Section 4.5.4: Minimization of Peron-Frobenius norm (GP)

% Boyd & Vandenberghe "Convex Optimization"
% (see page 165-167 for more details)
% Written for CVX 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 gr0wth 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)

% >> eig(A) (answer checks)
%
%    0.8041
%   -0.2841
%   -0.0100 + 0.2263i
%   -0.0100 - 0.2263i
 
Calling SeDuMi: 165 variables (6 free), 103 equality constraints
------------------------------------------------------------------------
SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
Split 6 free variables
eqs m = 103, order n = 60, dim = 284, blocks = 5
nnz(A) = 305 + 0, nnz(ADA) = 2373, nnz(L) = 1252
 it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
  0 :            7.81E-001 0.000
  1 : -2.32E+000 2.64E-001 0.000 0.3378 0.9000 0.9000   1.75  1  1  2.7E+000
  2 :  5.98E-001 8.41E-002 0.000 0.3188 0.9000 0.9000   2.58  1  1  4.3E-001
  3 : -1.43E-001 2.24E-002 0.000 0.2661 0.9000 0.9000   1.31  1  1  3.8E-001
  4 : -1.86E-001 8.18E-003 0.000 0.3658 0.9000 0.9000   1.15  1  1  1.1E-001
  5 : -1.89E-001 3.38E-003 0.000 0.4128 0.9000 0.9000   1.10  1  1  4.4E-002
  6 : -2.09E-001 7.95E-004 0.000 0.2355 0.9000 0.9000   1.11  1  1  9.1E-003
  7 : -2.16E-001 1.73E-004 0.000 0.2173 0.9000 0.9000   1.09  1  1  1.9E-003
  8 : -2.18E-001 9.83E-006 0.000 0.0569 0.9900 0.9900   1.04  1  1  1.0E-004
  9 : -2.18E-001 8.02E-008 0.000 0.0082 0.9901 0.9900   1.00  1  1  1.0E-006
 10 : -2.18E-001 6.75E-010 0.302 0.0084 0.9900 0.9413   1.00  2  3  4.4E-008
 11 : -2.18E-001 2.43E-011 0.000 0.0360 0.9902 0.9900   1.01  5  5  1.5E-009

iter seconds digits       c*x               b*y
 11      0.3   Inf -2.1807011411e-001 -2.1807010471e-001
|Ax-b| =  1.0e-008, [Ay-c]_+ =  6.6E-010, |x|= 2.8e+001, |y|= 4.7e+000

Detailed timing (sec)
   Pre          IPM          Post
1.001E-002    3.305E-001    1.001E-002    
Max-norms: ||b||=1.479659e+001, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 141.66.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.804069
 
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