# Euclidean projection on a hyperplane

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    % Section 8.1.1, Boyd & Vandenberghe "Convex Optimization"
    % Joelle Skaf - 10/04/05
    %
    % The projection of x0 on a hyperplane C = {x | a'*x = b} is given by
    %           minimize || x - x0 ||^2
    %               s.t.    a'*x = b
    % It is also given by P_C(x0) = x0 + (b - a'*x0)*a/||a||^2

    % Input data
    randn('seed',0);
    n  = 10;
    a  = randn(n,1);
    b  = randn(1);
    x0 = randn(n,1);

    % Analytical solution
    fprintf(1,'Computing the analytical solution ...');
    pc_x0 = x0 + (b - a'*x0)*a/norm(a)^2;
    fprintf(1,'Done! \n');

    % Solution via QP
    fprintf(1,'Computing the optimal solution by solving a QP ...');

    cvx_begin quiet
        variable x(n)
        minimize ( square_pos(norm(x - x0)) )
        a'*x == b;
    cvx_end

    fprintf(1,'Done! \n');

    % Verification
    disp('--------------------------------------------------------------------------------');
    disp('Verifying that p_C(x0) and x_star belong to the hyperplane C: ');
    disp(['a^T*p_C(x0) - b = ' num2str(a'*pc_x0 - b)]);
    disp(['a^T*x_star - b  = ' num2str(a'*x - b)]);
    disp('Computing the distance between x0 and the hyperplane in each case');
    disp(['||x0 - p_C(x0)|| = ' num2str(norm(x0 - pc_x0))]);
    disp(['||x0 - x_star || = ' num2str(norm(x0 - x))]);
    disp('Verifying that the analytical solution and the solution obtained via QP are equal: ');
    [pc_x0 x]

    Computing the analytical solution ...Done!
    Computing the optimal solution by solving a QP ...Done!
    --------------------------------------------------------------------------------
    Verifying that p_C(x0) and x_star belong to the hyperplane C:
    a^T*p_C(x0) - b = -2.2204e-16
    a^T*x_star - b  = 4.5741e-14
    Computing the distance between x0 and the hyperplane in each case
    ||x0 - p_C(x0)|| = 0.18218
    ||x0 - x_star || = 0.18218
    Verifying that the analytical solution and the solution obtained via QP are equal:

    ans =

       -0.6316   -0.6316
        1.2834    1.2834
       -0.6345   -0.6345
        0.5984    0.5984
       -0.4016   -0.4016
       -0.0343   -0.0343
       -1.3458   -1.3458
       -1.1631   -1.1631
        1.0003    1.0003
        0.0072    0.0072