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consensustest.m
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%The system
k11 = 2; k12 = 1; k21 = 1; k22 = 2;
L1 = 150; o1 = 30; L2 = 80; o2 = 0;
K = [k11, k12 ; k21 , k22];
L = [L1;L2]; o = [o1;o2];
%The cost function
c1 = 1; c2 = 3; q1 = 0.; q2 = 0.;
c = [c1 c2]; Q = [q1 0; 0 q2];
% SOLVE WITH MATLAB QUADPROG
A = -K; b = [o1-L1; o2-L2];
lb = [0;0]; ub = [100;100];
disp('Matlab solutions')
d = quadprog(Q,c,A,b,[],[],lb,ub)
l = K*d+o
% SOLVE WITH CONSENSUS
rho = 0.1;
%node 1 initialization
d1 = [0;0];
d1_av = [0;0];
d2_copy = [0;0];
y1 = [0;0];
k1 = [k11;k12];
%node 2 initialization
d2 = [0;0];
d2_av = [0;0];
d1_copy = [0;0];
y2 = [0;0];
k2 = [k21;k22];
%iterations
for i=1:50,
% node 1
d11_best = -1;
d12_best = -1;
min_best_1(i) = 100000; %big number
sol_unconstrained = 1;
sol_boundary_linear = 1;
sol_boundary_0 = 1;
sol_boundary_100 = 1;
sol_linear_0 = 1;
sol_linear_100 = 1;
z11 = -c1 - y1(1) + rho*d1_av(1);
z12 = -y1(2) + rho*d1_av(2);
u1 = o1-L1;
u2 = 0;
u3 = 100;
p11 = 1/(rho+q1);
p12 = 1/rho;
n = k11*k11*p11 + k12*k12*p12;
w1 = -k11*p11*z11-k12*z12*p12;
w2 = -z11*p11;
w3 = z11*p11;
%compute unconstrained minimum
d11u = p11*z11;
d12u = p12*z12;
%check feasibility of unconstrained minimum using local constraints
if (d11u < 0), sol_unconstrained = 0; end;
if (d11u > 100), sol_unconstrained = 0; end;
if (k11*d11u + k12*d12u < L1-o1), sol_unconstrained = 0; end;
% compute function value and if best store new optimum
if sol_unconstrained,
min_unconstrained = 0.5*q1*d11u^2 + c1*d11u + y1(1)*(d11u-d1_av(1)) + ...
y1(2)*(d12u-d1_av(2)) + rho/2*(d11u-d1_av(1))^2 + rho/2*(d12u-d1_av(2))^2;
if min_unconstrained < min_best_1(i),
d11_best = d11u;
d12_best = d12u;
min_best_1(i) = min_unconstrained;
end;
end;
%compute minimum constrained to linear boundary
d11bl = p11*z11+p11*k11/n*(w1-u1);
d12bl = p12*z12+p12*k12/n*(w1-u1);
%check feasibility of minimum constrained to linear boundary
if (d11bl < 0), sol_boundary_linear = 0; end;
if (d11bl > 100), sol_boundary_linear = 0; end;
% compute function value and if best store new optimum
if sol_boundary_linear,
min_boundary_linear = 0.5*q1*d11bl^2 + c1*d11bl + y1(1)*(d11bl-d1_av(1)) + ...
y1(2)*(d12bl-d1_av(2)) + rho/2*(d11bl-d1_av(1))^2 + rho/2*(d12bl-d1_av(2))^2;
if min_boundary_linear < min_best_1(i),
d11_best = d11bl;
d12_best = d12bl;
min_best_1(i) = min_boundary_linear;
end;
end;
%compute minimum constrained to 0 boundary
d11b0 = 0;
d12b0 = p12*z12;
%check feasibility of minimum constrained to 0 boundary
if (d11b0 > 100), sol_boundary_0 = 0; end;
if (k11*d11b0 + k12*d12b0 < L1-o1), sol_boundary_0 = 0; end;
% compute function value and if best store new optimum
if sol_boundary_0,
min_boundary_0 = 0.5*q1*d11b0^2 + c1*d11b0 + y1(1)*(d11b0-d1_av(1)) + ...
y1(2)*(d12b0-d1_av(2)) + rho/2*(d11b0-d1_av(1))^2 + rho/2*(d12b0-d1_av(2))^2;
if min_boundary_0 < min_best_1(i),
d11_best = d11b0;
d12_best = d12b0;
min_best_1(i) = min_boundary_0;
end;
end;
%compute minimum constrained to 100 boundary
d11b100 = 100;
d12b100 = p12*z12;
%check feasibility of minimum constrained to 100 boundary
if (d11b0 < 0), sol_boundary_100 = 0; end;
if (k11*d11b100 + k12*d12b100 < L1-o1), sol_boundary_100 = 0; end;
% compute function value and if best store new optimum
if sol_boundary_100,
min_boundary_100 = 0.5*q1*d11b100^2 + c1*d11b100 + y1(1)*(d11b100-d1_av(1)) + ...
y1(2)*(d12b100-d1_av(2)) + rho/2*(d11b100-d1_av(1))^2 + rho/2*(d12b100-d1_av(2))^2;
if min_boundary_100 < min_best_1(i),
d11_best = d11b100;
d12_best = d12b100;
min_best_1(i) = min_boundary_100;
end;
end;
% compute minimum constrained to linear and zero boundary
common = (rho+q1)/((rho+q1)*n-k11*k11);
det1 = common;
det2 = -k11*common;
det3 = det2;
det4 = n*(rho+q1)*common;
x1 = det1*w1 + det2*w2;
x2 = det3*w1 + det4*w2;
v1 = det1*u1 + det2*u2; %u2 = 0 so this can be simplified
v2 = det3*u1 + det4*u2; %u2 = 0 so this can be simplified
d11l0 = p11*z11+p11*k11*(x1-v1)+p11*(x2-v2);
d12l0 = p12*z12+p12*k12*(x1-v1);
%check feasibility
if (d11l0 > 100), sol_linear_0 = 0; end;
% compute function value and if best store new optimum
if sol_linear_0,
min_linear_0 = 0.5*q1*d11l0^2 + c1*d11l0 + y1(1)*(d11l0-d1_av(1)) + ...
y1(2)*(d12l0-d1_av(2)) + rho/2*(d11l0-d1_av(1))^2 + rho/2*(d12l0-d1_av(2))^2;
if min_linear_0 < min_best_1(i),
d11_best = d11l0;
d12_best = d12l0;
min_best_1(i) = min_linear_0;
end;
end;
% compute minimum constrained to linear and 100 boundary
common = (rho+q1)/((rho+q1)*n-k11*k11);
det1 = common;
det2 = k11*common;
det3 = det2;
det4 = n*(rho+q1)*common;
x1 = det1*w1 + det2*w3;
x2 = det3*w1 + det4*w3;
v1 = det1*u1 + det2*u3;
v2 = det3*u1 + det4*u3;
d11l100 = p11*z11+p11*k11*(x1-v1)-p11*(x2-v2);
d12l100 = p12*z12+p12*k12*(x1-v1);
%check feasibility
if (d11l100 < 0), sol_linear_100 = 0; end;
% compute function value and if best store new optimum
if sol_linear_100,
min_linear_100 = 0.5*q1*d11l100^2 + c1*d11l100 + y1(1)*(d11l100-d1_av(1)) + ...
y1(2)*(d12l100-d1_av(2)) + rho/2*(d11l100-d1_av(1))^2 + rho/2*(d12l100-d1_av(2))^2;
if min_linear_100 < min_best_1(i),
d11_best = d11u;
d12_best = d12u;
min_best_1_(i) = min_linear100;
end;
end;
%store data and save for next cycle
best_d11(i) = d11_best;
best_d12(i) = d12_best;
d1 = [d11_best;d12_best];
%check with matlab quadprog
Q = [q1+rho, 0; 0 rho];
c = [c1+y1(1)-rho*d1_av(1),y1(2)-rho*d1_av(2)];
A = [-k11 -k12; -1 0; 1 0];
b = [o1-L1, 0, 100];
d1_ = quadprog(Q,c,A,b,[],[],[],[]);
%
% node 2
d21_best = -1;
d22_best = -1;
min_best_2(i) = 100000; %big number
sol_unconstrained = 1;
sol_boundary_linear = 1;
sol_boundary_0 = 1;
sol_boundary_100 = 1;
sol_linear_0 = 1;
sol_linear_100 = 1;
z22 = -c2 - y2(2) + rho*d2_av(2);
z21 = -y2(1) + rho*d2_av(1);
u1 = o2-L2;
u2 = 0;
u3 = 100;
p22 = 1/(rho+q2);
p21 = 1/rho;
n = k22*k22*p22 + k21*k21*p21;
w1 = -k22*p22*z22-k21*z21*p21;
w2 = -z22*p22;
w3 = z22*p22;
%compute unconstrained minimum
d21u = p21*z21;
d22u = p22*z22;
%check feasibility of unconstrained minimum using local constraints
if (d22u < 0), sol_unconstrained = 0; end;
if (d22u > 100), sol_unconstrained = 0; end;
if (k21*d21u + k22*d22u < L2-o2), sol_unconstrained = 0; end;
% compute function value and if best store new optimum
if sol_unconstrained,
min_unconstrained = 0.5*q2*d22u^2 + c2*d22u + y2(1)*(d21u-d2_av(1)) + ...
y2(2)*(d22u-d2_av(2)) + rho/2*(d21u-d2_av(1))^2 + rho/2*(d22u-d2_av(2))^2;
if min_unconstrained < min_best_2(i),
d21_best = d21u;
d22_best = d22u;
min_best_2(i) = min_unconstrained;
end;
end;
%compute minimum constrained to linear boundary
d21bl = p21*z21+p21*k21/n*(w1-u1);
d22bl = p22*z22+p22*k22/n*(w1-u1);
%check feasibility of minimum constrained to linear boundary
if (d22bl < 0), sol_boundary_linear = 0; end;
if (d22bl > 100), sol_boundary_linear = 0; end;
% compute function value and if best store new optimum
if sol_boundary_linear,
min_boundary_linear = 0.5*q2*d22bl^2 + c2*d22bl + y2(1)*(d21bl-d2_av(1)) + ...
y2(2)*(d22bl-d2_av(2)) + rho/2*(d21bl-d2_av(1))^2 + rho/2*(d22bl-d2_av(2))^2;
if min_boundary_linear < min_best_2(i),
d21_best = d21bl;
d22_best = d22bl;
min_best_2(i) = min_boundary_linear;
end;
end;
%compute minimum constrained to 0 boundary
d22b0 = 0;
d21b0 = p21*z21;
%check feasibility of minimum constrained to 0 boundary
if (k21*d21b0 + k22*d22b0 < L2-o2), sol_boundary_0 = 0; end;
if (d22b0 > 100), sol_boundary_0 = 0; end;
% compute function value and if best store new optimum
if sol_boundary_0,
min_boundary_0 = 0.5*q2*d22b0^2 + c2*d22b0 + y2(1)*(d21b0-d2_av(1)) + ...
y2(2)*(d22b0-d2_av(2)) + rho/2*(d21b0-d2_av(1))^2 + rho/2*(d22b0-d2_av(2))^2;
if min_boundary_0 < min_best_2(i),
d21_best = d21b0;
d22_best = d22b0;
min_best_2(i) = min_boundary_0;
end;
end;
%compute minimum constrained to 100 boundary
d22b100 = 100;
d21b100 = p21*z21;
%check feasibility of minimum constrained to 100 boundary
if (k21*d21b100 + k22*d22b100 < L2-o2), sol_boundary_100 = 0; end;
if (d22b100 < 0), sol_boundary_100 = 0; end;
% compute function value and if best store new optimum
if sol_boundary_100,
min_boundary_100 = 0.5*q2*d22b100^2 + c2*d22b100 + y2(1)*(d21b100-d2_av(1)) + ...
y2(2)*(d22b100-d2_av(2)) + rho/2*(d21b100-d2_av(1))^2 + rho/2*(d22b100-d2_av(2))^2;
if min_boundary_100 < min_best_2(i),
d21_best = d21b100;
d22_best = d22b100;
min_best_2(i) = min_boundary_100;
end;
end;
% compute minimum constrained to linear and zero boundary
common = (rho+q2)/((rho+q2)*n-k22*k22);
det1 = common;
det2 = -k22*common;
det3 = det2;
det4 = n*(rho+q2)*common;
x1 = det1*w1 + det2*w2;
x2 = det3*w1 + det4*w2;
v1 = det1*u1 + det2*u2; %u2 = 0 so this can be simplified
v2 = det3*u1 + det4*u2; %u2 = 0 so this can be simplified
d22l0 = p22*z22+p22*k22*(x1-v1)+p22*(x2-v2);
d21l0 = p21*z21+p21*k21*(x1-v1);
%check feasibility
if(d22l0 > 100), sol_linear_0 = 0; end;
% compute function value and if best store new optimum
if sol_linear_0,
min_linear_0 = 0.5*q2*d22l0^2 + c2*d22l0 + y2(1)*(d21l0-d2_av(1)) + ...
y2(2)*(d22l0-d2_av(2)) + rho/2*(d21l0-d2_av(1))^2 + rho/2*(d22l0-d2_av(2))^2;
if min_linear_0 < min_best_2(i),
d21_best = d21l0;
d22_best = d22l0;
min_best_2(i) = min_linear_0;
end;
end;
% compute minimum constrained to linear and 100 boundary
common = (rho+q2)/((rho+q2)*n-k22*k22);
det1 = common;
det2 = k22*common;
det3 = det2;
det4 = n*(rho+q2)*common;
x1 = det1*w1 + det2*w3;
x2 = det3*w1 + det4*w3;
v1 = det1*u1 + det2*u3;
v2 = det3*u1 + det4*u3;
d22l100 = p22*z22+p22*k22*(x1-v1)-p22*(x2-v2);
d21l100 = p21*z21+p21*k21*(x1-v1);
%check feasibility
if (d22l100 < 0), sol_linear_100 = 0; end;
%now must choose the minimum among the feasible solutions
% compute function value and if best store new optimum
if sol_linear_100,
min_linear_100 = 0.5*q2*d22l100^2 + c2*d22l100 + y2(1)*(d21l100-d2_av(1)) + ...
y2(2)*(d22l100-d2_av(2)) + rho/2*(d21l100-d2_av(1))^2 + rho/2*(d22l100-d2_av(2))^2;
if min_linear_100 < min_best_2(i),
d21_best = d21u;
d22_best = d22u;
min_best_2(i) = min_linear100;
end;
end;
%store data and save for next cycle
best_d21(i) = d21_best;
best_d22(i) = d22_best;
d2 = [d21_best;d22_best];
%check with matlab quadprog
Q = [rho, 0; 0 rho+q2];
c = [y2(1)-rho*d2_av(1),c2+y2(2)-rho*d2_av(2)];
A = [-k21 -k22;0 -1; 0 1];
b = [o2-L2, 0, 100];
d2_ = quadprog(Q,c,A,b,[],[],[],[]);
%gather-scatter with consensus
d1_av = (d1+d2)/2;
d2_av = (d1+d2)/2;
y1 = y1 + rho*(d1-d1_av);
y2 = y2 + rho*(d2-d2_av);
av1(i) = d1_av(1);
av2(i) = d1_av(2);
end;
disp('Consensus Solutions')
d_ = d2_av
l_ = K*d_+o
%Plots
figure(10);
plot(1:50, av1, 1:50, av2);
legend('d_1','d_2');
title('primal vars');
xlabel('iter');
figure(15);
t = 0:100;
constr1 = L1-o1-2*t;
constr2 = (L2-o2)/2-t/2;
[x,y]=meshgrid(t,t);
hold on;
z = c1*x+c2*y+q1*x.^2+q2*y.^2;
contour(x,y,z);
plot(t,constr1,t,constr2,'LineWidth',2);
plot(t,zeros(size(t)),'k','LineWidth',2);
plot(zeros(size(t)),t,'k','LineWidth',2);
plot(t,100*ones(size(t)),'k','LineWidth',2);
plot(100*ones(size(t)),t,'k','LineWidth',2);
plot(av1,av2,'--','LineWidth',2);
plot(av1,av2,'bo');
title('trajectory');
xlabel('d_1');
ylabel('d_2');
plot(d(1),d(2),'r*')
axis([-10,110,-10,110]);
hold off;