第五章(优化设计).ppt

1、第五章第五章 优化设计优化设计5.1 概述（1）无约束优化问题例：（2）约束优化问题（3）线性规划问题目标函数和约束函数都是线性函数的优化问题。（4）多目标规划问题目标函数个数大于1的约束或无约束优化问题5.2 优化问题的求解方法最佳步长a*一般采取数值法求解下降方向d不同方法对应不同优化算法5.3优化问题的matlab求解函数5.3.1 fminunc函数：求无约束目标函数最小值用法：x=fminunc(fun,x0)x=fminunc(fun,x0,options)x,fval=fminunc(.)x,fval,exitflag=fminunc(.)x,fval,exitflag,outp

2、ut=fminunc(.)x,fval,exitflag,output,grad=fminunc(.)x,fval,exitflag,output,grad,hessian=fminunc(.)X:极小点Fval:极小值Fun:目标函数名字X0:初值,列 向量Option:选项Exitflag:=1求解成功,其他值表示不成功Output：保存输出状态的变量Grad：保存梯度计算值的变量Hessian：保存目标函数Hessian矩阵的变量强烈建议使用格式x y exit=fminunc(fun,x0,options)计算设置：op=optimset(MaxIter,10000);op=optim

3、set(op,TolFun,1.e-4);op=optimset(op,TolX,1.e-6);x y exit=fminunc(f2,1;2,op);function f=f2(X)y1=X(1)+cos(X(2);y2=sin(X(1)+X(2)+10;f=y1*y1+y2*y2;原选项名称，对原选项值进行修改Exit flag标志：1:Magnitude of gradient smaller than the specified tolerance.梯度幅度小于给定误差2:Change in x was smaller than the specified tolerance.设计变量

4、X变化小于给定误差3:Change in the objective function value was less than the specified tolerance.目标函数值变化小于给定误差0:Number of iterations exceeded options.MaxIter or number of function evaluations exceeded options.FunEvals.循环次数超出给定值-1:Algorithm was terminated by the output function.失败-2:Line search cannot find an

5、 acceptable point along the current search direction.失败使用fminunc函数时，强烈建议对循环次数、梯度误差、设计变量误差、目标函数误差进行设置。MaxIter：Maximum number of iterations allowed.TolFun：Termination tolerance on the function value.TolX：Termination tolerance on x例：（a）定义目标函数（b）调用fminunc函数(c)根据返回结果，如果计算不成功（可用计算点判断它是否满足约束条件），修改计算选项，直至计算

6、结果满意。5.3.2 线性规划求解函数数学模型：min:cTxs.t:AX=b，b的分量可以小于0 AeqX=beq Aeq:等式约束矩阵;beq:等式约束b向量 lb=X=ub，如果设计变量大于0，需要设置lb=0向量 lb:左边界向量,ub:右边界向量用法:x=linprog(c,A,b)x=linprog(c,A,b,Aeq,beq)x=linprog(c,A,b,Aeq,beq,lb,ub)x=linprog(c,A,b,Aeq,beq,lb,ub,x0)x=linprog(c,A,b,Aeq,beq,lb,ub,x0,options)x,fval=linprog(.)x,fval,e

7、xitflag=linprog(.)x,fval,exitflag,output=linprog(.)x,fval,exitflag,output,lambda=linprog(.)Set Aeq=and beq=if no equalities exist.Set A=and b=if no inequalities exist.Set lb=and ub=if no boudary exist英文Help说第2个参数为Lagrange乘子参数,是错误的，实际上仍为目标函数值1:function converged to a solution x.0:Number of iterations

8、 exceeded options.MaxIter.-2:No feasible point was found.-3:Problem is unbounded.-4:NaN value was encountered during execution of the algorithm.-5:Both primal and dual problems are infeasible.-7:Search direction became too small.No further progress could be made.例：某车间要100套钢架，每套钢架由2.0m,2.1m,1.5m长的钢材各

9、一段组成，现有钢材每条240m，如何分配才能使尾料最少。段数套1套2套3套4套52.9120102.1002211.531203尾料00.10.20.30.42.9x1+5.8x2+2.9x3=2404.2x3+4.2x4+2.1x5=2404.5x1+1.5x2+3x3+4.5x5=0Min0.1x2+0.2x3+0.3x4+0.4x55.3.3 二次规划求解函数quadprog数学模型：x=quadprog(G,q,A,b)x=quadprog(G,q,A,b,Aeq,beq)x=quadprog(G,q,A,b,Aeq,beq,lb,ub)x=quadprog(G,q,A,b,Aeq,b

10、eq,lb,ub,x0)x=quadprog(G,q,A,b,Aeq,beq,lb,ub,x0,options)返回标志：1:Function converged to a solution x.3:Change in the objective function value was smaller than the specified tolerance.4:Local minimizer was found.0:Number of iterations exceeded options.MaxIter.-2:Problem is infeasible.-3:Problem is unbou

11、nded.-4:Current search direction was not a direction of descent.No further progress could be made.-7:Magnitude of search direction became too small.No further progress could be made.x,fval=quadprog(.)x,fval,exitflag=quadprog(.)x,fval,exitflag,output=quadprog(.)x,fval,exitflag,output,lambda=quadprog(

12、.)例：5.3.4 约束优化函数fmincon返回向量的函数用法：x=fmincon(fun,x0,A,b)x=fmincon(fun,x0,A,b,Aeq,beq)x=fmincon(fun,x0,A,b,Aeq,beq,lb,ub)x=fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)x=fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)x,fval=fmincon(.)x,fval,exitflag=fmincon(.)x,fval,exitflag,output=fmincon(.)x,fval,exitf

13、lag,output,lambda=fmincon(.)x,fval,exitflag,output,lambda,grad=fmincon(.)x,fval,exitflag,output,lambda,grad,hessian=fmincon(.)Fun:目标函数Nonlcon:非线性约束函数 注意：（1）Set A=and b=if no inequalities exist（2）Set Aeq=and beq=if no equalities exist.Set lb(i)=-Inf if x(i)is unbounded below,and set ub(i)=Inf if x(i)

14、is unbounded above.（3）set lb=and/or ub=if no bounds exist.（4）Set nonlcon=if there are no nonlinear inequality or equality constraints.（5）inequalities c(x)or equalities ceq(x)is defined in nonlcon.用户不自己定义梯度求解函数时，目标函数定义：function f=myfun(x)f=.%Compute function value at x用户自己定义梯度求解函数，并且选项 GradObj 为 on即定

15、义了选项：options=optimset(GradObj,on)目标函数必须如下定义：function f,g=myfun(x)f=.%Compute the function value at xif nargout 1%fun called with two output arguments g=.%Compute the gradient evaluated at xendif the Hessian matrix can also be computed and the Hessian option is on,i.e.,options=optimset(Hessian,on),目标

16、函数必须如下定义：function f,g,H=myfun(x)f=.%Compute the objective function value at xif nargout 1%fun called with two output arguments g=.%Gradient of the function evaluated at xEndif nargout 2 H=.%Hessian evaluated at xend非线性约束用法：x=fmincon(myfun,x0,A,b,Aeq,beq,lb,ub,mycon)非线性约束必须如下定义：function c,ceq=mycon(x)c=.%Compute nonlinear inequalities at x.ceq=.%Compute nonlinear equalities at x.如果选项 GradConstr设置 为 on,as set byoptions=optimset(GradConstr,on)非线性约束函数必须如下定义：无不等式约束时，使用语句C=;或使用Ceq=function c,ceq,GC,GC