2016年大连理工大学优化方法上机大作业-标准化文件发布号：（9456-EUATWK-MWUB-WUNN-INNUL-DDQTY-KII2016年大连理工大学优化方法上机大作业学院：专业：班级：学号：姓名：上机大作业1：1.最速下降法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);endfunction x_star = steepest(x0,eps)gk = grad(x0);res = norm(gk);k = 0;while res > eps && k<=1000dk = -gk;ak =1; f0 = fun(x0);f1 = fun(x0+ak*dk);slope = dot(gk,dk);while f1 > f0 + 0.1*ak*slopeak = ak/4;xk = x0 + ak*dk;f1 = fun(xk);endk = k+1;x0 = xk;gk = grad(xk);res = norm(gk);fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk;end>> clear>> x0=[0,0]';>> eps=1e-4;>> x=steepest(x0,eps)2.牛顿法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad2(x)g = zeros(2,2);g(1,1)=2+400*(3*x(1)^2-x(2));g(1,2)=-400*x(1);g(2,1)=-400*x(1);g(2,2)=200;endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2));g(2) = 200*(x(2)-x(1)^2);endfunction x_star = newton(x0,eps)gk = grad(x0);bk = [grad2(x0)]^(-1);res = norm(gk);k = 0;while res > eps && k<=1000dk=-bk*gk;xk=x0+dk;k = k+1;x0 = xk;gk = grad(xk);bk = [grad2(xk)]^(-1);res = norm(gk);fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk;end>> clear>> x0=[0,0]';>> eps=1e-4;>> x1=newton(x0,eps)--The 1-th iter, the residual is 447.213595--The 2-th iter, the residual is 0.000000x1 =1.00001.00003.BFGS法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);endfunction x_star = bfgs(x0,eps)g0 = grad(x0);gk=g0;res = norm(gk);Hk=eye(2);k = 0;while res > eps && k<=1000dk = -Hk*gk;ak =1; f0 = fun(x0);f1 = fun(x0+ak*dk);slope = dot(gk,dk);while f1 > f0 + 0.1*ak*slopeak = ak/4;xk = x0 + ak*dk;f1 = fun(xk);endk = k+1;fa0=xk-x0;x0 = xk;go=gk;gk = grad(xk);y0=gk-g0;Hk=((eye(2)-fa0*(y0)')/((fa0)'*(y0)))*((eye(2)-(y0)*(fa0)')/((fa0)'*(y0)))+(fa0*(fa0)')/((fa0)'*(y0)); res = norm(gk);fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk;End>> clear>> x0=[0,0]';>> eps=1e-4;>> x=bfgs(x0,eps)4.共轭梯度法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);endfunction x_star =CG(x0,eps)gk = grad(x0);res = norm(gk);k = 0;dk = -gk;while res > eps && k<=1000ak =1; f0 = fun(x0);f1 = fun(x0+ak*dk);slope = dot(gk,dk);while f1 > f0 + 0.1*ak*slopeak = ak/4;xk = x0 + ak*dk;f1 = fun(xk);endk = k+1;x0 = xk;g0=gk;gk = grad(xk);res = norm(gk);p=(gk/g0)^2;dk1=dk;dk=-gk+p*dk1;fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk;end>> clear>> x0=[0,0]';>> eps=1e-4;>> x=CG(x0,eps)上机大作业2：function f= obj(x)f=4*x(1)-x(2)^2-12;endfunction [h,g] =constrains(x)h=x(1)^2+x(2)^2-25;g=zeros(3,1);g(1)=-10*x(1)+x(1)^2-10*x(2)+x(2)^2+34; g(2)=-x(1);g(3)=-x(2);endfunction f=alobj(x) %拉格朗日增广函数%N_equ等式约束个数?%N_inequ不等式约束个数N_equ=1;N_inequ=3;global r_al pena;%全局变量h_equ=0;h_inequ=0;[h,g]=constrains(x);%等式约束部分?for i=1:N_equh_equ=h_equ+h(i)*r_al(i)+(pena/2)*h(i).^2;end%不等式约束部分for i=1:N_inequh_inequ=h_inequ+(0.5/pena)*(max(0,(r_al(i)+pena*g(i))).^2-r_al(i).^2);end%拉格朗日增广函数值f=obj(x)+h_equ+h_inequ;function f=compare(x)global r_al pena N_equ N_inequ;N_equ=1;N_inequ=3;h_inequ=zeros(3,1);[h,g]=constrains(x);%等式部分for i=1:1h_equ=abs(h(i));end%不等式部分for i=1:3h_inequ=abs(max(g(i),-r_al(i+1)/pena));endh1 = max(h_inequ);f= max(abs(h_equ),h1); %sqrt(h_equ+h_inequ);function [ x,fmin,k] =almain(x_al)%本程序为拉格朗日乘子算法示例算法%函数输入：% x_al:初始迭代点% r_al：初始拉格朗日乘子N-equ:等式约束个数N_inequ:不等式约束个数%函数输出% X：最优函数点FVAL:最优函数值%============================程序开始================================ global r_al pena ; %参数（全局变量）pena=10; %惩罚系数r_al=[1,1,1,1];c_scale=2; %乘法系数乘数cta=0.5; %下降标准系数e_al=1e-4; %误差控制范围max_itera=25;out_itera=1; %迭代次数%===========================算法迭代开始============================= while out_itera<max_iterax_al0=x_al;r_al0=r_al;%判断函数?compareFlag=compare(x_al0);%无约束的拟牛顿法BFGS[X,fmin]=fminunc(@alobj,x_al0);x_al=X; %得到新迭代点%判断停止条件?if compare(x_al)<e_aldisp('we get the opt point');breakend%c判断函数下降度?if compare(x_al)<cta*compareFlagpena=1*pena; %可以根据需要修改惩罚系数变量elsepena=min(1000,c_scale*pena); %%乘法系数最大1000disp('pena=2*pena');end%%更新拉格朗日乘子[h,g]=constrains(x_al);for i=1:1%%等式约束部分r_al(i)= r_al0(i)+pena*h(i);endfor i=1:3%%不等式约束部分r_al(i+1)=max(0,(r_al0(i+1)+pena*g(i)));endout_itera=out_itera+1;end%+++++++++++++++++++++++++++迭代结束+++++++++++++++++++++++++++++++++ disp('the iteration number');k=out_itera;disp('the value of constrains');compare(x_al)disp('the opt point');x=x_al;fmin=obj(X);>> clear>> x_al=[0,0];>> [x,fmin,k]=almain(x_al)上机大作业3：1、>> clear alln=3; c=[-3,-1,-3]'; A=[2,1,1;1,2,3;2,2,1;-1,0,0;0,-1,0;0,0,-1];b=[2,5,6,0,0,0]';cvx_beginvariable x(n)minimize( c'*x)subject toA*x<=bcvx_endCalling SDPT3 4.0: 6 variables, 3 equality constraints------------------------------------------------------------num. of constraints = 3dim. of linear var = 6******************************************************************* SDPT3: Infeasible path-following algorithms******************************************************************* version predcorr gam expon scale_dataNT 1 0.000 1 0it pstep dstep pinfeas dinfeas gap prim-obj dual-obj cputime-------------------------------------------------------------------0|0.000|0.000|1.1e+01|5.1e+00|6.0e+02|-7.000000e+01 0.000000e+00| 0:0:00| chol 1 11|0.912|1.000|9.4e-01|4.6e-02|6.5e+01|-5.606627e+00 -2.967567e+01| 0:0:01| chol 1 12|1.000|1.000|1.3e-07|4.6e-03|8.5e+00|-2.723981e+00 -1.113509e+01| 0:0:01| chol 1 13|1.000|0.961|2.3e-08|6.2e-04|1.8e+00|-4.348354e+00 -6.122853e+00| 0:0:01| chol 1 14|0.881|1.000|2.2e-08|4.6e-05|3.7e-01|-5.255152e+00 -5.622375e+00| 0:0:01| chol 1 15|0.995|0.962|1.6e-09|6.2e-06|1.5e-02|-5.394782e+00 -5.409213e+00| 0:0:01| chol 1 16|0.989|0.989|2.7e-10|5.2e-07|1.7e-04|-5.399940e+00 -5.400100e+00| 0:0:01| chol 1 17|0.989|0.989|5.3e-11|5.8e-09|1.8e-06|-5.399999e+00 -5.400001e+00| 0:0:01| chol 1 18|1.000|0.994|2.8e-13|4.3e-11|2.7e-08|-5.400000e+00 -5.400000e+00| 0:0:01|stop: max(relative gap, infeasibilities) < 1.49e-08-------------------------------------------------------------------number of iterations = 8primal objective value = -5.39999999e+00dual objective value = -5.40000002e+00gap := trace(XZ) = 2.66e-08relative gap = 2.26e-09actual relative gap = 2.21e-09rel. primal infeas (scaled problem) = 2.77e-13rel. dual " " " = 4.31e-11rel. primal infeas (unscaled problem) = 0.00e+00rel. dual " " " = 0.00e+00norm(X), norm(y), norm(Z) = 4.3e+00, 1.3e+00, 1.9e+00norm(A), norm(b), norm(C) = 6.7e+00, 9.1e+00, 5.4e+00Total CPU time (secs) = 0.71CPU time per iteration = 0.09termination code = 0DIMACS: 3.6e-13 0.0e+00 5.8e-11 0.0e+00 2.2e-09 2.3e-09 -------------------------------------------------------------------------------------------------------------------------------Status: SolvedOptimal value (cvx_optval): -5.42、>> clear alln=2; c=[-2,-4]'; G=[0.5,0;0,1];A=[1,1;-1,0;0,-1]; b=[1,0,0]';cvx_beginvariable x(n)minimize( x'*G*x+c'*x)subject toA*x<=bcvx_endCalling SDPT3 4.0: 7 variables, 3 equality constraintsFor improved efficiency, SDPT3 is solving the dual problem.------------------------------------------------------------num. of constraints = 3dim. of socp var = 4, num. of socp blk = 1dim. of linear var = 3******************************************************************* SDPT3: Infeasible path-following algorithms*******************************************************************version predcorr gam expon scale_dataNT 1 0.000 1 0it pstep dstep pinfeas dinfeas gap prim-obj dual-obj cputime-------------------------------------------------------------------0|0.000|0.000|8.0e-01|6.5e+00|3.1e+02| 1.000000e+01 0.000000e+00| 0:0:00| chol 1 11|1.000|0.987|4.3e-07|1.5e-01|1.6e+01| 9.043148e+00 -2.714056e-01| 0:0:00| chol 1 12|1.000|1.000|2.6e-07|7.6e-03|1.4e+00| 1.234938e+00 -5.011630e-02| 0:0:00| chol 1 13|1.000|1.000|2.4e-07|7.6e-04|3.0e-01| 4.166959e-01 1.181563e-01| 0:0:00| chol 1 1 4|0.892|0.877|6.4e-08|1.6e-04|5.2e-02| 2.773022e-01 2.265122e-01| 0:0:00| chol 1 1 5|1.000|1.000|1.0e-08|7.6e-06|1.5e-02| 2.579468e-01 2.427203e-01| 0:0:00| chol 1 1 6|0.905|0.904|3.1e-09|1.4e-06|2.3e-03| 2.511936e-01 2.488619e-01| 0:0:00| chol 1 1 7|1.000|1.000|6.1e-09|7.7e-08|6.6e-04| 2.503336e-01 2.496718e-01| 0:0:00| chol 1 1 8|0.903|0.903|1.8e-09|1.5e-08|1.0e-04| 2.500507e-01 2.499497e-01| 0:0:00| chol 1 1 9|1.000|1.000|4.9e-10|3.5e-10|2.9e-05| 2.500143e-01 2.499857e-01| 0:0:00| chol 1 1 10|0.904|0.904|4.7e-11|1.3e-10|4.4e-06| 2.500022e-01 2.499978e-01| 0:0:00| chol 2 211|1.000|1.000|2.3e-12|9.4e-12|1.2e-06| 2.500006e-01 2.499994e-01| 0:0:00| chol 2 212|1.000|1.000|4.7e-13|1.0e-12|1.8e-07| 2.500001e-01 2.499999e-01| 0:0:00| chol 2 213|1.000|1.000|2.0e-12|1.0e-12|4.2e-08| 2.500000e-01 2.500000e-01| 0:0:00| chol 2 214|1.000|1.000|2.6e-12|1.0e-12|7.3e-09| 2.500000e-01 2.500000e-01| 0:0:00|stop: max(relative gap, infeasibilities) < 1.49e-08-------------------------------------------------------------------number of iterations = 14primal objective value = 2.50000004e-01dual objective value = 2.49999996e-01gap := trace(XZ) = 7.29e-09relative gap = 4.86e-09actual relative gap = 4.86e-09rel. primal infeas (scaled problem) = 2.63e-12rel. dual " " " = 1.00e-12rel. primal infeas (unscaled problem) = 0.00e+00rel. dual " " " = 0.00e+00norm(X), norm(y), norm(Z) = 3.2e+00, 1.5e+00, 1.9e+00norm(A), norm(b), norm(C) = 3.9e+00, 4.2e+00, 2.6e+00Total CPU time (secs) = 0.36CPU time per iteration = 0.03termination code = 0DIMACS: 3.7e-12 0.0e+00 1.3e-12 0.0e+00 4.9e-09 4.9e-09 -------------------------------------------------------------------------------------------------------------------------------Status: SolvedOptimal value (cvx_optval): -3。