% 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 is specific to the digital circuit shown in figure 4
% (page 28) of GP tutorial paper. All the constraints and
% the worst-case delay expression are hard-coded for this
% particular circuit.
%
% A more general code with more precise models for digital cicuit
% sizing is also available as part of the CVX examples library.

% number of cells
m = 7;

% 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 = 25;
Pmax = linspace(10,100,N);
Amax = [25 50 100];
min_delay = zeros(length(Amax),N);

disp('Generating the optimal tradeoff curve...')

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

          % input capacitance is an affine function of sizes
          cin = alpha + beta.*x;

          % load capacitance of a gate is the sum of its fan-out c_in's
          clear cload; % start with a fresh variable
          cload(1) = cin(4);
          cload(2) = cin(4) + cin(5);
          cload(3) = cin(5) + cin(7);
          cload(4) = cin(6) + cin(7);
          cload(5) = cin(7);
          % output gates have their load capacitances
          cload(6) = Cout6;
          cload(7) = Cout7;

          % gate delay is the product of its driving res. R = gamma./x and cload
          d = (cload').*gamma./x;

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

          % evaluate delay over all paths in the given circuit (there are 7 paths)
          path_delays = [ ...
            d(1) + d(4) + d(6); % delay of path 1
            d(1) + d(4) + d(7); % delay of path 2, etc...
            d(2) + d(4) + d(6);
            d(2) + d(4) + d(7);
            d(2) + d(5) + d(7);
            d(3) + d(5) + d(6);
            d(3) + d(7) ];

          % overall circuit delay
          circuit_delay = ( max(path_delays) );

          % objective is the worst-case delay
          minimize( circuit_delay )
          subject to
            % construct the constraints
            x >= 1;             % all sizes greater than 1 (normalized)
            power <= Pmax(n);   % power constraint
            area <= Amax(k);    % area constraint
        cvx_end
        fprintf( 'delay = %3.2f\n', cvx_optval );
        min_delay(k,n) = cvx_optval;
    end
end

% plot the tradeoff curve
plot(Pmax,min_delay(1,:), Pmax,min_delay(2,:), Pmax,min_delay(3,:));
xlabel('Pmax'); ylabel('Dmin');
disp('Optimal tradeoff curve plotted.')

Generating the optimal tradeoff curve...
Amax = 25:
    Pmax =  10.00: delay = 12.21
    Pmax =  13.75: delay = 9.81
    Pmax =  17.50: delay = 8.51
    Pmax =  21.25: delay = 7.63
    Pmax =  25.00: delay = 6.98
    Pmax =  28.75: delay = 6.80
    Pmax =  32.50: delay = 6.80
    Pmax =  36.25: delay = 6.80
    Pmax =  40.00: delay = 6.80
    Pmax =  43.75: delay = 6.80
    Pmax =  47.50: delay = 6.80
    Pmax =  51.25: delay = 6.80
    Pmax =  55.00: delay = 6.80
    Pmax =  58.75: delay = 6.80
    Pmax =  62.50: delay = 6.80
    Pmax =  66.25: delay = 6.80
    Pmax =  70.00: delay = 6.80
    Pmax =  73.75: delay = 6.80
    Pmax =  77.50: delay = 6.80
    Pmax =  81.25: delay = 6.80
    Pmax =  85.00: delay = 6.80
    Pmax =  88.75: delay = 6.80
    Pmax =  92.50: delay = 6.80
    Pmax =  96.25: delay = 6.80
    Pmax = 100.00: delay = 6.80
Amax = 50:
    Pmax =  10.00: delay = 12.21
    Pmax =  13.75: delay = 9.81
    Pmax =  17.50: delay = 8.51
    Pmax =  21.25: delay = 7.63
    Pmax =  25.00: delay = 6.98
    Pmax =  28.75: delay = 6.48
    Pmax =  32.50: delay = 6.08
    Pmax =  36.25: delay = 5.75
    Pmax =  40.00: delay = 5.48
    Pmax =  43.75: delay = 5.24
    Pmax =  47.50: delay = 5.03
    Pmax =  51.25: delay = 4.85
    Pmax =  55.00: delay = 4.71
    Pmax =  58.75: delay = 4.71
    Pmax =  62.50: delay = 4.71
    Pmax =  66.25: delay = 4.71
    Pmax =  70.00: delay = 4.71
    Pmax =  73.75: delay = 4.71
    Pmax =  77.50: delay = 4.71
    Pmax =  81.25: delay = 4.71
    Pmax =  85.00: delay = 4.71
    Pmax =  88.75: delay = 4.71
    Pmax =  92.50: delay = 4.71
    Pmax =  96.25: delay = 4.71
    Pmax = 100.00: delay = 4.71
Amax = 100:
    Pmax =  10.00: delay = 12.21
    Pmax =  13.75: delay = 9.81
    Pmax =  17.50: delay = 8.51
    Pmax =  21.25: delay = 7.63
    Pmax =  25.00: delay = 6.98
    Pmax =  28.75: delay = 6.48
    Pmax =  32.50: delay = 6.08
    Pmax =  36.25: delay = 5.75
    Pmax =  40.00: delay = 5.48
    Pmax =  43.75: delay = 5.24
    Pmax =  47.50: delay = 5.03
    Pmax =  51.25: delay = 4.85
    Pmax =  55.00: delay = 4.69
    Pmax =  58.75: delay = 4.55
    Pmax =  62.50: delay = 4.42
    Pmax =  66.25: delay = 4.30
    Pmax =  70.00: delay = 4.19
    Pmax =  73.75: delay = 4.09
    Pmax =  77.50: delay = 4.00
    Pmax =  81.25: delay = 3.92
    Pmax =  85.00: delay = 3.84
    Pmax =  88.75: delay = 3.77
    Pmax =  92.50: delay = 3.70
    Pmax =  96.25: delay = 3.63
    Pmax = 100.00: delay = 3.57
Optimal tradeoff curve plotted.