# Exercise 5.1d: Sensitivity analysis for a simple QCQP

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    % Boyd & Vandenberghe, "Convex Optimization"
    % Joëlle Skaf - 08/29/05
    % (a figure is generated)
    %
    % Let p_star(u) denote the optimal value of:
    %           minimize    x^2 + 1
    %               s.t.    (x-2)(x-2)<=u
    % Finds p_star(u) and plots it versus u.

    fprintf(1,'Computing p_star(u)...\n ');

    u = linspace(-0.9,10,50);
    p_star = zeros(1,length(u));
    for i = 1:length(u)
        disp(['for u = ' num2str(u(i))]);
        % perturbed problem
        cvx_begin quiet
            variable x(1)
            minimize ( quad_form(x,1) + 1 )
            quad_form(x,1) - 6*x + 8 <= u(i);
        cvx_end
        % optimal value
        p_star(i) = cvx_optval;
    end

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

    % Plots
    plot(u,p_star)
    axis([-2 10 -2 10])
    xlabel('u');
    ylabel('p^*(u)')
    title('Sensitivity analysis: p^*(u) vs u');

    Computing p_star(u)...
     for u = -0.9
    for u = -0.67755
    for u = -0.4551
    for u = -0.23265
    for u = -0.010204
    for u = 0.21224
    for u = 0.43469
    for u = 0.65714
    for u = 0.87959
    for u = 1.102
    for u = 1.3245
    for u = 1.5469
    for u = 1.7694
    for u = 1.9918
    for u = 2.2143
    for u = 2.4367
    for u = 2.6592
    for u = 2.8816
    for u = 3.1041
    for u = 3.3265
    for u = 3.549
    for u = 3.7714
    for u = 3.9939
    for u = 4.2163
    for u = 4.4388
    for u = 4.6612
    for u = 4.8837
    for u = 5.1061
    for u = 5.3286
    for u = 5.551
    for u = 5.7735
    for u = 5.9959
    for u = 6.2184
    for u = 6.4408
    for u = 6.6633
    for u = 6.8857
    for u = 7.1082
    for u = 7.3306
    for u = 7.5531
    for u = 7.7755
    for u = 7.998
    for u = 8.2204
    for u = 8.4429
    for u = 8.6653
    for u = 8.8878
    for u = 9.1102
    for u = 9.3327
    for u = 9.5551
    for u = 9.7776
    for u = 10
    Done!