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R1252/simulation-code/cvx/examples/cvxbook/Ch06_approx_fitting/penalty_comp_cvx.m
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penalty_comp_cvx.m
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% Figure 6.2: Penalty function approximation
% Section 6.1.2
% Boyd & Vandenberghe "Convex Optimization"
% Original by Lieven Vandenberghe
% Adapted for CVX Argyris Zymnis - 10/2005
%
% Comparison of the ell1, ell2, deadzone-linear and log-barrier
% penalty functions for the approximation problem:
% minimize phi(A*x-b),
%
% where phi(x) is the penalty function
% Log-barrier will be implemented in the future version of CVX
% Generate input data
randn('state',0);
m=100; n=30;
A = randn(m,n);
b = randn(m,1);
% ell_1 approximation
% minimize ||Ax+b||_1
disp('ell-one approximation');
cvx_begin
variable x1(n)
minimize(norm(A*x1+b,1))
cvx_end
% ell_2 approximation
% minimize ||Ax+b||_2
disp('ell-2');
x2=-A\b;
% deadzone penalty approximation
% minimize sum(deadzone(Ax+b,0.5))
% deadzone(y,z) = max(abs(y)-z,0)
dz = 0.5;
disp('deadzone penalty');
cvx_begin
variable xdz(n)
minimize(sum(max(abs(A*xdz+b)-dz,0)))
cvx_end
% log-barrier penalty approximation
%
% minimize -sum log(1-(ai'*x+bi)^2)
disp('log-barrier')
% parameters for Newton Method & line search
alpha=.01; beta=.5;
% minimize linfty norm to get starting point
cvx_begin
variable xlb(n)
minimize norm(A*xlb+b,Inf)
cvx_end
linf = cvx_optval;
A = A/(1.1*linf);
b = b/(1.1*linf);
for iters = 1:50
yp = 1 - (A*xlb+b); ym = (A*xlb+b) + 1;
f = -sum(log(yp)) - sum(log(ym));
g = A'*(1./yp) - A'*(1./ym);
H = A'*diag(1./(yp.^2) + 1./(ym.^2))*A;
v = -H\g;
fprime = g'*v;
ntdecr = sqrt(-fprime);
if (ntdecr < 1e-5), break; end;
t = 1;
newx = xlb + t*v;
while ((min(1-(A*newx +b)) < 0) | (min((A*newx +b)+1) < 0))
t = beta*t;
newx = xlb + t*v;
end;
newf = -sum(log(1 - (A*newx+b))) - sum(log(1+(A*newx+b)));
while (newf > f + alpha*t*fprime)
t = beta*t;
newx = xlb + t*v;
newf = -sum(log(1-(A*newx+b))) - sum(log(1+(A*newx+b)));
end;
xlb = xlb+t*v;
end
% Plot histogram of residuals
ss = max(abs([A*x1+b; A*x2+b; A*xdz+b; A*xlb+b]));
tt = -ceil(ss):0.05:ceil(ss); % sets center for each bin
[N1,hist1] = hist(A*x1+b,tt);
[N2,hist2] = hist(A*x2+b,tt);
[N3,hist3] = hist(A*xdz+b,tt);
[N4,hist4] = hist(A*xlb+b,tt);
range_max=2.0; rr=-range_max:1e-2:range_max;
figure(1), clf, hold off
subplot(4,1,1),
bar(hist1,N1);
hold on
plot(rr, abs(rr)*40/3, '-');
ylabel('p=1')
axis([-range_max range_max 0 40]);
hold off
subplot(4,1,2),
bar(hist2,N2);
hold on;
plot(rr,2*rr.^2),
ylabel('p=2')
axis([-range_max range_max 0 11]);
hold off
subplot(4,1,3),
bar(hist3,N3);
hold on
plot(rr,30/3*max(0,abs(rr)-dz))
ylabel('Deadzone')
axis([-range_max range_max 0 25]);
hold off
subplot(4,1,4),
bar(hist4,N4);
rr_lb=linspace(-1+(1e-6),1-(1e-6),600);
hold on
plot(rr_lb, -3*log(1-rr_lb.^2),rr,2*rr.^2,'--')
axis([-range_max range_max 0 11]);
ylabel('Log barrier'),
xlabel('r')
hold off
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