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
variables lambda b(4) s(3) v(4) c(2)
minimize( lambda )
subject to
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];
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
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(' ')
A = zeros(4,4);
A(1,:) = b';
A(2,1) = s(1);
A(3,2) = s(2);
A(4,3) = s(3);
disp('Eigenvalues of matrix A are: ')
eigA = eig(A)
Calling Mosek 9.1.9: 49 variables, 11 equality constraints
------------------------------------------------------------
MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 11
Cones : 4
Scalar variables : 49
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 1
Eliminator terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 2 time : 0.00
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.00
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 11
Cones : 4
Scalar variables : 49
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 4
Optimizer - Cones : 4
Optimizer - Scalar variables : 16 conic : 12
Optimizer - Semi-definite variables: 0 scalarized : 0
Factor - setup time : 0.00 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 10 after factor : 10
Factor - dense dim. : 0 flops : 1.48e+02
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 4.2e+00 1.3e+00 5.0e+00 0.00e+00 1.346912185e-01 -3.913555187e+00 1.0e+00 0.00
1 5.2e-01 1.6e-01 2.9e-01 4.33e-01 4.973840520e-01 -9.468557263e-02 1.2e-01 0.01
2 9.0e-02 2.8e-02 1.6e-02 1.30e+00 -8.974357115e-02 -1.849264324e-01 2.2e-02 0.01
3 1.1e-02 3.5e-03 6.9e-04 1.14e+00 -2.052065597e-01 -2.164218567e-01 2.7e-03 0.01
4 1.5e-03 4.6e-04 3.3e-05 1.02e+00 -2.164139095e-01 -2.178955683e-01 3.6e-04 0.01
5 5.3e-05 1.6e-05 2.2e-07 1.00e+00 -2.179991570e-01 -2.180516153e-01 1.3e-05 0.01
6 8.2e-07 2.5e-07 4.2e-10 1.00e+00 -2.180711074e-01 -2.180719161e-01 2.0e-07 0.01
7 1.0e-07 3.2e-08 1.9e-11 1.00e+00 -2.180720730e-01 -2.180721757e-01 2.5e-08 0.01
8 4.9e-09 1.6e-09 2.2e-13 1.00e+00 -2.180721939e-01 -2.180721991e-01 1.3e-09 0.01
Optimizer terminated. Time: 0.02
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -2.1807219386e-01 nrm: 4e+00 Viol. con: 5e-09 var: 2e-09 cones: 3e-11
Dual. obj: -2.1807219907e-01 nrm: 1e+00 Viol. con: 0e+00 var: 1e-09 cones: 0e+00
Optimizer summary
Optimizer - time: 0.02
Interior-point - iterations : 8 time: 0.01
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
------------------------------------------------------------
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 + 0.0000i
-0.2841 + 0.0000i
-0.0100 + 0.2263i
-0.0100 - 0.2263i