simplex method were used to solve a smooth problem approximated by perturbed FB smoothing function,we apply a CHKS smoothing function and an augmented Lagrangian multiplier method to solve NBLP problem,and focus on the detailed analysis on the condition for solution optimality.The remainder of this pa-per is organized as follows:Section2presents the formulation and basic definitions of bilevel programming,while Section3intro-duces the smoothing method for nonlinear complementarity prob-lems.In Section4we explain the augmented Lagrangian multiplier method in detail and derive the condition for asymptotic stability, solution feasibility,and solution optimality.Numerical examples are reported in Section5and Section6concludes the paper.2.Formulation of an NBLP problemThe general formulation of NBLP problems is as follows[25]: minxFðx;yÞ;s:t:Gðx;yÞ60;minyfðx;yÞ;s:t:gðx;yÞ60;x2X&R n1;y2Y&R n2;ð1Þwhere F;f:R n1ÂR n2!R represent the objective function of the leader and the follower,respectively,G:R n1ÂR n2!R m1denotes the constraints of the leader,g:R n1ÂR n2!R m2denotes the con-straints of the follower,and x2R n1;y2R n2denote the decision vari-ables of the leader and the follower,respectively.The following definitions are associated with the problem(1).The relaxed feasible set(or constrained region)is defined as S={(x,y):G(x,y)60,g(x, y)60},the lower level feasible set for eachfixed x as S(x)={y: g(x,y)60},the follower’s rational reaction set as PðxÞ¼f y:y2arg min f fðx;^yÞ;^y2SðxÞgg,and the inducible region as IR={(x,y):(x,y)2S,y2P(x)}.To guarantee that there is at least one solution to the NBLP problem,we assume that the constrained region S is nonempty and compact.At the same time,for somefixed x2X,we assume that the lower level problem is a convex optimization problem sat-isfying the Mangasarian–Fromovitz constraints qualification (MFCQ)for any y2S(x).Then we can reduce the bilevel program-ming problem to a one-level problem using the KKT condition.minx;y;kFðx;yÞ;s:t:Gðx;yÞ60;r y Lðx;y;kÞ¼0;k T gðx;yÞ¼0;gðx;yÞ60;k P0;ð2Þwhere Lðx;y;kÞ¼fðx;yÞþk T gðx;yÞ;k2R m2is a Lagrange function.3.Smoothing methodProblem(2)is a mathematical program with nonlinear comple-mentary constraints.It is non-convex,non-differentiable and the regularity assumptions,which are needed to handle smooth opti-mization programs successfully,are never satisfied.So it is not appropriate to use standard nonlinear programming software to solve it[26,27].We consider the following perturbation of problem(2):minx;y;kFðx;yÞ;s:t:Gðx;yÞ60;r y Lðx;y;kÞ¼0;k T gðx;yÞ¼ 2;gðx;yÞ60;k P0;ð3Þwhere >0is an identity element.For a given ,we solve problem (3)to get the solution(x ,y ,k ).As decreases slowly toward0,the optimum of F follows to the global optimum of F0.This set of constraint conditions is called the central path conditions as they define a trajectory approaching the solution set for ?0.The smoothing method is an effective way to deal with this issue.The main feature of smoothing method is to reformulate the problem (3)as a smooth nonlinear programming problem by replacing the difficult equilibrium constraint with a smoothing function.Conse-quently,the true problem(3)can be approximated by a sequence of smooth problem[21,28,29].The numerical evidence on the effi-ciency of the smoothing approach is very impressive.Hence,In this work,we smooth the nonlinear programming with complementary constraints by applying CHKS smoothing function.Define function/ :R3?R using the CHKS smoothing function / ða;bÞ¼aþbÀffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaÀbÞ2þ4 2q.Then,we can have the following proposition[21].Proposition1.For every 2R,we have/ (a,b)=0,if and only if a P0,b P0,ab= 2for P0,where is a parameter converging to zero.Note that for every P0,/ (a,b)is continuously differentiable with respect to(a,b).By applying the CHKS smoothing function / (x,y),problem(3)can be approximated by:minx;y;kFðx;yÞ;s:t:Gðx;yÞ60;r y Lðx;y;kÞ¼0;k jÀg jðx;yÞÀffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk iþg jðx;yÞÞ2þ4 2q¼0;j¼1;2;...;m2:ð4ÞProblem(4)can be viewed as a perturbation of problem(2). Problem(4)overcomes the difficulty due to problem(2)not satis-fying any regularity assumption and paves the way for using an augmented Lagrangian multiplier approach to solve problem(2).10Y.Jiang et al./Knowledge-Based Systems55(2014)9–14The idea of the algorithm is as followed:the smoothing method starts with a large ,finds the optimum of F ,decreases gradually, and recomputes the optimum at each step-starting from the previ-ous solution-until is small enough.We claim that this procedure yields near-optimal solutions.To simply our discussion and for convenience,we introduce the following notations:Let h(x,y,k)=G(x,y),Hðx;y;kÞ¼r y Lðx;y;kÞ/ ðk j;Àgðx;yÞÞj¼1;2;...;m2!and z=(x,y,k),then problem(4)canbe written as:minzFðzÞ;s:t:hðzÞ60;HðzÞ¼0:ð5ÞDefinition1.Let z be a feasible point of problem(4).We say that z is a regular point of problem(5)if gradients r z h(z)and r z H(z)are linearly independent.Similar to the main result,Theorem3,in[21],the theorem be-low follows directly.Theorem1.Let{z }be a sequence of solutions of problem(5).If sequence{z }is contained in a compact set and converges to some z for ?0,then z is a global solution of problem(2).4.Augmented Lagrangian multiplier method4.1.Definition of augmented Lagrangian multiplier methodDuring the past few decades,considerable attention has been focused onfinding methods for solving nonlinear programming problem(5)using unconstrained minimization u-ally,a penalty function method,which requires that an infinite se-quence of penalty problems be solved,is used to solve this type of problem.The main drawback of the sequential penalty function method is that when the penalty parameter approaches its limiting value,the corresponding penalty problems are ill-conditioned, resulting in various shortcomings such as numerical instability and slow convergence.At the same time,owing to the presence of non-convexity in the smooth approximate problems(5),multi-ple local optimal solutions may exist and thus,the traditional Lagrangian method is not guaranteed tofind a global solution of the approximation problem.To overcome these shortcomings,we apply an augmented Lagrangian multiplier algorithm[30,31], which combines the Lagrangian multiplier method and penalty method,to solve nonlinear programming problems.Numerical studies indicate that the augmented Lagrangian multiplier method offers significant advantages over the usual penalty methods.The augmented Lagrangian function is defined as:Uðz;w;v;rÞ¼FðzÞþ12r X m1j¼1½maxð0;w jþr h jðzÞÞ 2Àw2j n oþXm2þn2j¼1v j H jðzÞþr2Xm2þn2j¼1H jðzÞ2;ð6Þwhere w2R m1,and v2R n2þm2are multipliers corresponding to the inequality and equality,separately,and r is a positive penalty factor.The augmented Lagrangian multiplier method for solving non-linear programming problems can be stated as follows.Suppose that w(k)and v(k)are Lagrangian multiplier vectors and z(k)is the minimum of U(z,w(k),v(k),r)at iteration k.Then,the update formulas for multiplier w(k+1)and v(k+1)at iteration k+1are given as:wðkþ1Þ¼maxð0;wðkÞþr hðzðkÞÞÞ;ð7Þvðkþ1Þ¼vðkÞÀr HðzðkÞÞ:ð8ÞDenoting the solution by z(k+1),we solve the unconstrained pro-gram min U(z,w,v,r).After a certain number of iterations,if wðkÞ! w and vðkÞ! v,then zðkÞ! z.If{w(k)}and{v(k)}do not con-verge or converge very slowly,we increase the value of penalty parameter r and repeat the above steps.The steps of the augmented Lagrangian multiplier algorithm for NBLP problems are given as follows:Step1(Initialization).Choose initial value z(0),Lagrangian mul-tiplier factors w(1),v(1)and penalty parameter r.Give expansion coefficient a>1,reduced coefficient b2(0,1)and allowed error e>0.Set k=1.Step2(Solution of the problem(7)).Solve the unconstrained programming problem min U(z(kÀ1),w(k),v(k),r)and obtain the optimal solution z(k).Step3(Termination check).If k h(z(k))<e k and k H(z(k))<e k, then the algorithm terminates.z(k)is taken as the approximate optimal solution and F(z(k))is the approximate optimal value. Otherwise,go to Step4.Step4(Updating penalty parameter).If k hðzðkÞÞkk hðzðkÀ1ÞÞkP b or k HðzðkÞÞkk HðzðkÀ1ÞÞkP b,let r=a r,and go to Step5.Step5(Updating multipliers).Update the multipliers using Eqs.(7)and(8).Let k=k+1and go to Step2.The relevant feature of the augmented Lagrangian function method is that,given suitable assumptions,the solution of the con-strained optimization problem can be obtained by recursively updating the multiplier vector,without increasing penalty param-eter r to infinity.4.2.Optimality resultsNext we study the relationship between the solution of problem (5)and the unconstrained minimum points of U(z,w,v,r).If we use min instead of max,we have:Uðz;w;v;rÞ¼FðzÞþX m1j¼1w j h jðzÞþr2X m1j¼1h jðzÞ2À12rX m1j¼1min0;h jðzÞþw jrh i2þXm2þn2j¼1v j H jðzÞþr2Xm2þn2j¼1H jðzÞ2:ð9ÞDefine d2iðz;w;rÞ¼Àmin0;h iðzÞþw i rÈÉ;i¼1;...;m1and de-note dðz;w;rÞ¼ðd1ðz;w;rÞ;...;d m1ðz;w;rÞÞT,and D(z,w, r)=diag(d i(z,w,r)).It can easily be verified that:Uðz;w;v;rÞ¼FðzÞþw T½hðzÞþDðz;w;rÞdðz;w;rÞ þr2k hðzÞþDðz;w;rÞdðz;w;rÞk2þv T HðzÞþr2k HðzÞk2: The Lagrangian function L0for problem(5)is defined as:L0ðz;w;vÞ¼FðzÞþw T hðzÞþv T HðzÞ:Under the assumption that F,h and H are continuously differen-tiable on R n1þn2,we obtain the following results.Theorem2.For any r>0,letð z; w; vÞbe a stationary point of U(z,w, v,r).Then,ð z; w; vÞis a KKT pair for problem(5)and Fð zÞ¼Uð z; w; v;rÞ.Y.Jiang et al./Knowledge-Based Systems55(2014)9–1411Proof.We have the following expressions for the components of the gradient of U (z ,w ,v ,r ):r z U ðz ;w ;v ;r Þ¼r z L 0ðz ;w ;v Þþ2r @h ðz ÞT@z½h ðz ÞþD ðz ;w ;r Þd ðz ;w ;r Þ þ2r @H ðz ÞT@zH ðz Þ;r w U ðz ;w ;v ;r Þ¼h ðz ÞþD ðz ;w ;r Þd ðz ;w ;r Þ;r v U ðz ;w ;v ;r Þ¼H ðz Þ:If ð z ; w ; v Þis a stationary point of U (z ,w ,v ,r ),then r z U ðz ; w ; v ;r Þ¼0;r w U ð z ; w ; v ;r Þ¼0,and r v U ð z ; w ; v ;r Þ¼0.This implies that r z L 0ð z ; w ; v Þ¼0;h ð z ÞþD ð z ; w;r Þd ðz ;w ;r Þ¼0,and H ðz Þ¼0.Then we have:h i ðz Þþd 2i ð z ; w ;r Þ¼h i ð z ÞÀmin 0;h i ð z Þþw irh i¼max h i ðz Þ;Àw i rh i¼0:This implies that h i ðz Þ60and w T h ð z Þ¼0,which shows that ð z ; w ; v Þis a KKT pair for problem (5).By substitutingh ðz ÞþD ð z ; w ;r Þd ðz ;w ;r Þ¼0and H ð z Þ¼0into (10),we get F ð z Þ¼U ð z ; w ; v ;r Þ.This completes the proof.h Theorem 3.Let ðz ; w ; v Þbe an unconstrained global minimum point of U (z,w,v,r )of ?0on subset Z ÂW ÂV #R n 1þn 2þm 2ÂR m 1ÂR m 2þn 2.Then,ðz ; w ; v Þis a KKT pair for problem (5),and z is a global minimum point of problem (5)and also a global minimum point of the NBLP problem.Proof.The fact that ðz ; w ; v Þis an unconstrained global minimum point of U (z ,w ,v ,r )implies thatz is an unconstrained global min-imum point of U ðz ; w ; v ;r Þ,r z U ð z ; w ; v ;r Þ¼0;r w U ð z ; w; v ;r Þ¼0,and r v U ð z ; w ; v ;r Þ¼0.Therefore,ð z ; w; v Þis a stationary point of U (z ,w ,v ,r ).And by Theorem 2,ðz ; w ; v Þis a KKT pair for problem (5)and F ð z Þ¼U ð z ; w; v ;r Þ.Since z is an unconstrained global minimum point of U ðz ; w; v ;r Þon subset Z ÂW ÂV ,there exists a neighborhood X ofz ,such that F ðz Þ6U ðz ; w ; v ;r Þ;8z 2X :By Eq.(9),the above inequality yields:F ðz Þ6F ðz Þ;8z 2X \f z :h ðz Þ60;H ðz Þ¼0g :So,z is a global minimum point of problem (5).From Theorem 1above,we can complete the proof.h 5.Numerical experimentsIn this section,we present three examples to illustrate the fea-sibility of the augmented Lagrangian multiplier algorithm for NBLP problems.Example 1[32].Consider the following NBLP problem,where x 2R 2,y 2R 2.min x P 0F ðx ;y Þ¼Àx 21À3x 2À4y 1þy 22;s :t :x 21þ2x 264;min y P 0f ðx ;y Þ¼2x 21þy 21þx 22þy 21À5y 2;s :t :x 21À2x 1þx 22À2y 1þy 2P À3;x 2þ3y 1À4y 2P 4:ð10ÞFor mixed x ,the lower level problem satisfies MFCQ.Applying the KKT condition,problem (10)can be transformed into the fol-lowing single-level programming problem:minx ;y ;k P 0F ðx ;y Þ¼Àx 21À3x 2À4y 1þy 22;s :t :x 21þ2x 264;2y 1þ2k 1À3k 2¼0;À5Àk 1þ4k 2¼0;k 1ðx 21À2x 1þx 22À2y 1þy 2þ3Þ¼0;k 2ðx 2þ3y 1À4y 2À4Þ¼0;x 21À2x 1þx 22À2y 1þy 2P À3;x 2þ3y 1À4y 2P 4:ð11ÞAdoptingthe CHSK smoothing function,/ ða ;b Þ¼a þb Àffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða Àb Þ2þ4 2q ,problem (11)can be approxi-mated by the following problem:minxy ;k P 0F ðx ;y Þ¼Àx 21À3x 2À4y 1þy 22;s :t :x 21þ2x 264;2y 1þ2k 1À3k 2¼0;À5Àk 1þ4k 2¼0;k 1þðx 21À2x 1þx 22À2y 1þy 2þ3ÞÀffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk 1Àðx 21À2x 1þx 22À2y 1þy 2þ3ÞÞ2þ4 2q ¼0;k 2þðx 2þ3y 1À4y 2À4ÞÀffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk 2Àðx 2þ3y 1À4y 2À4ÞÞ2þ4 2q ¼0:ð12ÞLet z =(x 1,x 2,y 1,y 2,k 1,k 2)T .Set the initial point z (0)=(1,1,1,1,1,1)T ,r =100,w (0)=0.5,v (0)=(0.5,0.5,0.5,0.5)T ,a =10,b =0.25,and e =0.001.According to the above steps,the results listed in Table 1are obtained for different values of .Example 2[33].Consider the following NBLP problem,where x 2R 1,y 2R 1.min x P 0F ðx ;y Þ¼x 2þðy À10Þ2;s :t :06x 615;min y P 0f ðx ;y Þ¼ðx þ2y À30Þ2;s :t :x Ày 2P 0;20Àx Ày 2P 0:ð13ÞAfter applying the KKT condition and CHKS smoothing function,the above problem can be transformed into a problem similar to problem (5)which can be solved by using the augmented Lagrang-ian multiplier method.Let z =(x ,y ,k 1,k 2)T and set the initial point z (0)=(1,1,1,1)T .Parameters r ,w (0),v (0),a ,b ,and e are selected as in Example 1.Table 1presents the different optimal solutions of Example 2for varying small values of .Example 3[34].Consider the following nonlinear BLP problem,where x 2R 1,y 2R 2.min x P 0F ðx ;y Þ¼ðx À1Þ2þ2y 21À2x ;min y P 0f ðx ;y Þ¼ð2y 1À4Þ2þð2y 2À1Þ2þxy 1;s :t :4x þ5y 1þ4y 2612;À4x À5y 1þ4y 26À4;4x À4y 1þ5y 264;À4x þ4y 1þ5y 264:ð14Þ12Y.Jiang et al./Knowledge-Based Systems 55(2014)9–14For Example3,following the same procedure as for Example2,we study the different optimal solutions for varying values of .Let z=(x1,y1,y2,k1,k2,k3,k4)T and set the initial point z(0)=(1,1,1,1,1,1,1)T.Parameters r,w(0),v(0),a,b,and e are selected as in Example1.Parameter is set to different small values and the results are also presented in Table1.From Table1,wefind that the computed results converge to an optimal solution with decreasing values of .Moreover,the solutions obtained by the proposed algorithm for the three examples with different values of ,except for the solution in Example2for =0.1,are better than those presented in literature based on a comparison of the objective value for different values of .Thus we arrive at the conclusion that the proposed algorithm is feasible for solving NBLP problems.6.ConclusionsBilevel programming problems are intrinsically non-convex and difficult to solve for a global optimum solution.In this paper,we presented an augmented Lagrangian multiplier method to solve NBLP problems.In this algorithm,the KKT condition is used in the lower level problem to transform the NBLP problem into a single nonlinear programming problem with complementary constraints.A CHKS smoothing function is adopted to avoid the dif-ficulty of dealing with a non-differentiable problem because of the complementary condition.An augmented Lagrangian multiplier method is then applied to solve the smoothed nonlinear program-ming problem to obtain the optimal solution of the NBLP problem. We analyzed the asymptotic properties of the method and derived the condition for solution optimality.Numerical results show that the proposed algorithm is feasible for solving NBLP problems. AcknowledgementsThis work was supported by the National Natural Science Foundation of China(Nos.50809004and41071323),the State Key Development Program for Basic Research of China(No. 2009CB421104)and the National Major Science and Technology Projects for Water Pollution Control and Management(Nos. 2012ZX07029-002,2012ZX07203-003).References[1]J.Lu,C.G.Shi,G.Q.Zhang,On bilevel multi-follower decision making:generalframework and solutions,Information Sciences176(2006)1607–1627.[2]O.Ben-Ayed,C.Blair,Computational difficulties of bilevel linear programming,Operations Research38(3)(1990)556–560.[3]J.F.Bard,Some properties of the bilevel linear programming,Journal ofOptimization Theory and Applications68(7)(1991)146–164.[4]L.Vicente,G.Savard,J.Judice,Descent approaches for quadratic bilevelprogramming,Journal of Optimization Theory and Applications81(1994) 379–399.[5]I.C.Herminia,G.Carmen,On linear bilevel problems with multiple objectivesat the lower 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optimal solution to the bilevel nonlinear programming.b The upper level’s objective function value of the bilevel nonlinear programming.Y.Jiang et al./Knowledge-Based Systems55(2014)9–1413。

基于可靠性的结构动态拓扑优化方法唐东峰;游世辉【摘要】Actual engineering structures are often designed using deterministic parameters,which may lead to high failure probability.This paper proposed a reliability-based structural dynamic topology method,in which structural reliability analysis was incorporated into the topology optimization procedure.The geometry dimensions and material volume were considered as uncertain parameters,and it was assumed that they obey a Gaussian distribution.It is a two-nested optimization problem when the structural reliability analysis is considered as constraints into the topology optimization,which results in low efficiency and cannot be used in practice.To this end,a new decouple strategy was proposed to decouple the reliability analysis from the topology optimization procedure.In this case,structure reliability analysis and dynamic topology optimization become two independent optimization cycles,and the computational efficiency is improved enormously.The design problem was then constructed so as to maximize the first eigenfrequency and to meet the volume and reliability requirement.SIMP and MMA were combined to successfully solve the design problem.The proposed method can produce various topologies that satisfy different reliability requirement,and its validity is demonstrated by one benchmark example.%实际工程结构设计往往在确定性范畴内进行,所得结构存在较大失效可能性.基于此,提出一种基于可靠性的连续体动态拓扑优化方法,将结构可靠性分析方法嵌套到连续体拓扑优化中.考虑了结构几何尺寸和材料体积的不确定性,并用高斯分布来度量.将结构可靠度作为约束嵌套到连续体拓扑优化中,属于二次嵌套优化问题,但计算效率低下,不适于工程应用.提出一种解耦策略将结构可靠性分析从连续体拓扑优化中解耦出来,使结构可靠性分析与动态拓扑优化为两个独立的优化循环,大大提高了计算效率,建立以结构基频最大为优化目标,满足一定体积约束和可靠度要求的优化问题,利用各向同性材料惩罚模型(SIMP)和移动渐进方法(MMA)求解该优化问题.所提方法可以得到满足不同可靠度要求的一系列最优结构,并用标准算例验证其有效性.【期刊名称】《湖南大学学报（自然科学版）》【年(卷),期】2017(044)010【总页数】6页(P62-67)【关键词】可靠性;不确定性;动态拓扑优化;解耦【作者】唐东峰;游世辉【作者单位】湘潭大学土木与力学学院,湖南湘潭411105;湖南科技大学信息与电气学院,湖南湘潭411201;湘潭大学土木与力学学院,湖南湘潭411105【正文语种】中文【中图分类】TM30连续体拓扑优化方法旨在满足一定约束条件下寻求材料最优分布,相对于尺寸优化[1]和形状优化[2],它有更多的设计自由度,同时也是更具挑战性的研究领域.自从1988年Bendsøe和Kikuchi[3]提出基于均匀化法的结构拓扑优化理论以来,连续体拓扑优化方法[4-10]经过近30年的发展,已经成功应用到各种领域,如航天[11]、碰撞[12]、汽车[13]和桥梁[14]等.其中,动态连续体拓扑优化问题[15-17]更具有挑战性,其难点在于优化过程中需要克服局部模态和频率交换现象,研究相对较少.尽管传统连续拓扑优化方法可以得到性能优越的设计结果,但是,其未考虑结构参数的不确定性.不确定性是工程结构的固有属性,在结构设计时不容忽略,否则设计出的产品会存在较大的失效风险,甚至可能造成灾难性的后果[18-19].考虑结构不确定性的优化设计方法一般分为两类：稳健性优化设计方法[20]和基于可靠性优化设计方法[21].前者为了降低不确定性变量对结构响应的敏感性,而后者使结构优化设计增加一个可靠度约束.基于可靠性的优化设计方法可以得到满足设计人员一系列不同可靠度要求的结构.而基于可靠性的连续体拓扑优化方法的研究,由于其设计变量数量庞大,且功能函数往往为隐式,更具有挑战性.Kharmanda等[22]首次将结构可靠性分析引入到连续体拓扑优化中,研究发现所提方法设计出的结构比传统确定性方法所得结构更加可靠.在其研究基础上,该领域越来越受到国内外学者的关注,出现大量研究成果[23-27].最近,Zhao等[28]提出一种高效的基于可靠性连续体拓扑优化方法,利用随机响应面显式表达结构失效功能函数.Liu等[29]结合一次二阶矩法和描述函数拓扑优化方法探讨了一种可以获得光滑边界且满足一定可靠度要求的优化方法.Jalalpour和Tootkaboni[30]考虑材料的不确定性,提出了一种高效的基于可靠性的连续体拓扑优化方法,假设材料服从相关对数正态分布,并利用随机摄动法近似得到结构响应,使结构功能函数近似显式化.然而,上述研究均是静力学方面的研究,动力学方面的研究甚少.动态拓扑优化问题研究本身就较静力学拓扑优化繁琐,将可靠性分析与动态拓扑优化相结合更加具有挑战性.工程中存在大量承受动载荷的结构,这些结构较之静力下的结构更容易失效,因此,考虑结构可靠性的动态连续体拓扑优化方法的研究具有重要理论意义和工程实际价值.基于此,本文提出一种基于可靠性的动态连续体拓扑优化方法,尝试解决自由振动结构拓扑优化时考虑结构可靠度的难题.实际工程制造误差可能造成结构几何尺寸和体积的不确定性,假设这些不确定性变量服从高斯分布,利用一次二阶矩法计算结构的可靠度(可靠性指标),并作为约束参与到动态拓扑优化循环中.而结构可靠性分析也属于一个优化过程,因此,整个优化过程属于二次嵌套优化.考虑到连续体拓扑优化设计变量数量庞大,导致计算效率十分低下,无法实际应用，为突破该瓶颈,提出一种解耦策略,将二次嵌套优化循环解耦成两个独立的单独优化循环,大大提高了计算效率.建立以结构第一阶固有频率最大为优化目标,以体积和可靠度指标为约束的连续拓扑优化问题,用各向同性材料惩罚模型惩罚连续设计变量,并利用移动渐进方法求解该优化问题.所提方法可以得到一系列满足设计人员不同可靠度需求的拓扑结构,具有重要的工程意义.工程结构受到动态载荷时,设计人员往往希望结构固有频率偏离驱动频率,以防止共振发生.未考虑阻尼时,自由振动系统的微分方程为：式中:M为系统质量矩阵;K为系统刚度矩阵;u和分别为系统位移和加速度;F为系统所受外力,自由振动系统中其为一个零向量.对式(1)进行拉普拉斯变换：MU(s)s2+KU(s)=0令s=jω,可以得到：式中:ωi为系统第i阶固有频率;ui是响应的特征向量.固有频率可以写成：式中:ki和mi分别为模态刚度矩阵和模态质量矩阵.在动力学拓扑优化研究中,比较常用的研究目标是使结构的第一阶固有频率最大,并受一定的体积约束,即:to find:ρ1,…,ρe,…,ρNsubject to::0<ρmin≤ρe≤1,e=1,…,N.式中:ρe为单元密度；ρmin的设定是为了防止发生奇异解,一般取ρmin=0.001;N 为单元个数;ve和V0分别为单元体积和设计允许的最终体积.利用各向同性材料惩罚模型对连续密度进行惩罚,文中分别对单元刚度矩阵和单元质量矩阵进行惩罚,惩罚策略如下:Ke=(ρe)p1K0Me=(ρe)p2M0式中:Ke和Me分别为单元刚度矩阵和单元质量矩阵;K0和M0分别为实体单元刚度矩阵和单元质量矩阵;ρ1和ρ2为值均大于1的惩罚系数,为了在优化过程中抑制中间密度的产生,传统方法视ρ1和ρ2为相同的值,但是研究发现,这样对中间密度的抑制并不理想.因此,本文采用一种新的惩罚策略,取ρ1和ρ2为不同的值,即:当ρ≤ρe≤1,p1=3;当ρmin≤ρe≤0.1,p1=0.01;当0<ρmin≤ρe≤1,p2=1.问题(5)是一个非凸优化问题,可用移动渐进方法(MMA)[31]进行高效求解.MMA将初始非凸优化问题近似为许多独立的、小的凸优化子问题,并对之进行高效求解.利用MMA需要首先得到目标函数对设计变量的一阶偏导数:==将材料惩罚策略代入式(8)中,得到:=以式(9)为梯度信息,MMA可以快速且高效求解式(5)的动态拓扑优化问题.但是,不确定性是结构设计中无法避免的,例如,由于制造误差造成的结构尺寸不确定性、材料自身的不确定性和结构使用过程中加载力的不确定性等.而上述动态拓扑优化问题并未考虑结构的不确定性,所设计出的结构往往存在较高的失效概率.因此,文章尝试将结构不确定性引入到动态拓扑优化中,设计出满足一定结构可靠度要求的结构. 基于可靠性的连续体拓扑优化方法有较多学者进行了研究,但大多数局限于静力学结构的优化设计.将可靠性分析与动态拓扑优化问题相结合更具有挑战性.2.1 问题提出基于可靠性的动态拓扑优化方法旨在提高所设计结构的可靠度水平,当结构可靠度作为拓扑优化的一个约束时,利用可靠性指标法,优化问题(5)变为:to find:ρ1,…,ρe,…,ρNsuject to:0<ρmin≤ρe≤1,e=1,…,N,其中,结构可靠性指标的求解如下:‖η‖suject to:Gj(d,η)=0式中:g1和g2为不等式约束,分别控制体积和结构可靠度和βj分别为第j个结构可靠性指标目标值和第j个结构可靠性指标;m指结构系统一共有m个可靠度约束;μγ为结构不确定性变量均值;Gj(·)为在标准正态空间中第j个结构的功能函数,当Gj(·)>0时认为结构处于可靠状态,当Gj(·)<0时认为结构处于失效状态,Gj(·)=0时为结构极限状态;‖·‖指向量的范数;η为不确定性变量在标准正态坐标系中的形式. 利用伴随法可以求得式(10)优化问题的敏感系数,然后利用MMA进行求解.然而,该问题求解属于二次嵌套优化问题,即结构动态拓扑优化每次迭代循环都需要判断该次迭代结果是否满足所需结构可靠度要求,而结构可靠性分析也是一个优化问题.直接求解计算效率十分低下,不适合实际工程应用.因此,文章提出一种解耦策略,将二次嵌套优化问题转化为两个单独的优化循环,大大提高了计算效率.2.2 解耦策略不确定性变量用Y度量,Y=(Y1,…,Yt),结构拥有t个不确定变量,其均值和方差分别为μγ和这些不确定性变量对结构可靠性分析具有重要影响,然而,它们对结构固有频率并不一定有很大的影响.因此,需要首先计算目标函数对不确定性变量的敏感性,选择对目标函数影响大的不确定性变量,去除影响小的不确定性变量.这里使用有限差分法[32]:==式中：=0.005.确定对目标函数贡献大的不确定性变量后,需要确定可靠性指标对不确定性变量的梯度信息以每个迭代步更新可靠性指标.不确定性变量首先转化为标准正态分布:η=此时,结构的可靠性指标为:β=可靠性指标对不确定变量的一阶导数可以显式表达:ε==利用式(15)的梯度信息可以在每一迭代步更新结构可靠性指标的值.为了将结构可靠性分析解耦出来,首先给出一个初始猜测值η,将该猜测值代入式(14)中计算可靠性指标.若该可靠性指标大于目标值,利用式(15)梯度信息更新其数值,直到满足目标值为止.此时,得到一个新的η*,进而可以得到在物理坐标系下的不确定性变量:Y*=η*σY+μγ上式得到的是一个确定性向量,使用其作为拓扑优化的输入量所得到的拓扑结构,可以满足所需结构可靠度要求.这样,结构可靠性分析成功地从拓扑优化循环中解耦出来,大大提高了计算效率.需要指出的是,由于文章只关心线弹性结构的优化设计,故该解耦策略的精度可以接受.使用解耦策略的算法流程如图1所示.用一个标准算例来验证文章所提方法的有效性,同时说明动态拓扑优化设计考虑结构可靠性的重要性.这里只是为了验证算法的有效性,所有单位均无量纲处理.同时,使用了防止优化过程中棋盘格现象的策略[33],得到边界光滑的拓扑结构.两端固定梁的动态拓扑优化设计问题,如图2所示,长和宽分别为100和20,假设均服从正态分布,且离差均为0.02.初始域被离散为100×20个矩形单元.设计域材料的属性为：弹性模量为1,泊松比为0.3,密度为1，体积约束设为0.9,假设服从正态分布量,离差为0.02.确定性动态拓扑优化所得结果如图3所示,拓扑结构边界光滑,便于实际结构制造.图4表明利用MMA求解动态拓扑优化问题收敛性良好,迭代106步后收敛,结构的第一阶固有频率为0.574 7,图中也给出了结构的第二阶和第三阶固有频率.然而,在实际制造加工中误差无法避免,由误差导致的结构不确定性，在图3结构中并未考虑,该结构在实际工程服役过程中往往会存在较大的失效概率.利用文中所提方法优化设计该梁结构,设定结构可靠性指标值β0=4.0,利用一次二阶矩可得结构的失效概率为pf=Φ(-β0)=0.000 032,所得拓扑结构(图5所示)有很高的可靠度,可以很好地应对复杂多变的不确定性服役环境,如制造误差等.所得拓扑结构明显与图3不同,可以更好地抵抗结构不确定性的变化.图6显示所提算法收敛性良好,目标函数迭代90步收敛为0.520 3.需要指出的是,图5中结构最终材料体积为0.916,大于确定性优化结果的0.9,这是由于结构需要更多的材料来提高自身的可靠度,这也和文献[26,27]研究结果一致.文章将结构可靠性分析引入到连续体动态拓扑优化中,提出一种基于可靠性的连续体拓扑优化方法.同时,为了提高计算效率,提出一种新的解耦策略,将结构可靠度计算约束从拓扑优化循环中解耦出来,大大提高了计算效率.得出以下结论;1)考虑了由于制造误差等造成的几何尺寸和体积不确定性,设计出的结构可以满足设计人员的可靠度要求,可以更好地应对复杂的不确定性的服役环境.2)设计出的结构相对于确定性优化结构拥有更多的材料,拥有更高的可靠度.3)所提算法有很好的收敛性,迭代过程平稳,并得到边界光滑的拓扑结构.†通讯联系人,E-mail:*********************【相关文献】[1] LI C,KIM I Y,JESWIET J. 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数学中的凸优化与凸分析凸优化（Convex Optimization）是数学中一个重要的研究领域，旨在解决凸函数的优化问题。

凸分析（Convex Analysis）则是凸优化的理论基础，探讨凸集合和凸函数的性质。

本文将介绍凸优化与凸分析的基本概念和原理，以及其在各个领域中的应用。

一、凸集合与凸函数1.1 凸集合在数学中，凸集合是指任意两点之间的连线上的点也属于该集合。

具体地，对于一个集合A，若对于该集合中的任意两点x和y，以及任意的t（0≤t≤1），都有tx + (1-t)y ∈ A，则该集合A为凸集合。

凸集合具有许多良好的性质，例如，凸集合的交集仍为凸集合，凸集合加凸集合的运算结果仍为凸集合。

1.2 凸函数凸函数是定义在凸集合上的实值函数，满足函数图像上的任意两点之间的连线位于函数图像上方。

具体地，对于一个凸集合A上的函数f(x)，若对于该凸集合上的任意两点x和y，以及任意的t（0≤t≤1），都有f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y)，则该函数f(x)为凸函数。

凸函数具有许多重要的性质，例如，凸函数的局部最小值就是全局最小值，凸函数加凸函数仍为凸函数。

二、凸优化问题凸优化问题是指在满足一定约束条件下，求解凸函数的最优值问题。

一般形式的凸优化问题可以表示为：minimize f(x)subject to g_i(x) ≤ 0, i = 1,2,...,mh_i(x) = 0, i = 1,2,...,p其中，f(x)为目标函数，g_i(x)和h_i(x)分别为不等式约束和等式约束。

凸优化具有许多良好的性质，例如，任意局部最小值就是全局最小值。

凸优化问题可以通过各种数值方法进行求解，常用的方法包括梯度下降法、牛顿法和内点法等。

这些方法对于大规模的凸优化问题具有较高的收敛速度和求解精度。

三、凸优化与凸分析的应用凸优化与凸分析在众多领域中具有广泛的应用，下面将列举几个典型的应用领域。

理解凸优化导⾔凸优化（convex optimization）是最优化问题中⾮常重要的⼀类，也是被研究的很透彻的⼀类。

对于机器学习来说，如果要优化的问题被证明是凸优化问题，则说明此问题可以被⽐较好的解决。

在本⽂中，SIGAI将为⼤家深⼊浅出的介绍凸优化的概念以及在机器学习中的应⽤。

凸优化简介在SIGAI之前的公众号⽂章“理解梯度下降法”中我们介绍了最优化的基本概念以及梯度下降法。

如果读者对⽬标函数，优化变量，可⾏域，等式约束，不等式约束，局部极⼩值，全局极⼩值的概念还不清楚，请先阅读那篇⽂章。

求解⼀个⼀般性的最优化问题的全局极⼩值是⾮常困难的，⾄少要⾯临的问题是：函数可能有多个局部极值点，另外还有鞍点问题。

对于第⼀个问题，我们找到了⼀个梯度为0的点，它是极值点，但不是全局极值，如果⼀个问题有多个局部极值，则我们要把所有局部极值找出来，然后⽐较，得到全局极值，这⾮常困难，⽽且计算成本相当⾼。

第⼆个问题更严重，我们找到了梯度为0的点，但它连局部极值都不是，典型的是x3这个函数，在0点处，它的导数等于0，但这根本不是极值点：梯度下降法和⽜顿法等基于导数作为判据的优化算法，找到的都导数/梯度为0的点，⽽梯度等于0只是取得极值的必要条件⽽不是充分条件。

如果我们将这个必要条件变成充分条件，即：问题将会得到简化。

如果对问题加以限定，是可以保证上⾯这个条件成⽴的。

其中的⼀种限制⽅案是：对于⽬标函数，我们限定是凸函数；对于优化变量的可⾏域（注意，还要包括⽬标函数定义域的约束），我们限定它是凸集。

同时满⾜这两个限制条件的最优化问题称为凸优化问题，这类问题有⼀个⾮常好性质，那就是局部最优解⼀定是全局最优解。

接下来我们先介绍凸集和凸函数的概念。

凸集对于n维空间中点的集合C，如果对集合中的任意两点x和y，以及实数0 ≤ θ ≤ 1，都有：则称该集合称为凸集。

如果把这个集合画出来，其边界是凸的，没有凹进去的地⽅。

直观来看，把该集合中的任意两点⽤直线连起来，直线上的点都属于该集合。

DOI : 10.11992/tis.202006046非光滑凸情形Adam 型算法的最优个体收敛速率黄鉴之1，丁成诚1，陶蔚2，陶卿1（1. 中国人民解放军陆军炮兵防空兵学院 信息工程系，安徽 合肥 230031; 2. 中国人民解放军陆军工程大学 指挥控制工程学院，江苏 南京 210007）l 1摘 要：Adam 是目前深度神经网络训练中广泛采用的一种优化算法框架，同时使用了自适应步长和动量技巧，克服了SGD 的一些固有缺陷。

但即使对于凸优化问题，目前Adam 也只是在线学习框架下给出了和梯度下降法一样的regret 界，动量的加速特性并没有得到体现。

这里针对非光滑凸优化问题，通过巧妙选取动量和步长参数，证明了Adam 的改进型具有最优的个体收敛速率，从而说明了Adam 同时具有自适应和加速的优点。

通过求解 范数约束下的hinge 损失问题，实验验证了理论分析的正确性和在算法保持稀疏性方面的良好性能。

关键词：机器学习；AdaGrad 算法；RMSProp 算法；动量方法；Adam 算法；AMSGrad 算法；个体收敛速率；稀疏性中图分类号：TP181 文献标志码：A 文章编号：1673−4785(2020)06−1140−07中文引用格式：黄鉴之, 丁成诚, 陶蔚, 等. 非光滑凸情形Adam 型算法的最优个体收敛速率[J]. 智能系统学报, 2020, 15(6):1140–1146.英文引用格式：HUANG Jianzhi, DING Chengcheng, TAO Wei, et al. Optimal individual convergence rate of Adam-type al-gorithms in nonsmooth convex optimization[J]. CAAI transactions on intelligent systems, 2020, 15(6): 1140–1146.Optimal individual convergence rate of Adam-type algorithms innonsmooth convex optimizationHUANG Jianzhi 1，DING Chengcheng 1，TAO Wei 2，TAO Qing 1(1. Department of Information Engineering, Army Academy of Artillery and Air Defense of PLA, Hefei 230031, China; 2. Command and Control Engineering, Army Engineering University of PLA, Nanjing 210007, China)Abstract : Adam is a popular optimization framework for training deep neural networks, which simultaneously employs adaptive step-size and momentum techniques to overcome some inherent disadvantages of SGD. However, even for the convex optimization problem, Adam proves to have the same regret bound as the gradient descent method under online optimization circumstances; moreover, the momentum acceleration property is not revealed. This paper focuses on nonsmooth convex problems. By selecting suitable time-varying step-size and momentum parameters, the improved Adam algorithm exhibits an optimal individual convergence rate, which indicates that Adam has the advantages of both adaptation and acceleration. Experiments conducted on the l 1-norm ball constrained hinge loss function problem verify the correctness of the theoretical analysis and the performance of the proposed algorithms in keeping the sparsity.Keywords : machine learning; AdaGrad algorithm; RMSProp algorithm; momentum methods; Adam algorithm; AMS-Grad algorithm; individual convergence rate; sparsityAdam 是目前深度学习中广泛采用的一种优化算法[1]。

双层规划一、双层规划的定义及背景双层规划（，简称）是一种具有二层递阶结构的系统优化问题，上层问题和下层问题都有各自的决策变量、约束条件和目标函数。

双层系统优化研究的是具有两个层次系统的规划与管理问题。

上层决策者只是通过自己的决策去指导下层决策者，并不直接干涉下层的决策；而下层决策者只需要把上层的决策作为参数，他可以在自己的可能范围内自由决策。

这种决策机制使得上层决策者在选择策略以优化自己的目标达成时，必须考虑到下层决策者可能采取的策略对自己的不利影响。

首先提出层次规划模型的是，上世纪50年代，为了更好的描述现实中的经济模式，在他的专著中首次提出了层次规划这种概念，虽然多层规划与之有共同点，但各层决策者依次做出决策，并且各自的策略集也不必再是分离的。

20世纪60年代，和提出了大规模线性规划的分解算法，承认有一个核心决策者，它的目标高于一切，但与多层规划有很大区别，多层规划承认有最高决策者，大不是绝对的，他允许下层决策者有各自不同的利益。

20世纪70年代发展起来的多目标规划通常寻求的是一个决策者的互相矛盾的多个目标额折衷解，而多层规划强调下层决策对上层目标的影响，并且多层规划问题通常不能逐层独立求解。

上世纪70年代以来，在解决实际问题的过程中，人们才逐渐形成多层规划的概念和方法。

多层规划（）一词是和在奶制品工业模型和墨西哥农业模型的研究报告中首先提出来的。

上世纪70年代，人们对多目标规划进行了深入的研究，也形成了一些求解多目标规划的有效方法，如分层优化技术，这种技术也可以用来求解层次问题，但这种技术建立在下层的决策不影响上层的目标基础上，而多层规划正是强调下层决策对上层目标的影响。

因此多层规划同城不同于多目标规划。

在过去的几十年中，多层规划的理论、方法及应用都有了很大的发展，并且已经成为规划论中的一个新的重要分支，而在多多层规划的研究中，双层规划是一个重要的研究对象，这是因为双层规划是多层规划中的一个特例，同时多层规划可以看作是一系列的双层规划的复合。