# Example 6.6: Comparison of worst-case robust, Tikhonov, and nominal least squares

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    % Section 6.4.2, Figure 6.16
    % Boyd & Vandenberghe "Convex Optimization"
    % Original by Lieven Vandenberghe
    % Adapted for CVX Argyris Zymnis - 11/27/05
    % (a figure is generated)
    %
    % Consider the least-squares problem:
    %       minimize ||(A0 + u1*A1 + u2*A2)x - b||_2
    % where u = [u1 u2]' is an uncertain parameter and ||u||_2 <= 1
    % Three approximate solutions are found:
    %   1- nominal optimal (i.e. letting u=0)
    %   2- Tikhonov Regularized Solution:
    %           minimize ||A0*x - b||_2 + delta*||x||_2
    %      for some delta (in this case we set delta = 0.1)
    %   3- worst-case robust approximation:
    %           minimize sup{||u||_2 <= 1} ||(A0 + u1*A1 + u2*A2)x - b||_2)
    %      (reduces to solving an SDP, see pages 323-324 in the book)

    m = 50;
    n = 20;
    randn('state',0);
    rand('state',0);

    A0 = randn(m,n);
    [U,S,V] = svd(A0);
    S= diag(fliplr(logspace(-0.7,1,n)));
    A0 = U(:,1:n)*S*V';
    A1 = randn(m,n);  A1 = A1/norm(A1);
    A2 = randn(m,n);  A2 = A2/norm(A2);

    Aperb0 = [A1;A2];
    p = 2;

    b = U(:,1:n)*randn(n,1) + .1*randn(m,1);

    % we consider LS problems || (A0 + u1*A1 + u2*A2) x - b||
    % where  ||u|| leq rho

    % Nominal Solution
    xnom = A0\b;

    % Tikhonov Regularized Solution
    delta = .1;
    xtych =  [A0; sqrt(delta)*eye(n)] \ [b; zeros(n,1)];

    % Robust Least Squares solution
    cvx_begin sdp quiet
        variables t lambda xrob(n)
        minimize(t+lambda)
        subject to
            [eye(m) A1*xrob A2*xrob A0*xrob-b; ...
             [A1*xrob A2*xrob]' lambda*eye(2) zeros(2,1); ...
             [A0*xrob-b]' zeros(1,2) t] >= 0;
    cvx_end

    % Generate Random Trials
    notrials=100000;
    r = sqrt(rand(notrials,1));     % random on [0,1] with pdf g(r) = 2r;
    theta = 2*pi*rand(notrials,1);  % uniform on [0,2pi]
    v = [r.*cos(theta)  r.*sin(theta)];
    ls_res = zeros(1,notrials);
    rob2_res = zeros(1,notrials);
    rob_res = zeros(1,notrials);
    tych_res = zeros(1,notrials);

    for i =1:notrials

      A = A0 + v(i,1)*A1 + v(i,2)*A2;
      ls_res(i) = norm(A*xnom-b);
      rob_res(i) = norm(A*xrob-b);
      tych_res(i) = norm(A*xtych-b);

    end;

    % Plot histograms
    figure
    %subplot(211)
    [N1, hist1] = hist(ls_res,[min(ls_res):.1:max(ls_res)]);
    freq1 = N1/notrials;
    [N2, hist2] = hist(rob_res,hist1);
    freq2 = N2/notrials;
    [N3, hist3] = hist(tych_res,hist1);
    freq3 = N3/notrials;

    h = bar(hist3,freq3);
    text(3, 0.07, 'Tikhonov');
    set(h,'FaceColor',0.90*[1 1 1]);
    hold on

    h = bar(hist2,freq2);
    text(4.2, 0.05, 'Nominal');
    set(h,'FaceColor',0.80*[1 1 1]);

    h = bar(hist2,freq2);
    set(h,'FaceColor','none');
    text(2.6, 0.2, 'Robust LS');

    h = bar(hist3,freq3);
    set(h,'FaceColor','none');
    h = bar(hist1,freq1);
    set(h,'FaceColor','none');

    xlabel('||(A0 + u1*A1 + u2*A2)*x - b||_2')
    ylabel('Frequency')
    hold off