n = 200;
K = spdiags([-ones(n,1), ones(n,1)],0:1, n,n);
K(end,end)=0;
n1=40;
n2 = 60;
n3 = 100;
q(1:n1) = linspace(1/n1,1,n1);
q(n1+1:n1+n2) = linspace(1-2/n2,-1,n2);
q(1+n1+n2:n1+n2+n3) = linspace(-1+1/n3,0,n3);
q = reshape(q, [n,1]);
u = K'*q;

isSubgradient = @(u,K,q)((abs(sum(abs(K*u))-sum((K'*q).*u))<0.000001) & (max(abs(q))<=1));
isSubgradient(u,K,q)
figure, plot(u./(max(abs(u))), 'b-', 'linewidth',3), hold on, plot(q, 'r--', 'linewidth',3)
leg = legend('q', 'u=K^T q');
set(leg, 'fontsize', 16);