# Example 6.9: Bounding consumer preference

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    % Section 6.5.5, Figures 6.25-6.26
    % Boyd & Vandenberghe "Convex Optimization"
    % Original by Lieven Vandenberghe
    % Adapted for CVX Argyris Zymnis - 11/30/2005
    %
    % We are given a set of consumer preference data for bundles
    % of two goods x_1 and x_2. These points are generated by
    % taking 40 random points and comparing them using the
    % utility function: u(x_1,x_2) = (1.1*sqrt(x_1)+0.8*sqrt(x_2))/1.9
    % Then, if we have u(i) >= u(j) we say that (i,j) is in Pweak.
    %
    % Given this, we wish to compare the point (0.5,0.5) to each
    % of the bundles in the given dataset. I.e. for each point k in the
    % dataset, we wish to decide wether u(k) >= u(0) or u(k) <= u(0),
    % or both, in which case we cannot make any conclusions about
    % consumer preferences.
    %
    % To do this, we have to solve two LPs for each point:
    %       minimize     u(k) - u(0)
    %       subject to   g_i  >= 0
    %                    u(j) <= u(i) + g_i^T(a_j - a_i), for all i,j
    %                    u(i) >= u(j), for all (i,j) in Pweak
    %
    % and:
    %       maximize     u(k) - u(0)
    %       subject to   g_i  >= 0
    %                    u(j) <= u(i) + g_i^T(a_j - a_i), for all i,j
    %                    u(i) >= u(j), for all (i,j) in Pweak
    %
    % If the second LP has a strictly negative solution, we can deduce that
    % u(k) < u(0). If on the other hand the second LP has a nonnegative
    % solution and the first LP has a strictly positive solution, we can
    % deduce that u(k) > u(0). Finally if none of the two previous cases
    % holds, we cannot make a decision between the two bundles.
    %
    % NOTE: This file requires the auxilliary function utilfun.m to run.

    data= [...
       4.5e-01   9.6e-01
       2.1e-01   3.4e-01
       9.6e-01   3.0e-02
       8.0e-02   9.2e-01
       2.0e-02   2.2e-01
       0.0e+00   3.9e-01
       2.6e-01   6.4e-01
       3.5e-01   9.7e-01
       9.1e-01   7.8e-01
       1.2e-01   1.4e-01
       5.8e-01   8.4e-01
       4.9e-01   2.7e-01
       7.0e-02   8.0e-01
       9.3e-01   8.7e-01
       4.4e-01   8.6e-01
       3.3e-01   4.2e-01
       8.9e-01   9.0e-01
       4.9e-01   7.0e-02
       9.5e-01   3.3e-01
       6.6e-01   2.6e-01
       9.5e-01   7.3e-01
       4.2e-01   9.1e-01
       6.8e-01   2.0e-01
       5.2e-01   6.2e-01
       7.7e-01   6.3e-01
       2.0e-02   2.9e-01
       9.8e-01   2.0e-02
       5.0e-02   7.9e-01
       7.9e-01   1.9e-01
       6.2e-01   6.0e-02
       2.8e-01   8.7e-01
       6.9e-01   1.0e-01
       6.9e-01   3.7e-01
       0.0e+00   7.2e-01
       8.7e-01   1.7e-01
       6.3e-01   4.0e-02
       3.2e-01   7.3e-01
       4.0e-02   4.6e-01
       3.6e-01   9.5e-01
       8.2e-01   6.7e-01 ];

    % objective point
    obj=[0.5,0.5];

    figure(1);
    % display the utility function's level sets on some data points.

    plot(data(:,1),data(:,2),'o');
    hold on;

    [X,Y] = meshgrid(0:.01:1,0:.01:1);
    Z=(1.1*X.^(1/2)+0.8*Y.^(1/2))/1.9;

    [C,h] = contour(X,Y,Z,[.1,.2,.3,.4,.5,.6,.7,.8,.9],'--');
    clear X Y Z C
    hold off;
    xlabel('x_1');
    ylabel('x_2');
    hold off

    m = size(data,1);  % number of baskets, including 0,1

    % add preference data
    Pweak = zeros(m+1,m+1);
    for i=1:m,
       for j=1:m
          if (i~=j) & (1.1*data(i,1).^(1/2)+0.8*data(i,2).^(1/2))/1.9 >= ...
                 (1.1*data(j,1).^(1/2)+0.8*data(j,2).^(1/2))/1.9,
             Pweak(i,j) = 1;
          end;
       end;
    end;

    % Find consumer preferences
    data = [data; 0.5 0.5];
    bounds = zeros(m,2);
    for k = 1:m
        fprintf(1,'Deciding on bundle %d of %d: ',k,m);

        % Check for u(k) >= u(0.5,0.5)
        cvx_begin quiet
            variables u(m+1) g_x(m+1) g_y(m+1)
            minimize(u(k)-u(m+1))
            subject to
                g_x >= 0;
                g_y >= 0;
                ones(m+1,1)*u' <= u*ones(1,m+1)+(g_x*ones(1,m+1)).*...
                  (ones(m+1,1)*data(:,1)'-data(:,1)*ones(1,m+1))+...
                  (g_y*ones(1,m+1)).*(ones(m+1,1)*data(:,2)'-data(:,2)*ones(1,m+1));
                (u*ones(1,m+1)).*Pweak >= (ones(m+1,1)*u').*Pweak;
        cvx_end
        bounds(k,1) = cvx_optval;
        fprintf( 1,'%g', round(cvx_optval) );

        % Check for u(0.5,0.5) >= u(k)
        cvx_begin quiet
            variables u(m+1) g_x(m+1) g_y(m+1)
            maximize(u(k)-u(m+1))
            subject to
                g_x >= 0;
                g_y >= 0;
                ones(m+1,1)*u' <= u*ones(1,m+1) + (g_x*ones(1,m+1)).*...
                  (ones(m+1,1)*data(:,1)'-data(:,1)*ones(1,m+1))+...
                  (g_y*ones(1,m+1)).*(ones(m+1,1)*data(:,2)'-data(:,2)*ones(1,m+1));
                (u*ones(1,m+1)).*Pweak >= (ones(m+1,1)*u').*Pweak;
        cvx_end
        bounds(k,2) = cvx_optval;
        fprintf( 1,' %g\n', round(cvx_optval) );

    end

    figure(2);
    hold off

    % plot data pt and contour line through it
    val = 1.1*sqrt(0.5)+ 0.8*sqrt(.5);   % value at center
    t = linspace(((val-.8)/1.1)^2, 1, 1000);
    y = ( (val - 1.1*(t.^(1/2)))/.8 ).^2;
    plot(t,y,'--', [.5 .5], [0 1], ':', [0 1], [.5 .5], ':');
    axis([0 1 0 1]);
    hold on

    for k=1:m
       if bounds(k,2) < 1e-5,  % preferred over (.5,.5)
          dot = plot(data(k,1),data(k,2),'o');
          %'MarkerSize',8);
       elseif bounds(k,1) > -1e-5,  % rejected in favor of (.5,.5)
          dot = plot(data(k,1),data(k,2),'o','MarkerFaceColor',[0 0 0]);
       else % no conclusion
          dot = plot(data(k,1),data(k,2),'square', 'LineWidth',1.0,...
               'MarkerSize',10);
       end;
    end;
    xlabel('x_1');  ylabel('x_2');

    Deciding on bundle 1 of 40: -0 Inf
    Deciding on bundle 2 of 40: -Inf 0
    Deciding on bundle 3 of 40: -Inf -0
    Deciding on bundle 4 of 40: -Inf -0
    Deciding on bundle 5 of 40: -Inf 0
    Deciding on bundle 6 of 40: -Inf -0
    Deciding on bundle 7 of 40: -Inf -0
    Deciding on bundle 8 of 40: -0 Inf
    Deciding on bundle 9 of 40: 0 Inf
    Deciding on bundle 10 of 40: -Inf 0
    Deciding on bundle 11 of 40: 0 Inf
    Deciding on bundle 12 of 40: -Inf -0
    Deciding on bundle 13 of 40: -Inf 0
    Deciding on bundle 14 of 40: -0 Inf
    Deciding on bundle 15 of 40: -0 Inf
    Deciding on bundle 16 of 40: -Inf 0
    Deciding on bundle 17 of 40: 0 Inf
    Deciding on bundle 18 of 40: -Inf 0
    Deciding on bundle 19 of 40: 0 Inf
    Deciding on bundle 20 of 40: -Inf -0
    Deciding on bundle 21 of 40: -0 Inf
    Deciding on bundle 22 of 40: -0 Inf
    Deciding on bundle 23 of 40: -Inf -0
    Deciding on bundle 24 of 40: -0 Inf
    Deciding on bundle 25 of 40: -0 Inf
    Deciding on bundle 26 of 40: -Inf -0
    Deciding on bundle 27 of 40: -Inf -0
    Deciding on bundle 28 of 40: -Inf -0
    Deciding on bundle 29 of 40: -Inf Inf
    Deciding on bundle 30 of 40: -Inf -0
    Deciding on bundle 31 of 40: -Inf Inf
    Deciding on bundle 32 of 40: -Inf 0
    Deciding on bundle 33 of 40: -Inf Inf
    Deciding on bundle 34 of 40: -Inf -0
    Deciding on bundle 35 of 40: -Inf Inf
    Deciding on bundle 36 of 40: -Inf 0
    Deciding on bundle 37 of 40: -Inf Inf
    Deciding on bundle 38 of 40: -Inf 0
    Deciding on bundle 39 of 40: 0 Inf
    Deciding on bundle 40 of 40: -0 Inf