matlab在科学计算中的应用7+-+代数方程与最优化问题的求解.ppt

第七章代数方程与最优化问题的求解,代数方程的求解无约束最优化问题的计算机求解有约束最优化问题的计算机求解整数规划问题的计算机求解,7.1代数方程的求解7.1.1代数方程的图解法,一元方程的图解法例：ezplot(exp(-3*t)*sin(4*t+2)+4*exp(-0.5*t)*cos(2*t)-0.5,05)holdon,line(0,5,0,0)%同时绘制横轴,验证：symst;t=3.52028;vpa(exp(-3*t)*sin(4*t+2)+4*exp(-0.5*t)*cos(2*t)-0.5)ans=-.19256654148425145223200161126442e-4,二元方程的图解法例：ezplot(x2*exp(-x*y2/2)+exp(-x/2)*sin(x*y)%第一个方程曲线holdon%保护当前坐标系ezplot(x2*cos(x+y2)+y2*exp(x+y),方程的图解法仅适用于一元、二元方程的求根问题。,7.1.2多项式型方程的准解析解法,例：ezplot(x2+y2-1);holdon%绘制第一方程的曲线ezplot(0.75*x3-y+0.9)%绘制第二方程为关于x的6次多项式方程应有6对根。图解法只能显示求解方程的实根。,一般多项式方程的根可为实数，也可为复数。可用MATLAB符号工具箱中的solve()函数。最简调用格式：S=solve(eqn1,eqn2,eqnn)（返回一个结构题型变量S，如S.x表示方程的根。）直接得出根：（变量返回到MATLAB工作空间）x,=solve(eqn1,eqn2,eqnn)同上，并指定变量x,=solve(eqn1,eqn2,eqnn，x,),例：symsxy;x,y=solve(x2+y2-1=0,75*x3/100-y+9/10=0)x=-.98170264842676789676449828873194-.55395176056834560077984413882735-.35471976465080793456863789934944*i-.55395176056834560077984413882735+.35471976465080793456863789934944*i.35696997189122287798839037801365.86631809883611811016789809418650-1.2153712664671427801318378544391*i.86631809883611811016789809418650+1.2153712664671427801318378544391*iy=.19042035099187730240977756415289.92933830226674362852985276677202-.21143822185895923615623381762210*i.92933830226674362852985276677202+.21143822185895923615623381762210*i.93411585960628007548796029415446-1.4916064075658223174787216959259-.70588200721402267753918827138837*i-1.4916064075658223174787216959259+.70588200721402267753918827138837*i,验证eval(x.2+y.2-1)eval(75*x.3/100-y+9/10)ans=0,00,00,0-.1e-31,0.5e-30+.1e-30*i,0.5e-30-.1e-30*i,0由于方程阶次可能太高，不存在解析解。然而，可利用MATLAB的符号工具箱得出原始问题的高精度数值解，故称之为准解析解。,例：x,y,z=solve(x+3*y3+2*z2=1/2,x2+3*y+z3=2,x3+2*z+2*y2=2/4);size(x)ans=271x(22),y(22),z(22)ans=-1.0869654762986136074917644096117ans=.37264668450644375527750811296627e-1ans=.89073290972562790151300874796949,验证：err=x+3*y.3+2*z.2-1/2,x.2+3*y+z.3-2,x.3+2*z+2*y.2-2/4;norm(double(eval(err)eval可对字符型变量进行符号求解ans=1.4998e-027多项式乘积形式也可，如把第三个方程替换一下。x,y,z=solve(x+3*y3+2*z2=1/2,x2+3*y+z3=2,x3+2*z*y2=2/4);err=x+3*y.3+2*z.2-1/2,x.2+3*y+z.3-2,x.3+2*z.*y.2-2/4;norm(double(eval(err)%将解代入求误差ans=5.4882e-028,例：求解（含变量倒数）symsxy;x,y=solve(x2/2+x+3/2+2/y+5/(2*y2)+3/x3=0,.y/2+3/(2*x)+1/x4+5*y4,x,y);size(x)ans=261err=x.2/2+x+3/2+2./y+5./(2*y.2)+3./x.3,y/2+3./(2*x)+1./x.4+5*y.4;验证norm(double(eval(err)ans=8.9625e-030,例：求解（带参数方程）symsabxy;x,y=solve(x2+a*x2+6*b+3*y2=0,y=a+(x+3),x,y)x=1/2/(4+a)*(-6*a-18+2*(-21*a2-45*a-27-24*b-6*a*b-3*a3)(1/2)1/2/(4+a)*(-6*a-18-2*(-21*a2-45*a-27-24*b-6*a*b-3*a3)(1/2)y=a+1/2/(4+a)*(-6*a-18+2*(-21*a2-45*a-27-24*b-6*a*b-3*a3)(1/2)+3a+1/2/(4+a)*(-6*a-18-2*(-21*a2-45*a-27-24*b-6*a*b-3*a3)(1/2)+3,7.1.3一般非线性方程数值解,非线性方程的标准形式为f(x)=0函数fzero格式x=fzero(fun,x0)%用fun定义表达式f(x)，x0为初始解。x=fzero(fun,x0,options)x,fval=fzero()%fval=f(x)x,fval,exitflag=fzero()x,fval,exitflag,output=fzero()说明该函数采用数值解求方程f(x)=0的根。,例:求的根解：fun=x3-2*x-5;z=fzero(fun,2)%初始估计值为2z=2.0946formatlongopt=optimset(Tolx,1.0e-8);y=fzero(x3-2*x-5,2,opt)y=2.09455148211709,7.1.4一般非线性方程组数值解,格式：最简求解语句x=fsolve(Fun,x0)一般求解语句x,f,flag,out=fsolve(Fun,x0,opt,p1,p2,)若返回的flag大于0，则表示求解成功，否则求解出现问题，opt求解控制参数，结构体数据。获得默认的常用变量opt=optimset;用这两种方法修改参数（解误差控制量）opt.Tolx=1e-10;或set(opt.Tolx,1e-10),例：自编函数：functionq=my2deq(p)q=p(1)*p(1)+p(2)*p(2)-1;0.75*p(1)3-p(2)+0.9;OPT=optimset;OPT.LargeScale=off;x,Y,c,d=fsolve(my2deq,1;2,OPT)Optimizationterminatedsuccessfully:First-orderoptimalityislessthanoptions.TolFun.x=0.35700.9341Y=1.0e-009*0.12150.0964,c=1d=iterations:7funcCount:21algorithm:trust-regiondoglegfirstorderopt:1.3061e-010解回代的精度调用inline()函数：f=inline(p(1)*p(1)+p(2)*p(2)-1;0.75*p(1)3-p(2)+0.9,p);x,Y=fsolve(f,1;2,OPT);%结果和上述完全一致，从略。Optimizationterminatedsuccessfully:First-orderoptimalityislessthanoptions.TolFun.若改变初始值x0=-1,0T,x,Y,c,d=fsolve(f,-1,0,OPT);x,Y,kk=d.funcCountOptimizationterminatedsuccessfully:First-orderoptimalityislessthanoptions.TolFun.x=-0.98170.1904Y=1.0e-010*0.5619-0.4500kk=15初值改变有可能得出另外一组解。故初值的选择对解的影响很大，在某些初值下甚至无法搜索到方程的解。,例：用solve()函数求近似解析解symst;solve(exp(-3*t)*sin(4*t+2)+4*exp(-0.5*t)*cos(2*t)-0.5)ans=.67374570500134756702960220427474不允许手工选择初值，只能获得这样的一个解。可先用用图解法选取初值，再调用fsolve()函数数值计算,formatlongy=inline(exp(-3*t).*sin(4*t+2)+4*exp(-0.5*t).*cos(2*t)-0.5,t);ff=optimset;ff.Display=iter;t,f=fsolve(y,3.5203,ff)NormofFirst-orderTrust-regionIterationFunc-countf(x)stepoptimalityradius121.8634e-0095.16e-0051243.67694e-0193.61071e-0057.25e-0101Optimizationterminatedsuccessfully:First-orderoptimalityislessthanoptions.TolFun.t=3.52026389294877f=-6.063776702980306e-010,重新设置相关的控制变量，提高精度。ff=optimset;ff.TolX=1e-16;ff.TolFun=1e-30;ff.Display=iter;t,f=fsolve(y,3.5203,ff)NormofFirst-orderTrust-regionIterationFunc-countf(x)stepoptimalityradius121.8634e-0095.16e-0051243.67694e-0193.61071e-0057.25e-01013605.07218e-01001Optimizationterminatedsuccessfully:First-orderoptimalityislessthanoptions.TolFun.t=3.52026389244155f=0,例:求的根解：化为标准形式设初值点为x0=-5-5。functionF=myfun(x)F=2*x(1)-x(2)-exp(-x(1);-x(1)+2*x(2)-exp(-x(2);x0=-5;-5;%初始点options=optimset(Display,iter);%显示输出信息x,fval=fsolve(myfun,x0,options),NormofFirst-orderTrust-regionIterationFunc-countf(x)stepoptimalityradius0347071.22.29e+00411612003.415.75e+0031293147.0211.47e+0031312854.45213881415239.5271107151867.0412130.8162116.704219.0517242.4278812.2618270.0326580.7595110.2062.59307.03149e-0060.1119270.002942.510333.29525e-0130.001691326.36e-0072.5Optimizationterminated:first-orderoptimalityislessthanoptions.TolFun.x=0.567143031397360.56714303139736fval=1.0e-006*-0.40590960570519-0.40590960570519,例:求矩阵x使其满足方程，并设初始解向量为x=1,1;1,1。解：functionF=myfun(x)F=x*x*x-1,2;3,4;x0=ones(2,2);%初始解向量options=optimset(Display,off);%不显示优化信息x,Fval,exitflag=fsolve(myfun,x0,options)x=-0.12910.86021.29031.1612Fval=1.0e-003*0.1541-0.11630.0109-0.0243exitflag=1,7.2无约束最优化问题求解7.2.1解析解法和图解法,例：symst;y=exp(-3*t)*sin(4*t+2)+4*exp(-0.5*t)*cos(2*t)-0.5;y1=diff(y,t);%求取一阶导函数ezplot(y1,0,4)%绘制出选定区间内一阶导函数曲线,t0=solve(y1)%求出一阶导数等于零的点t0=1.4528424981725411893375778048840y2=diff(y1);b=subs(y2,t,t0)%并验证二阶导数为正b=7.8553420253333601379464405534591t=0:0.4:4;y=exp(-3*t).*sin(4*t+2)+4*exp(-0.5*t).*cos(2*t)-0.5;plot(t,y),单变量函数求最小值的,标准形式为:s.t.函数fminbnd格式x=fminbnd(fun,x1,x2)x=fminbnd(fun,x1,x2,options)x,fval=fminbnd()%fval为目标函数的最小值x,fval,exitflag=fminbnd()exitflag为终止迭代的条件,若exitflag0，收敛于x，exitflag=0，表示超过函数估计值或迭代的最大数字，exitflagx,fval,exitflag,output=fminbnd(x3+cos(x)+x*log(x)/exp(x),0,1)x=0.5223fval=0.3974exitflag=1output=iterations:9funcCount:11algorithm:goldensectionsearch,parabolicinterpolation,例:在0,5上求下面函数的最小值解：functionf=myfun(x)f=(x-3).2-1;x=fminbnd(myfun,0,5)x=3,7.2.2基于MATLAB的数值解法,例：f=inline(x(1)2-2*x(1)*exp(-x(1)2-x(2)2-x(1)*x(2),x);x0=0;0;ff=optimset;ff.Display=iter;x=fminsearch(f,x0,ff)IterationFunc-countminf(x)Procedure13-0.000499937initial24-0.000499937reflect72137-0.641424contractoutsideOptimizationterminatedsuccessfully:x=0.6111-0.3056,x=fminunc(f,0;.0,ff)DirectionalIterationFunc-countf(x)Step-sizederivative12-2e-0080.001-429-0.5846690.3043530.343316-0.6413970.9503220.00191422-0.6414240.984028-1.45e-008x=0.6110-0.3055比较可知fminunc()函数效率高于fminsearch()函数，但当所选函数高度不连续时，使用fminsearch效果较好。故若安装了最优化工具箱则应调用fminunc()函数。,7.2.3全局最优解与局部最优解,例：f=inline(exp(-2*t).*cos(10*t)+exp(-3*(t+2).*sin(2*t),t);%目标函数t0=1;t1,f1=fminsearch(f,t0);t1f1ans=0.9228-0.1547t0=0.1;t2,f2=fminsearch(f,t0);t2f2ans=0.2945-0.5436,symst;y=exp(-2*t)*cos(10*t)+exp(-3*(t+2)*sin(2*t);ezplot(y,0,2.5);set(gca,Ylim,-0.6,1)在t0,2.5内的曲线ezplot(y,-0.5,2.5);set(gca,Ylim,-2,1.2)在-0.5,2.5曲线t0=-0.2;t3,f3=fminsearch(f,t0);t3f3ans=-0.3340-1.9163,7.2.4利用梯度求解最优化问题,例：x,y=meshgrid(0.5:0.01:1.5);z=100*(y.2-x).2+(1-x).2;contour3(x,y,z,100),set(gca,zlim,0,310)%测试算法的函数,f=inline(100*(x(2)-x(1)2)2+(1-x(1)2,x);ff=optimset;ff.TolX=1e-10;ff.TolFun=1e-20;ff.Display=iter;x=fminunc(f,0;0,ff)Warning:Gradientmustbeprovidedfortrust-regionmethod;usingline-searchmethodinstead.DirectionalIterationFunc-countf(x)Step-sizederivative1210.5-4442023.01749e-0123.40397e-009-1.77e-013x=1.000001736959721.00000347608069,求梯度向量(比较引入梯度的作用，但梯度的计算也费时间）symsx1x2;f=100*(x2-x12)2+(1-x1)2;J=jacobian(f,x1,x2)J=-400*(x2-x12)*x1-2+2*x1,200*x2-200*x12functiony,Gy=c6fun3(x)目标函数编写：y=100*(x(2)-x(1)2)2+(1-x(1)2;%目标函数Gy=-400*(x(2)-x(1)2)*x(1)-2+2*x(1);200*x(2)-200*x(1)2;%梯度ff.GradObj=on;x=fminunc(c6fun3,0;0,ff)NormofFirst-orderIterationf(x)stepoptimalityCG-iterations196.38977e-0292.12977e-0121.6e-0141x=0.999999999999990.99999999999998,7.2.5非线性最小二乘,函数lsqnonlin格式x=lsqnonlin(fun,x0)x=lsqnonlin(fun,x0,lb,ub)x=lsqnonlin(fun,x0,lb,ub,options)%x0为初始解向量；fun为，i=1,2,m，lb、ub定义x的下界和上界,options为指定优化参数，若x没有界，则lb=，ub=。,例:求下面非线性最小二乘问题初始解向量为x0=0.3,0.4。解：先建立函数文件，并保存为myfun.m.由于lsqnonlin中的fun为向量形式而不是平方和形式，因此，myfun函数应由建立：k=1,2,10,functionF=myfun10(x)k=1:10;F=2+2*k-exp(k*x(1)-exp(k*x(2);然后调用优化程序：x0=0.30.4;x,resnorm=lsqnonlin(myfun10,x0)Optimizationterminated:normofthecurrentstepislessthanOPTIONS.TolX.x=0.25780.2578resnorm=124.3622,7.3有约束最优化问题的计算机求解6.3.1约束条件与可行解区域,有约束最优化问题的一般描述：对于一般的一元问题和二元问题，可用图解法直接得出问题的最优解。,例：用图解方法求解：x1,x2=meshgrid(-3:.1:3);%生成网格型矩阵z=-x1.2-x2;%计算出矩阵上各点的高度i=find(x1.2+x2.29);z(i)=NaN;%找出x12+x229的坐标，并置函数值为NaNi=find(x1+x21);z(i)=NaN;%找出x1+x21的坐标,置为NaNsurf(x1,x2,z);shadinginterp;max(z(:),view(0,90)ans=3,7.3.2线性规划问题的计算机求解,例：求解f=-21431;A=02142;345-1-1;B=54;62;Ae=;Be=;xm=0,0,3.32,0.678,2.57;ff=optimset;ff.LargeScale=off;%不使用大规模问题求解ff.TolX=1e-15;ff.TolFun=1e-20;ff.TolCon=1e-20;ff.Display=iter;x,f_opt,key,c=linprog(f,A,B,Ae,Be,xm,ff),Optimizationterminatedsuccessfully.x=19.78500.00003.320011.38502.5700f_opt=-89.5750key=1求解成功c=iterations:5algorithm:medium-scale:activesetcgiterations:,例：求解f=-3/4,150,-1/50,6;Aeq=;Beq=;A=1/4,-60,-1/50,9;1/2,-90,-1/50,3;B=0;0;xm=-5;-5;-5;-5;xM=Inf;Inf;1;Inf;ff=optimset;ff.TolX=1e-15;ff.TolFun=1e-20;TolCon=1e-20;ff.Display=iter;x,f_opt,key,c=linprog(f,A,B,Aeq,Beq,xm,xM,0;0;0;0,ff),Residuals:PrimalDualUpperDualityTotalInfeasInfeasBoundsGapRelA*x-bA*y+z-w-fx+s-ubx*z+s*wError-Iter0:9.39e+0031.43e+0021.94e+0026.03e+0042.77e+001Iter1:6.38e-0121.21e+0010.00e+0003.50e+0031.78e+000Iter10:0.00e+0006.15e-0260.00e+0001.70e-0244.10e-028Optimizationterminatedsuccessfully.x=-5.0000-0.19471.0000-5.0000f_opt=-55.4700key=1c=iterations:10cgiterations:0algorithm:lipsol,7.3.3二次型规划的求解,例：求解H=diag(2,2,2,2);f=-2,-4,-6,-8;OPT=optimset;OPT.LargeScale=off;%不使用大规模问题算法A=1,1,1,1;3,3,2,1;B=5;10;Aeq=;Beq=;LB=zeros(4,1);x,f_opt=quadprog(H,f,A,B,Aeq,Beq,LB,OPT)Optimizationterminatedsuccessfully.x=0.00000.66671.66672.6667f_opt=-23.6667（解的目标值应为30（-23.6667）6.3333),例:求解下面二次规划问题解：则,H=1-1;-12;f=-2;-6;A=11;-12;21;b=2;2;3;lb=zeros(2,1);x,fval,exitflag,output,lambda=quadprog(H,f,A,b,lb)x=%最优解0.66671.3333fval=%最优值-8.2222exitflag=%收敛1output=iterations:3algorithm:medium-scale:active-setfirstorderopt:cgiterations:,例:求二次规划的最优解maxf(x1,x2)=x1x2+3sub.tox1+x2-2=0解：化成标准形式：H=0,-1;-1,0;f=0;0;Aeq=11;x,fval,exitflag,output=quadprog(H,f,Aeq),?,6.3.4一般非线性规划问题的求解,例：求解目标函数：functiony=opt_fun1(x)y=1000-x(1)*x(1)-2*x(2)*x(2)-x(3)*x(3)-x(1)*x(2)-x(1)*x(3);非线性约束函数：functionc,ceq=opt_con1(x)ceq=x(1)*x(1)+x(2)*x(2)+x(3)*x(3)-25;8*x(1)+14*x(2)+7*x(3)-56;c=;,ff=optimset;ff.LargeScale=off;ff.Display=iter;ff.TolFun=1e-30;ff.TolX=1e-15;ff.TolCon=1e-20;x0=1;1;1;xm=0;0;0;xM=;A=;B=;Aeq=;Beq=;x,f_opt,c,d=fmincon(opt_fun1,x0,A,B,Aeq,Beq,xm,xM,opt_con1,ff)；x,f_opt,kk=d.funcCountx=3.51210.21703.5522f_opt=961.7152kk=%目标函数调用的次数108,A,B,Aeq,Beq只用于线性方程和不等式,简化非线约束函数functionc,ceq=opt_con2(x)ceq=x(1)*x(1)+x(2)*x(2)+x(3)*x(3)-25;c=;求解：x0=1;1;1;Aeq=8,14,7;Beq=56;x,f_opt,c,d=fmincon(opt_fun1,x0,A,B,Aeq,Beq,xm,xM,opt_con2,ff);maxDirectionalFirst-orderIterF-countf(x)constraintStep-sizederivativeoptimalityProcedure111955.33622.90.25-29518.3infeasible21116961.71501-6.3e-0156.97e-005HessianmodifiedtwiceOptimizationterminatedsuccessfully:Searchdirectionlessthan2*options.TolXandmaximumconstraintviolationislessthanoptions.TolConActiveConstraints:12x,f_opt,kk=d.funcCount%从略（计算结果同上一样）,例：利用梯度求解梯度函数：symsx1x2x3;f=1000-x1*x1-2*x2*x2-x3*x3-x1*x2-x1*x3;J=jacobian(f,x1,x2,x3)J=-2*x1-x2-x3,-4*x2-x1,-2*x3-x1改写目标函数：functiony,Gy=opt_fun2(x)y=1000-x(1)*x(1)-2*x(2)*x(2)-x(3)*x(3)-x(1)*x(2)-x(1)*x(3);Gy=-2*x(1)+x(2)+x(3);4*x(2)+x(1);2*x(3)+x(1);,x0=1;1;1;xm=0;0;0;xM=;A=;B=;Aeq=;Beq=;ff=optimset;ff.GradObj=on;若知道梯度函数ff.Display=iter;ff.LargeScale=off;ff.TolFun=1e-30;ff.TolX=1e-15;ff.TolCon=1e-20;x,f_opt,c,d=fmincon(opt_fun2,x0,A,B,Aeq,Beq,xm,xM,opt_con1,ff);x,f_opt,kk=d.funcCountx=3.51210.21703.5522f_opt=961.7152kk=95,6.3.5约束线性最小二乘,有约束线性最小二乘的标准形式为:其中：C、A、Aeq为矩阵；d、b、beq、lb、ub、x是向量。,函数lsqlin格式x=lsqlin(C,d,A,b)x=lsqlin(C,d,A,b,Aeq,beq,lb,ub,x0,options)%求在约束条件下，方程Cx=d的最小二乘解x。Aeq、beq满足等式约束，若没有不等式约束，则设A=，b=。lb、ub满足，若没有等式约束，则Aeq=，beq=。x0为初始解向量，若x没有界，则lb=，ub=。options为指定优化参数x,resnorm,residual,exitflag,output,lambda=lsqlin()%resnorm=norm(C*x-d)2，即2-范数。residual=C*x-d，即残差。exitflag为终止迭代的条件output表示输出优化信息lambda为解x的Lagrange乘子,例:求解下面系统的最小二乘解系统：Cx=d约束:先输入系统系数和x的上下界：C=0.95010.76200.61530.4057;0.23110.45640.79190.9354;0.60680.01850.92180.9169;0.48590.82140.73820.4102;0.89120.44470.17620.8936;d=0.0578;0.3528;0.8131;0.0098;0.1388;A=0.20270.27210.74670.4659;0.19870.19880.44500.4186;0.60370.01520.93180.8462;,b=0.5251;0.2026;0.6721;lb=-0.1*ones(4,1);ub=2*ones(4,1);然后调用最小二乘命令：x,resnorm,residual,exitflag,output,lambda=lsqlin(C,d,A,b,lb,ub)x=-0.1000-0.10000.21520.3502resnorm=0.1672,residual=0.04550.0764-0.35620.16200.0784exitflag=1%说明解x是收敛的output=iterations:4algorithm:medium-scale:active-setfirstorderopt:cgiterations:,lambda=lower:4x1doubleupper:4x1doubleeqlin:0 x1doubleineqlin:3x1double%通过lambda.ineqlin可查看非线性不等式约束是否有效。lambda.ineqlinans=00.23920,7.4整数规划问题的计算机求解7.4.1整数线性规划问题的求解,A、B线性等式和不等式约束,，约束式子由ctype变量控制，intlist为整数约束标示，how0表示结果最优，2为无可行解，其余失败。,例：f=-21431;A=02142;345-1-1;intlist=ones(5,1);B=54;62;ctype=-1;-1;xm=0,0,3.32,0.678,2.57;xM=inf*ones(5,1);res,b=ipslv_mex(f,A,B,intlist,xM,xm,ctype)%因为返回的b=0，表示优化成功res=1904105b=0,x1,x2,x3,x4,x5=ndgrid(1:20,0:20,4:20,1:20,3:20);i=find(2*x2+x3+4*x4+2*x5f=-2*x1(i)-x2(i)-4*x3(i)-3*x4(i)-x5(i);fmin,ii=sort(f);index=i(ii(1);x=x1(index),x2(index),x3(index),x4(index),x5(index)x=1904105当把20换为30，一般计算机配置下实现不了，故求解整数规划时不适合采用穷举算法。,次最优解fmin(1:10)ans=-89-88-88-88-88-88-88-88-87-87in=i(ii(1:4);x=x1(in),x2(in),x3(in),x4(in),x5(in),fmin(1:4)x=1904105-891804113-881705104-881506104-88intlist=1,0,0,1,1;混合整数规划res,b=ipslv_mex(f,A,B,intlist,xM,xm,ctype)res=19.000003.800011.00003.0000b=0,7.4.2一般非线性整数规划问题与求解,err字符串为空，则返回最优解。该函数尚有不完全之处，解往往不是精确整数，可用下面语句将其化成整数。,例：functionf=c6miopt(x)f=-21431*x;A=02142;345-1-1;intlist=ones(5,1);Aeq=;Beq=;B=54;62;ctype=-1;-1;xm=0,0,4,1,3;xM=20000*ones(5,1);x0=xm;errmsg,f,X=bnb20(c6miopt,x0,intlist,xm,xM,A,B,Aeq,Beq);X=XX=19.000004.000010.00005.0000,intlist=1,0,0,1,1;xm=0,0,3.32,1,3;errmsg,f,X=bnb20(c6miopt,x0,intlist,xm,xM,A,B,Aeq,Beq);XX=19.000003.800011.00003.0000,例：编写函数：functiony=c7fun1(x)y=100*(x(2)-x(1)2)2+(4.5543-x(1)2;x0=1;1;xm=-100*1;1;xM=100*1;1;A=;B=;Aeq=;Beq=;intlist=1,1;errmsg,f,x=bnb20(c6fun1,x0,intlist,xm,xM,A,B,Aeq,Beq);xans=5.00000000000