# Section 4.5.4: Minimum spectral radius via Peron-Frobenius theory (GP)

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    % Boyd & Vandenberghe "Convex Optimization"
    % Joelle Skaf - 01/29/06
    % Updated to use CVX mode by Almir Mutapcic 02/08/06
    %
    % The goal is to minimize the spectral radius of a square matrix A
    % which is elementwise nonnegative, Aij >= 0 for all i,j. In this
    % case A has a positive real eigenvalue lambda_pf (the Perron-Frobenius
    % eigenvalue) which is equal to the spectral radius, and thus gives
    % the fastest decay rate or slowest growth rate.
    % The problem of minimizing the Perron-Frobenius eigenvalue of A,
    % possibly subject to posynomial inequalities in some underlying
    % variable x can be posed as a GP (for example):
    %
    %   minimize   lambda_pf( A(x) )
    %       s.t.   f_i(x) <= 1   for i = 1,...,p
    %
    % where matrix A entries are some posynomial functions of variable x,
    % and f_i are posynomials.
    %
    % We consider a specific example in which we want to find the fastest
    % decay or slowest growth rate for the bacteria population governed
    % by a simple dynamic model (see page 166). The problem is a GP:
    %   minimize   lambda
    %       s.t.   b1*v1 + b2*v2 + b3*v3 + b4*v4 <= lambda*v1
    %              s1*v1 <= lambda*v2
    %              s2*v2 <= lambda*v3
    %              s3*v3 <= lambda*v4
    %              1/2 <= ci <= 2
    %              bi == bi^{nom}*(c1/c1^{nom})^alpha_i*(c2/c2^{nom})^beta_i
    %              si == si^{nom}*(c1/c1^{nom})^gamma_i*(c2/c2^{nom})^delta_i
    %
    % with variables bi, si, ci, vi, lambda.

    % constants
    c_nom = [1 1]';
    b_nom = [2 3 2 1]';
    alpha = [1 1 1 1]'; beta  = [1 1 1 1]';
    s_nom = [1 1 3]';
    gamma = [1 1 1]'; delta = [1 1 1]';

    cvx_begin gp
      % optimization variables
      variables lambda b(4) s(3) v(4) c(2)

      % objective is the Perron-Frobenius eigenvalue
      minimize( lambda )
      subject to
        % inequality constraints
        b'*v      <= lambda*v(1);
        s(1)*v(1) <= lambda*v(2);
        s(2)*v(2) <= lambda*v(3);
        s(3)*v(3) <= lambda*v(4);
        [0.5; 0.5] <= c; c <= [2; 2];
        % equality constraints
        b == b_nom.*((ones(4,1)*(c(1)/c_nom(1))).^alpha).*...
                    ((ones(4,1)*(c(2)/c_nom(2))).^beta);
        s == s_nom.*((ones(3,1)*(c(1)/c_nom(1))).^gamma).*...
                    ((ones(3,1)*(c(2)/c_nom(2))).^delta);
    cvx_end

    % displaying results
    disp(' ')
    if lambda < 1
      fprintf(1,'The fastest decay rate of the bacteria population is %3.2f.\n', lambda);
    else
      fprintf(1,'The slowest growth rate of the bacteria population is %3.2f.\n', lambda);
    end
    disp(' ')
    fprintf(1,'The concentration of chemical 1 achieving this result is %3.2f.\n', c(1));
    fprintf(1,'The concentration of chemical 2 achieving this result is %3.2f.\n', c(2));
    disp(' ')

    % construct matrix A
    A = zeros(4,4);
    A(1,:) = b';
    A(2,1) = s(1);
    A(3,2) = s(2);
    A(4,3) = s(3);

    % eigenvalues of matrix A
    disp('Eigenvalues of matrix A are: ')
    eigA = eig(A)

    Successive approximation method to be employed.
       sedumi will be called several times to refine the solution.
       Original size: 21 variables, 11 equality constraints
       4 exponentials add 32 variables, 20 equality constraints
    -----------------------------------------------------------------
     Cones  |             Errors              |
    Mov/Act | Centering  Exp cone   Poly cone | Status
    --------+---------------------------------+---------
      4/  4 | 3.608e+00  9.922e-01  2.166e-07 | Solved
      4/  4 | 1.314e-01  1.230e-03  5.761e-07 | Solved
      3/  3 | 8.388e-03  4.893e-06  3.197e-07 | Solved
      0/  2 | 5.249e-04  2.081e-08  4.268e-08 | Solved
    -----------------------------------------------------------------
    Status: Solved
    Optimal value (cvx_optval): +0.804067

    The fastest decay rate of the bacteria population is 0.80.

    The concentration of chemical 1 achieving this result is 0.50.
    The concentration of chemical 2 achieving this result is 0.50.

    Eigenvalues of matrix A are:

    eigA =

       0.8041
      -0.2841
      -0.0100 + 0.2263i
      -0.0100 - 0.2263i