```% Boyd, Kim, Vandenberghe, and Hassibi, "A Tutorial on Geometric Programming"
% Written for CVX by Almir Mutapcic 02/08/06
% (a figure is generated)
%
% Solves the problem of choosing gate scale factors x_i to give
% minimum ckt delay, subject to limits on the total area and power.
%
%   minimize   D
%       s.t.   P <= Pmax, A <= Amax
%              x >= 1
%
% where variables are scale factors x.
%
% This code uses matrices in order to evaluate signal paths
% through the circuit (thus, it uses vectorize Matlab features).
% It is specific to the digital circuit shown in figure 4 (page 28)
% of GP tutorial paper.

% digital circuit shown in figure 4 (page 28) of GP tutorial paper
m = 7;  % number of cells
n = 8;  % number of edges
A = sparse(m,n);

% A is standard cell-edge incidence matrix of the circuit
% A_ij = 1 if edge j comes out of cell i, -1 if it comes in, 0 otherwise
A(1,1) =     1;
A(2,2) =     1;
A(2,3) =     1;
A(3,4) =     1;
A(3,8) =     1;
A(4,1) =    -1;
A(4,2) =    -1;
A(4,5) =     1;
A(4,6) =     1;
A(5,3) =    -1;
A(5,4) =    -1;
A(5,7) =     1;
A(6,5) =    -1;
A(7,6) =    -1;
A(7,7) =    -1;
A(7,8) =    -1;

% decompose A into edge outgoing and edge-incoming part
Aout = double(A > 0);
Ain = double(A < 0);

% problem constants
f = [1 0.8 1 0.7 0.7 0.5 0.5]';
e = [1 2 1 1.5 1.5 1 2]';
Cout6 = 10;
Cout7 = 10;

a     = ones(m,1);
alpha = ones(m,1);
beta  = ones(m,1);
gamma = ones(m,1);

% varying parameters for an optimal trade-off curve
N = 20;
Pmax = linspace(10,100,N);
Amax = [25 50 100];
min_delay = zeros(length(Amax),N);

for k = 1:length(Amax)
fprintf( 'Amax = %d:\n', Amax(k) );
for n = 1:N
fprintf( '    Pmax = %6.2f: ', Pmax(n) );
cvx_begin gp quiet
% optimization variables
variable x(m)                 % scale factors
variable t(m)                 % arrival times

% objective is the upper bound on the overall delay
% and that is the max of arrival times for output gates 6 and 7
minimize( max( t(6),t(7) ) )
subject to
% input capacitance is an affine function of sizes
cin = alpha + beta.*x;

% load capacitance is the input capacitance times the fan-out matrix
% given by Fout = Aout*Ain'

% delay is the product of its driving resistance R = gamma./x and cload

% power and area definitions
power = (f.*e)'*x;
area = a'*x;

% scale size, power, and area constraints
x >= 1;
power <= Pmax(n);
area <= Amax(k);

% create timing constraints
% these constraints enforce t_j + d_j <= t_i over all gates j that drive gate i
Aout'*t + Ain'*d <= Ain'*t;

% for gates with inputs not connected to other gates we enforce d_i <= t_i
d(1:3) <= t(1:3);
cvx_end
fprintf( 'delay = %3.2f\n', cvx_optval );
min_delay(k,n) = cvx_optval;
end
end

plot(Pmax,min_delay(1,:), Pmax,min_delay(2,:), Pmax,min_delay(3,:));
xlabel('Pmax'); ylabel('Dmin');
```
```Generating the optimal tradeoff curve...
Amax = 25:
Pmax =  10.00: delay = 12.21
Pmax =  14.74: delay = 9.41
Pmax =  19.47: delay = 8.01
Pmax =  24.21: delay = 7.11
Pmax =  28.95: delay = 6.80
Pmax =  33.68: delay = 6.80
Pmax =  38.42: delay = 6.80
Pmax =  43.16: delay = 6.80
Pmax =  47.89: delay = 6.80
Pmax =  52.63: delay = 6.80
Pmax =  57.37: delay = 6.80
Pmax =  62.11: delay = 6.80
Pmax =  66.84: delay = 6.80
Pmax =  71.58: delay = 6.80
Pmax =  76.32: delay = 6.80
Pmax =  81.05: delay = 6.80
Pmax =  85.79: delay = 6.80
Pmax =  90.53: delay = 6.80
Pmax =  95.26: delay = 6.80
Pmax = 100.00: delay = 6.80
Amax = 50:
Pmax =  10.00: delay = 12.21
Pmax =  14.74: delay = 9.41
Pmax =  19.47: delay = 8.01
Pmax =  24.21: delay = 7.11
Pmax =  28.95: delay = 6.46
Pmax =  33.68: delay = 5.97
Pmax =  38.42: delay = 5.59
Pmax =  43.16: delay = 5.27
Pmax =  47.89: delay = 5.01
Pmax =  52.63: delay = 4.79
Pmax =  57.37: delay = 4.71
Pmax =  62.11: delay = 4.71
Pmax =  66.84: delay = 4.71
Pmax =  71.58: delay = 4.71
Pmax =  76.32: delay = 4.71
Pmax =  81.05: delay = 4.71
Pmax =  85.79: delay = 4.71
Pmax =  90.53: delay = 4.71
Pmax =  95.26: delay = 4.71
Pmax = 100.00: delay = 4.71
Amax = 100:
Pmax =  10.00: delay = 12.21
Pmax =  14.74: delay = 9.41
Pmax =  19.47: delay = 8.01
Pmax =  24.21: delay = 7.11
Pmax =  28.95: delay = 6.46
Pmax =  33.68: delay = 5.97
Pmax =  38.42: delay = 5.59
Pmax =  43.16: delay = 5.27
Pmax =  47.89: delay = 5.01
Pmax =  52.63: delay = 4.79
Pmax =  57.37: delay = 4.60
Pmax =  62.11: delay = 4.43
Pmax =  66.84: delay = 4.28
Pmax =  71.58: delay = 4.15
Pmax =  76.32: delay = 4.03
Pmax =  81.05: delay = 3.92
Pmax =  85.79: delay = 3.82
Pmax =  90.53: delay = 3.73
Pmax =  95.26: delay = 3.65
Pmax = 100.00: delay = 3.57