CVX求解几何规划样例1：

Section 4.5.4: Design of a cantilever beam: recursive formulation (GP)

% Section 4.5.4: Design of a cantilever beam: recursive formulation (GP)
% Boyd & Vandenberghe "Convex Optimization"
% (a figure is generated)
% Almir Mutapcic 02/08/06
%
% We have a segmented cantilever beam with N segments. Each segment
% has a unit length and variable width and height (rectangular profile).
% The goal is minimize the total volume of the beam, over all segment
% widths w_i and heights h_i, subject to constraints on aspect ratios,
% maximum allowable stress in the material, vertical deflection y, etc.
%
% The problem can be posed as a geometric program (posynomial form)
%     minimize   sum( w_i* h_i)
%         s.t.   w_min <= w_i <= w_max,       for all i = 1,...,N
%                h_min <= h_i <= h_max
%                S_min <= h_i/w_i <= S_max
%                6*i*F/(w_i*h_i^2) <= sigma_max
%                y_1 <= y_max
%
% with variables w_i and h_i (i = 1,...,N).
% For other definitions consult the book.
% (See exercise 4.31 for a non-recursive formulation.)% optimization variables
clear all
clcN = 8;
% constants
wmin = .1;   wmax = 100;
hmin = .1;   hmax = 6;
Smin = 1/5;  Smax = 5;
sigma_max = 1;
ymax = 10;
E = 1; F = 1;cvx_begin gp  % quiet % optimization variablesvariables w(N) h(N)% setting up variables relations% (recursive formulation)v = cvx( zeros(N+1,1) );y = cvx( zeros(N+1,1) );%   v =  zeros(N+1,1) ;
%   y =  zeros(N+1,1) ;for i = N:-1:1fprintf(1,'Building recursive relations for index: %d\n',i);v(i) = 12*(i-1/2)*F/(E*w(i)*h(i)^3) + v(i+1);y(i) = 6*(i-1/3)*F/(E*w(i)*h(i)^3)  + v(i+1) + y(i+1);end% objective is the total volume of the beam% obj = sum of (widths*heights*lengths) over each section% (recall that the length of each segment is set to be 1)minimize( (w.^(-1))'*(h.^6) + (10*w.^(-2))'*(h.^2) )subject to% constraint setwmin <= w    <= wmax;hmin <= h    <= hmax;hmin <= h(1:4)    <= hmax;Smin <= h./w <= Smax;6*F*[1:N]'./(w.*(h.^2)) <= sigma_max;y(1) <= ymax;
cvx_end% display results
disp('The optimal widths and heights are: ');
w, h
fprintf(1,'The optimal minimum volume of the beam is %3.4f.\n', sum(w.*h))% plot the 3D model of the optimal cantilever beam
% figure, clf
% cantilever_beam_plot([h; w])

CVX求解几何规划样例2：

Exercise 4.31: Design of a cantilever beam (GP)

% Boyd & Vandenberghe "Convex Optimization"
% Almir Mutapcic - 01/30/06
% Updated to use GP mode 02/08/06
% (a figure is generated)
%
% We have a segmented cantilever beam with N segments. Each segment
% has a unit length and variable width and height (rectangular profile).
% The goal is minimize the total volume of the beam, over all segment
% widths w_i and heights h_i, subject to constraints on aspect ratios,
% maximum allowable stress in the material, vertical deflection y, etc.
%
% The problem can be posed as a geometric program (posynomial form)
%     minimize    sum( w_i* h_i)
%         s.t.    w_min <= w_i <= w_max,       for all i = 1,...,N
%                 h_min <= h_i <= h_max
%                 S_min <= h_i/w_i <= S_max
%                 6*i*F/(w_i*h_i^2) <= sigma_max
%                 6*F/(E*w_i*h_i^3) == d_i
%                 (2*i - 1)*d_i + v_(i+1) <= v_i
%                 (i - 1/3)*d_i + v_(i+1) + y_(i+1) <= y_i
%                 y_1 <= y_max
%
% with variables w_i, h_i, d_i, (i = 1,...,N) and v_i, y_i (i = 1,...,N+1).
% (Consult the book for other definitions and a recursive formulation of
% this problem.)% optimization variablesclear all
clcN = 8;
% constants
wmin = .1; wmax = 100;
hmin = .1; hmax = 6;
Smin = 1/5; Smax = 5;
sigma_max = 1;
ymax = 10;
E = 1; F = 1;cvx_begin gp% optimization variablesvariables w(N) h(N) v(N+1) y(N+1);% objective is the total volume of the beam% obj = sum of (widths*heights*lengths) over each section% (recall that the length of each segment is set to be 1)minimize( w'*h )subject to% non-recursive formulationd = 6*F*ones(N,1)./(E*ones(N,1).*w.*h.^3);for i = 1:N(2*i-1)*d(i) + v(i+1) <= v(i);(i-1/3)*d(i) + v(i+1) + y(i+1) <= y(i);end% constraint setwmin <= w    <= wmax;hmin <= h    <= hmax;Smin <= h./w <= Smax;6*F*[1:N]'./(w.*(h.^2)) <= sigma_max;y(1) <= ymax;
cvx_end% display results
disp('The optimal widths and heights are: ');
w, h
fprintf(1,'The optimal minimum volume of the beam is %3.4f.\n', sum(w.*h))% % plot the 3D model of the optimal cantilever beam
% figure, clf
% cantilever_beam_plot([h; w])

网址链接： CVX Example library

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