```% Boyd, Kim, Vandenberghe, and Hassibi, "A tutorial on geometric programming"
% Joshi, Boyd, and Dutton, "Optimal doping profiles via geometric programming"
% Written for CVX by Almir Mutapcic 02/08/06
% (a figure is generated)
%
% Determines the optimal doping profile that minimizes base transit
% time subject to a lower bound constraint on the current gain (beta).
% This problem can be posed as a GP:
%
%   minimize   tau_B
%       s.t.   Nmin <= v <= Nmax
%              y_(i+1) + v_i^const1 <= y_i
%              w_(i+1) + v_i^const2 <= w_i, etc...
%              beta => beta_min
%
% where variables are v_i, y_i, and w_i.

% problem size
M = 20;

% problem constants
g1 = 0.42;
g2 = 0.69;
Nmax = 5*10^18;
Nmin = 5*10^16;
Nref = 10^17;
Dn0 = 20.72;
ni0 = 1.4*(10^10);
WB = 10^(-5);
C =  WB^2/((M^2)*(Nref^g1)*Dn0);

% minimum current gain values
beta_min_GE = [1 1.4 1.8 2.2 2.6 3.0 3.4 3.43]*(1e-11);

% exponent powers
pwi = g2 -1;
pwj = 1+g1-g2;

v_array = [];
for k = 1:length(beta_min_GE)
fprintf( 'beta_min_GE = %g: ', beta_min_GE(k) );
cvx_begin gp quiet
% optimization variables
variables v(M) y(M) w(M)

% objective function is the base transmit time
tau_B = C*w(1);

minimize( tau_B )
subject to
% fixed problem constraints
Nmin <= v <= Nmax;

for i=1:M-1
y(i+1) + v(i)^pwj <= y(i);
w(i+1) + y(i)*v(i)^pwi <= w(i);
end

% equalities
y(M) == v(M)^pwj;
w(M) == y(M)*v(M)^pwi;

% changing constraint
(WB*beta_min_GE(k)/(M*Nref^(g1-g2)*Dn0))*y(1) <= 1;
cvx_end
fprintf( '%s\n', cvx_status );
% keep the optimal solution
v_array = [v_array v];
end

% plot the basic optimal doping profile
figure, clf
nbw = 0:1/M:1-1/M;
for k = 1:length(beta_min_GE)
semilogy(nbw,v_array(:,k),'LineWidth',2); hold on;
end
axis([0 1 1e16 1e19]);
xlabel('base');
ylabel('doping');
text(0,Nmin,'Nmin ', 'HorizontalAlignment','right');
text(0,Nmax,'Nmax ', 'HorizontalAlignment','right');
hold off;
```
```beta_min_GE = 1e-11: Solved
beta_min_GE = 1.4e-11: Solved
beta_min_GE = 1.8e-11: Solved
beta_min_GE = 2.2e-11: Solved
beta_min_GE = 2.6e-11: Solved
beta_min_GE = 3e-11: Solved
beta_min_GE = 3.4e-11: Solved
beta_min_GE = 3.43e-11: Solved
```