# 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])

    Successive approximation method to be employed.
       sedumi will be called several times to refine the solution.
       Original size: 225 variables, 144 equality constraints
       48 exponentials add 384 variables, 240 equality constraints
    -----------------------------------------------------------------
     Cones  |             Errors              |
    Mov/Act | Centering  Exp cone   Poly cone | Status
    --------+---------------------------------+---------
      8/  8 | 2.949e+00  6.560e-01  1.365e-07 | Solved
      8/  8 | 2.417e-01  4.179e-03  1.926e-07 | Solved
      8/  8 | 1.636e-02  1.932e-05  2.106e-07 | Solved
      7/  8 | 1.028e-03  2.853e-07  2.123e-07 | Solved
      0/  8 | 6.427e-05  2.153e-07  2.127e-07 | 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.