多目標(biāo)優(yōu)化課件

1、多目標(biāo)優(yōu)化與決策的基本方法基于Matlab的一種現(xiàn)代方法Elementary Method for Multiobjective Optimization & DecisionA Modern Approach in Matlab系統(tǒng)工程概論單元討論目標(biāo)Unit Object教科書多目標(biāo)優(yōu)化問題非劣解及約束法目標(biāo)規(guī)劃法擴(kuò)展到工程領(lǐng)域與非線性優(yōu)化Matlab R14 SP1的Optimization Toolbox詳細(xì)講解fgoalattain，大致對應(yīng)目標(biāo)規(guī)劃法，但是擴(kuò)展到非線性多目標(biāo)優(yōu)化的數(shù)學(xué)描述Multiobjective optimization model注意，這里以及之后的所有講述同

2、時適合于線性和非線性的多目標(biāo)優(yōu)化多目標(biāo)優(yōu)化的非劣解集Noninferior solution for the model例如：A,B點屬于非劣解，因為不滿足定義條件目標(biāo)規(guī)劃法Goal Attainment Method 引入目標(biāo)概念：F*，令非劣解集到目標(biāo)的距離（或稱范數(shù)）最小，選出一個非劣解。Wi引入了一個松弛度的概念，松弛度最小的一個非劣解就是對于目標(biāo)F*的最可行解。優(yōu)點：不漏解，目標(biāo)明確，計算量小。缺點：對于非線性規(guī)劃設(shè)計：運用連續(xù)二次形規(guī)劃(SQP - sequential quadratic programming)，線性的權(quán)值松弛在局部搜索范圍內(nèi)，會導(dǎo)致拒絕可大幅改進(jìn)總體目標(biāo)的小步

3、搜索。只針對連續(xù)問題，可能只能給出局部最優(yōu)解。改進(jìn)：閱讀Matlab Optimization Toolbox 3.0.1 Users Guide中Algorithm Improvements for Goal Attainment Method一節(jié)內(nèi)容。Matlab優(yōu)化工具箱Matlab Optimization Toolbox優(yōu)化工具箱提供了一般和大型的非線性優(yōu)化函數(shù)，同時還提供了線性規(guī)劃，二次規(guī)劃，非線性最小二乘以及非線性方程求解的工具。主要特性：無約束非線性極小化問題約束性線性極小化、極大極小、多目標(biāo)優(yōu)化，半無窮極小化問題。二次規(guī)劃和線性規(guī)劃問題非線性最小二乘和邊界曲線擬合問題非線性系

4、統(tǒng)方程求解問題約束線性最小二乘問題大型問題的特殊算法fgoalattain實例線性系統(tǒng)輸出反饋極點配置BC+AuxyK+線性系統(tǒng)理論P283頁結(jié)論6.15對于完全能控和能觀測n維聯(lián)系LTI系統(tǒng)，設(shè)rank(B)=p, rank(C)=q，則采用輸出反饋u=Ky+v，可隊數(shù)目為minn, p+q-1的閉環(huán)系統(tǒng)極點進(jìn)行“任意接近”式配置，即使其可任意地接近人給的期望極點位置。vPractical fgoalattain seeking KA = -0.5 0 0; 0 -2 10; 0 1 -2 ;B = 1 0; -2 2; 0 1 ;C = 1 0 0; 0 0 1 ; % Suppose w

5、e wish to design an output feedback controller, x, to have % poles to the left of the location -5, -3, -1 in the complex plane. % The controller must not have any gain element exceeding an absolute % value of 4.goal = -5, -3, -1 % Set the weights equal to the goals to ensure same percentage % under-

6、 or over-attainment in the goals.weight = abs(goal) % Initialize output feedback controllerx0 = -1 -1; -1 -1; % Set upper and lower bounds on the controllerlb = repmat(-4,size(x0) ub = repmat(4,size(x0) % Create a vector-valued function eigfun that returns the eigenvalues of the % closed loop system

7、. This function requires additional parameters (namely, % the matrices A, B, and C); the most convenient way to pass these is through % an anonymous function:eigfun = (x) sort(eig(A+B*x*C) % To begin the optimization we call FGOALATTAIN:x,fval,attainfactor,exitflag,output,lambda = . fgoalattain(eigf

8、un,x0,goal,weight,lb,ub,options); % The attainment factor indicates the level of goal achievement.% A negative attainment factor indicates over-achievement, positive% indicates under-achievement. The value attainfactor we obtained in % this run indicates that the objectives have been over-achieved b

9、y % about 39 percent: attainfactor % Suppose we now require the eigenvalues to be as near as possible% to the goal values, -5, -3, -1. % Set options.GoalsExactAchieve to the number of objectives that should % as near as possible to the goals (i.e., do not try to over-achieve): % All three objectives

10、 should be as near as possible to the goals.options = optimset(options,GoalsExactAchieve,3); % We are ready to call the optimization solver: x,fval,attainfactor,exitflag,output,lambda = . fgoalattain(eigfun,x0,goal,weight,lb,ub,options); % This time the eigenvalues of the closed loop system are as f

11、ollows:eigfun(x) % These values are also held in output fval % The attainment factor is the level of goal achievement. A negative % attainment factor indicates over-achievement, positive indicates % under-achievement. The low attainfactor obtained indicates that the% eigenvalues have almost exactly

12、met the goals: attainfactor教科書上工廠生產(chǎn)車輛優(yōu)化問題% fun_optim.mfunction y= fun_optim(x)y=zeros(1,2);y(1)= -(100*x(1)+90*x(2)+80*x(3)+70*x(4);y(2)=3*x(2)+2*x(4);% factory_goal.mA= -1 -1 0 0; 0 0 -1 -1; 3 0 2 0; 0 3 0 2;b=-30 -30 120 48;lb=zeros(1,4);x0=20,10,30,0;y0=10000, 40;x_opt=18 12 33 0;x fval=fgoalattain(fun_optim, x0, y0, 1 2e-4, A, b, , , lb, )教科書有誤，按照這個解才能算出書上(-5520, 36)的答案。工