% ex2_2.m% Chapter 2: Optimization with Matlab% Dr. P.Venkataraman % Example 2 Sec.2.3%% graphical solution using matlab (two design variables)% Unconstrained function illustrating global minimum% Example will introduce 3D plots, 3D contours, filled 2D% contours and gradient information%--------- -------------------------------------------------- -----x1=-1:0.01:1;% the semi-colon at the end prevents the echo x2=-1:0.01:1;% these are also the side constraints% x1 and x2 are vectors filled with numbers starting% at 0 and ending at 10.0 with values at intervals of 0.1[X1 X2] = meshgrid(x1,x2);% generates matrices X1 and X2 correspondin% vectors x1 and x2% filled contour with default colormap% help graph3d gives you the choices for colormapf1 = obj_ex2(X1,X2);% the objecive function is evaluated over the entire mesh[C1,h1] = contourf(x1,x2,f1,... [0 0.1 0.3 0.6 0.8 1.0 1.2 1.5 1.8 2.0 2.2 2.4 2.6 2.8 3.0]);clabel(C1,h1);colorbar% illustrates the scalegridset(gca,'xtick',[-1 -0.5 0.0 0.5 1.0])set(gca,'ytick',[- 1 -0.5 0.0 0.5 1.0])gridxlabel(' x_1 values','FontName','times','FontSize',12);% label for x-axesylabel(' x_2 values','FontName','times', 'FontSize',12);title({'Filled Labelled Contour','default color'}, ...'FontName','times','FontSize',10)grid
% a new figue is used% basic contour plot superimposed with gradient information% information is generated on a larger mesh to keep the% figure tidy. grid is removed for clarityfigurey1 =- 1:0.1:1.0;y2 = -1:0.1:1;[Y1,Y2] = meshgrid(y1,y2); f2 = obj_ex2(Y1,Y2);[C2,h2] = contour(y1,y2,f2 ,[0 0.5 0.75 1.0 1.5 1.75 2.0 2.5 2.5 3.0]);clabel(C2,h2)[GX, GY] = gradient(f2,0.2);%[...] = gradient(F,h), where h is a scalar, uses h as the spacing between points in each direction.hold on quiver(Y1 ,Y2,GX,GY);%quiver(x,y,u,v) A quiver plot displays velocity vectors as arrows with components (u,v) at the points (x,y).hold offset(gca,'xtick ',[-1 -0.5 0.0 0.5 1.0])set(gca,'ytick',[-1 -0.5 0.0 0.5 1.0])xlabel(' x_1 values','FontName','times','FontSize',12 );% label for x-axesylabel(' x_2 values','FontName','times','FontSize',12);title({'2D Contour','with Gradient Vectors'}, ...'FontName' ,'times','FontSize',10)
% this is a stacked contour mapfigurecontour3(x1,x2, f1,[0 0.3 0.6 0.8 1.0 1.5 1.8 2.0 2.2 2.4 2.6 2.8 3.0]);gridset(gca,'xtick ',[-1 -0.5 0.0 0.5 1.0])set(gca,'ytick',[-1 -0.5 0.0 0.5 1.0])% change color map set color barcolormap(spring)colorbarxlabel(' x_1 values','FontName' ,'times','FontSize',12);% label for x-axesylabel(' x_2 values','FontName','times','FontSize',12)title(('Stacked (3D) Contour',' colormap-spring'}, ...'FontName','times','FontSize',10)grid
% a mesh of the functionfigurecolormap(cool)mesh(y1,y2,f2)set(gca,'xtick',[-1 -0.5 0.0 0.5 1.0])set(gca,'ytick',[-1 -0.5 0.0 0.5 1.0] )colorbarxlabel(' x_1 values','FontName','times','FontSize',12);% label for x-axesylabel(' x_2 values','FontName','times','FontSi ze',12);title({'Coarse Mesh Plot','colormap-cool'}, ...'FontName','times','FontSize',10)grid
% surface plot with default colormapfigurecolormap(jet)surf(y1,y2,f2)gridcolorbarset(gca,'xtick',[-1 -0.5 0.0 0.5 1.0])set (gca,'ytick',[-1 -0.5 0.0 0.5 1.0])title({'Coarse Surface Plot','colormap-jet/default'}, ...'FontName','times','FontSize', 10)gridxlabel(' x_1 values','FontName','times','FontSize',12);% label for x-axesylabel(' x_2 values','FontName','times','FontSize',12) ;hold off
%obj_ex2.m function retval = obj_ex2(X1,X2)% Optimization with Matlab% Dr. P. Venkataraman% Chapter 2. Example 2% Problem appeared in Reference 2.3%% f(x1,x2) = a*x1^2 + b*x2^ 2 -ccos(aa*x1)-d*cos(bb*x2) + c + da = 1; b = 2; c = 0.3; d = 0.4; aa = 3.0*pi; bb = 4.0*pi; retval = a*X1.*X1 + b*X2.*X2 -c*cos(aa*X1)-... d*cos(bb*X2) + c + d;