% 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
N = 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 variables
variables w(N) h(N)

% setting up variables relations
% (recursive formulation)
v = cvx( zeros(N+1,1) );
y = cvx( zeros(N+1,1) );
for i = N:-1:1
fprintf(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'*h )
subject to
% constraint set
wmin <= 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])
Building recursive relations for index: 8
Building recursive relations for index: 7
Building recursive relations for index: 6
Building recursive relations for index: 5
Building recursive relations for index: 4
Building recursive relations for index: 3
Building recursive relations for index: 2
Building recursive relations for index: 1

Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 127 variables, 88 equality constraints
23 exponentials add 184 variables, 115 equality constraints
-----------------------------------------------------------------
Cones  |             Errors              |
Mov/Act | Centering  Exp cone   Poly cone | Status
--------+---------------------------------+---------
8/  8 | 2.949e+00  6.560e-01  7.202e-08 | Solved
8/  8 | 2.417e-01  4.179e-03  1.569e-07 | Solved
8/  8 | 1.641e-02  1.922e-05  1.253e-08 | Solved
8/  8 | 1.033e-03  8.954e-08  1.041e-08 | Solved
0/  8 | 6.465e-05  1.084e-08  1.031e-08 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +40.9668
The optimal widths and heights are:

w =

0.6214
0.7830
0.8963
0.9865
1.0627
1.1292
1.1888
1.3333

h =

3.1072
3.9149
4.4814
4.9324
5.3133
5.6462
5.9439
6.0000

The optimal minimum volume of the beam is 40.9668. 