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工程硕士研究生《最优化方法》试卷

工程硕士研究生《最优化方法》试卷
工程硕士研究生《最优化方法》试卷

华东理工大学工程硕士研究生《最优化方法》试卷

专业 班级 学号 姓名 成绩

2010年5月8日

一、简答题(20分,每小题5分)

1· 函数3()2710f x x x =-+在(,)-? 上有最优解吗?如果有,请说明是局部最优解还是全局最优解。

2·请给出最优化方法中下降迭代算法的一般步骤。

3·何谓下降方向?请给出一种判别方法?负梯度方向是下降最快的方向吗?为什么? 4·何谓可行方向?在最优点处有可行下降方向吗?为什么?

二·(15分)你学过哪些一维优化算法(一维搜索法)?请你说明它们之间的关系和特点。如果用黄金分割法求解该问题在区间[-1,1]上的最优解,要求近似精度达到0.01,那么需要迭代多少步就能找到所需要的解。

三·(10分)请用FR 共轭梯度法求解如下优化问题,只需迭代一次,并给出第二次迭代的搜索

方向,初始点 0(1,1)T x =

22212min ()23x R

f x x x ?=++ 四·(20分)1、请简述共轭方向法的一般步骤,并给出你所学过的共轭方向法。

2、简述拟牛顿算法的基本步骤,说明拟牛顿算法的特点。

3、请给出你所知道的直接算法,并说明每个算法的主要思想和特点,它们之间有关系吗? 五·(10分)试用惩罚函数法求解如下的优化问题

2..3min ()365s t x f x x x 3=-+

六·(15分)请考虑下述线性规划问题

12

121212max ()3423..23

,0f x x x x x s t x x x x =+ì+ ??í?+ ??3

1·请给出上述问题的基本可行解与相应顶点的对应关系。上述线性规划问题有不可行的基本解吗?如果有,请给出几何解释;

2·试求上述线性规划问题的最优解。

七·(10分)已知某工程问题中反应因素y 与因素x 和时间t 之间的关系为

),,,;,(=21k a a a t x f y

其中k a a a ,,,21 是待定参数,为确定这k 个参数,实验测得有关y x t ,,的m 组数据为:),,2,1=(==,=m i y y t t x x i i i 时,当

1. 请给出确定参数k a a a ,,,21 的优化问题的模型;

2.对于上述优化问题,你知道哪些优化方法可求解该问题?

最优化方法简明教程—centre

①图与网 破圈法:任取一个圈,去掉一条权最大的边,直到最小树。 避圈法:选最小权的边,避圈前进,直到最小树。 最短路算法: Dijkstra法:从V s给定P标号T标号λ标号(T标号变为P标号λ标号记位置) 反向追踪:列表,d1(V1,V j)→d k(V1,V j)=min(ωij+d k(V1,V i))据最小权反向追踪 网络优化: 最小截集最大流:找到最小截集(弧的集合) 标号法:开始,为的标号, 最小费用最大流: 邮递员问题:通过消灭奇点,找欧拉回路 网络计划图: 最早开始最晚开始机动时间 最早结束最晚结束自由时差 工期优化:人力,费用,工期优化。 费用率=(最短时间费用-正常时间费用)/(正常时间-最短时间)②排队论(保证服务质量,又减少费用) 顾客源→(排队规则)队列→(服务规则)服务机构→离去 服务规则:FCFS,LCFS,随机服务,PR

M(顾客到达)|A(服务时间)|1(服务台数)|∞(容量)|∞(顾客源) N(t)队长N q (t)排队长T(t)顾客逗留时间T q (t)顾客等待时间 L 平均队长L q 平均等待队长W 平均逗留时间W q 平均等待时间 R 为系统利用率 泊松流(M):无后效性;平稳性;单个性; P 1(t,t+Δt)=λΔt+o(Δt); o(Δt)=∑∞ 2P n (t,t+Δt);E ξ=D ξ=λt (t 时刻n 个顾客的概率) 负指数分布(M):无记忆性(P(T>t+s/t>s)=P(T>t));[0,t)至少到达一 个顾客1-P 0(t )=1-e -t λ,t>0 !)()(K t e t V K t k λλ-= ,2,1,0=K ?? ?<≥-=-0,00,1)(t t e t F t i λξ),2,1( =i 爱尔朗分布(E K ):(相当于泊松流到达后被k 个服务台均分顾客形成) (其中,t>0,E(T)=1/μ,Var(T)=1/μ2k ) )! 1()()(1 >-= --t e k t t f t k μμμ K=1为M ,k=∞定长分布D,k ≥30正态分布近似 G 表示一般相互独立的随机分布 Little 公式:(四者知一即可) μ1 + =q W W W L λ= q q W L λ= ρ+=q L L ∑∞ ==0 n n nP L ∑∑∞=∞ =+=-=s n n m s n q nP P s n L 0 )( 服务率:ρ=λ/μ(λ为到达μ为服务) 排队系统分析:

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