Exercise 4.31: Design of a cantilever beam (GP)

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% 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 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) 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 formulation
    d = 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 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])

 
Calling Mosek 9.1.9: 323 variables, 144 equality constraints
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 144             
  Cones                  : 48              
  Scalar variables       : 323             
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 16
Eliminator terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 2                 time                   : 0.00            
Lin. dep.  - tries                  : 1                 time                   : 0.00            
Lin. dep.  - number                 : 0               
Presolve terminated. Time: 0.00    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 144             
  Cones                  : 48              
  Scalar variables       : 323             
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 56
Optimizer  - Cones                  : 49
Optimizer  - Scalar variables       : 185               conic                  : 146             
Optimizer  - Semi-definite variables: 0                 scalarized             : 0               
Factor     - setup time             : 0.00              dense det. time        : 0.00            
Factor     - ML order time          : 0.00              GP order time          : 0.00            
Factor     - nonzeros before factor : 276               after factor           : 306             
Factor     - dense dim.             : 0                 flops                  : 4.18e+03        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   9.3e+00  1.3e+00  4.0e+01  0.00e+00   0.000000000e+00   -3.864489608e+01  1.0e+00  0.00  
1   5.1e+00  7.0e-01  1.7e+01  3.58e-01   1.743951235e+00   -2.266239385e+01  5.4e-01  0.01  
2   1.1e+00  1.6e-01  1.6e+00  7.70e-01   4.249122204e+00   -1.363745584e+00  1.2e-01  0.01  
3   5.6e-01  7.7e-02  5.5e-01  1.15e+00   4.459481801e+00   1.911303501e+00   6.0e-02  0.01  
4   1.6e-01  2.2e-02  7.6e-02  1.15e+00   4.122703747e+00   3.463984524e+00   1.7e-02  0.01  
5   4.4e-02  6.0e-03  1.1e-02  1.08e+00   3.828909022e+00   3.650940885e+00   4.7e-03  0.01  
6   1.1e-02  1.5e-03  1.4e-03  1.04e+00   3.771851182e+00   3.729072170e+00   1.1e-03  0.01  
7   2.4e-03  3.3e-04  1.5e-04  1.01e+00   3.753436111e+00   3.743849430e+00   2.6e-04  0.01  
8   9.7e-04  1.3e-04  3.9e-05  9.85e-01   3.749690488e+00   3.745774943e+00   1.0e-04  0.01  
9   3.5e-04  4.8e-05  8.6e-06  9.89e-01   3.748107109e+00   3.746682810e+00   3.8e-05  0.01  
10  2.3e-04  3.2e-05  4.6e-06  9.64e-01   3.747708483e+00   3.746767716e+00   2.4e-05  0.01  
11  4.8e-05  6.6e-06  4.5e-07  9.88e-01   3.747223797e+00   3.747025029e+00   5.1e-06  0.01  
12  2.2e-05  3.1e-06  1.5e-07  9.68e-01   3.747136513e+00   3.747041537e+00   2.4e-06  0.02  
13  4.4e-06  6.2e-07  1.3e-08  9.88e-01   3.747083302e+00   3.747064323e+00   4.8e-07  0.02  
14  2.1e-06  3.0e-07  4.5e-09  9.70e-01   3.747074316e+00   3.747064994e+00   2.3e-07  0.02  
15  4.6e-07  6.5e-08  4.7e-10  9.88e-01   3.747068818e+00   3.747066752e+00   5.1e-08  0.02  
16  2.5e-07  3.4e-08  1.8e-10  9.74e-01   3.747067830e+00   3.747066722e+00   2.7e-08  0.02  
17  7.0e-08  7.2e-09  1.8e-11  9.90e-01   3.747067132e+00   3.747066899e+00   5.6e-09  0.02  
18  7.0e-08  7.2e-09  1.8e-11  9.72e-01   3.747067132e+00   3.747066899e+00   5.6e-09  0.02  
19  7.0e-08  7.2e-09  1.8e-11  9.72e-01   3.747067132e+00   3.747066899e+00   5.6e-09  0.02  
20  3.9e-08  3.7e-09  6.8e-12  9.72e-01   3.747067021e+00   3.747066898e+00   2.9e-09  0.02  
21  4.8e-08  3.7e-09  6.8e-12  9.91e-01   3.747067021e+00   3.747066898e+00   2.9e-09  0.02  
22  3.1e-08  3.4e-09  5.8e-12  1.00e+00   3.747067011e+00   3.747066900e+00   2.6e-09  0.02  
23  4.4e-08  3.0e-09  5.0e-12  1.00e+00   3.747067001e+00   3.747066901e+00   2.4e-09  0.02  
24  3.6e-08  2.9e-09  4.6e-12  1.00e+00   3.747066997e+00   3.747066902e+00   2.3e-09  0.02  
25  3.3e-08  2.8e-09  4.3e-12  1.00e+00   3.747066993e+00   3.747066902e+00   2.2e-09  0.03  
26  4.4e-08  2.5e-09  3.7e-12  1.00e+00   3.747066985e+00   3.747066903e+00   1.9e-09  0.03  
27  4.3e-08  2.5e-09  3.6e-12  9.83e-01   3.747066984e+00   3.747066903e+00   1.9e-09  0.03  
28  4.5e-08  2.2e-09  3.1e-12  1.00e+00   3.747066977e+00   3.747066904e+00   1.7e-09  0.03  
29  4.7e-08  2.2e-09  3.0e-12  1.00e+00   3.747066975e+00   3.747066904e+00   1.7e-09  0.03  
30  4.8e-08  2.2e-09  3.0e-12  9.83e-01   3.747066975e+00   3.747066904e+00   1.7e-09  0.03  
31  4.7e-08  1.9e-09  2.5e-12  1.00e+00   3.747066969e+00   3.747066905e+00   1.5e-09  0.03  
Optimizer terminated. Time: 0.03    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: 3.7470669691e+00    nrm: 2e+01    Viol.  con: 8e-08    var: 9e-09    cones: 1e-08  
  Dual.    obj: 3.7470669053e+00    nrm: 1e+00    Viol.  con: 0e+00    var: 1e-09    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.03    
    Interior-point          - iterations : 31        time: 0.03    
      Basis identification  -                        time: 0.00    
        Primal              - iterations : 0         time: 0.00    
        Dual                - iterations : 0         time: 0.00    
        Clean primal        - iterations : 0         time: 0.00    
        Clean dual          - iterations : 0         time: 0.00    
    Simplex                 -                        time: 0.00    
      Primal simplex        - iterations : 0         time: 0.00    
      Dual simplex          - iterations : 0         time: 0.00    
    Mixed integer           - relaxations: 0         time: 0.00    

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +42.3965
 
The optimal widths and heights are: 

w =

    0.6214
    0.7830
    0.9060
    1.0124
    1.1004
    1.1762
    1.2000
    1.3333


h =

    3.1072
    3.9149
    4.5298
    5.0620
    5.5019
    5.8811
    6.0000
    6.0000

The optimal minimum volume of the beam is 42.3965.