Figure 6.2: Penalty function approximation
randn('state',0);
m=100; n=30;
A = randn(m,n);
b = randn(m,1);
disp('ell-one approximation');
cvx_begin
variable x1(n)
minimize(norm(A*x1+b,1))
cvx_end
disp('ell-2');
x2=-A\b;
dz = 0.5;
disp('deadzone penalty');
cvx_begin
variable xdz(n)
minimize(sum(max(abs(A*xdz+b)-dz,0)))
cvx_end
disp('log-barrier')
alpha=.01; beta=.5;
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
ss = max(abs([A*x1+b; A*x2+b; A*xdz+b; A*xlb+b]));
tt = -ceil(ss):0.05:ceil(ss);
[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
ell-one approximation
Calling sedumi: 230 variables, 100 equality constraints
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 100, order n = 203, dim = 232, blocks = 102
nnz(A) = 100 + 3000, nnz(ADA) = 100, nnz(L) = 100
Handling 31 + 1 dense columns.
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 1.53E+02 0.000
1 : 4.19E+01 7.27E+01 0.000 0.4759 0.9000 0.9000 3.41 1 1 1.8E+00
2 : 5.00E+01 2.82E+01 0.000 0.3883 0.9000 0.9000 1.71 1 1 6.5E-01
3 : 5.33E+01 9.83E+00 0.000 0.3483 0.9000 0.9000 1.21 1 1 2.3E-01
4 : 5.45E+01 3.25E+00 0.000 0.3302 0.9000 0.9000 1.07 1 1 7.9E-02
5 : 5.49E+01 9.31E-01 0.000 0.2868 0.9000 0.9000 1.02 1 1 2.3E-02
6 : 5.51E+01 2.79E-01 0.000 0.2998 0.9000 0.9039 1.01 1 1 6.7E-03
7 : 5.51E+01 5.33E-02 0.000 0.1909 0.9230 0.9000 1.00 1 1 1.7E-03
8 : 5.51E+01 9.92E-03 0.000 0.1862 0.9000 0.9101 1.00 1 1 2.5E-04
9 : 5.51E+01 2.00E-04 0.000 0.0202 0.9900 0.9901 1.00 1 1 4.1E-06
10 : 5.51E+01 5.64E-07 0.000 0.0028 0.9990 0.9990 1.00 4 4 9.6E-09
iter seconds digits c*x b*y
10 0.1 7.9 5.5128922241e+01 5.5128921562e+01
|Ax-b| = 1.7e-08, [Ay-c]_+ = 1.9E-10, |x|= 1.2e+01, |y|= 8.8e+00
Detailed timing (sec)
Pre IPM Post
1.000E-02 6.000E-02 0.000E+00
Max-norms: ||b||=1.957607e+00, ||c|| = 1,
Cholesky |add|=16, |skip| = 0, ||L.L|| = 1.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +55.1289
ell-2
deadzone penalty
Calling sedumi: 300 variables, 130 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 130, order n = 301, dim = 301, blocks = 101
nnz(A) = 3200 + 0, nnz(ADA) = 7000, nnz(L) = 3565
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 1.33E+01 0.000
1 : -5.72E+00 4.39E+00 0.000 0.3312 0.9000 0.9000 5.07 1 1 1.5E+00
2 : -1.61E+01 1.39E+00 0.000 0.3176 0.9000 0.9000 1.35 1 1 3.2E-01
3 : -1.99E+01 4.23E-01 0.000 0.3032 0.9000 0.9000 1.09 1 1 9.0E-02
4 : -2.10E+01 1.45E-01 0.000 0.3426 0.9000 0.9000 1.02 1 1 3.1E-02
5 : -2.13E+01 4.69E-02 0.000 0.3241 0.9000 0.9000 1.01 1 1 9.9E-03
6 : -2.13E+01 4.22E-05 0.000 0.0009 0.9000 0.0000 1.00 1 1 4.3E-03
7 : -2.14E+01 7.31E-06 0.000 0.1730 0.9133 0.9000 1.00 1 1 7.8E-04
8 : -2.15E+01 4.83E-07 0.000 0.0661 0.9902 0.9900 1.00 1 1 5.1E-05
9 : -2.15E+01 1.32E-08 0.000 0.0273 0.9900 0.9829 1.00 1 1 1.5E-06
10 : -2.15E+01 4.27E-13 0.000 0.0000 1.0000 1.0000 1.00 1 1 4.9E-11
iter seconds digits c*x b*y
10 0.1 Inf -2.1468211790e+01 -2.1468211790e+01
|Ax-b| = 1.4e-10, [Ay-c]_+ = 4.0E-11, |x|= 1.1e+01, |y|= 4.4e+00
Detailed timing (sec)
Pre IPM Post
1.000E-02 6.000E-02 0.000E+00
Max-norms: ||b||=1, ||c|| = 1.957607e+00,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 3.41276.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +21.4682
log-barrier
Calling sedumi: 200 variables, 31 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 31, order n = 201, dim = 201, blocks = 101
nnz(A) = 3100 + 0, nnz(ADA) = 961, nnz(L) = 496
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 7.10E+01 0.000
1 : -8.67E+00 5.16E+00 0.000 0.0727 0.9900 0.9900 -0.54 1 1 4.6E+01
2 : -3.94E+00 2.60E+00 0.000 0.5034 0.9000 0.9000 2.91 1 1 1.0E+01
3 : -1.63E+00 1.22E+00 0.000 0.4698 0.9000 0.9000 4.10 1 1 2.0E+00
4 : -1.33E+00 4.53E-01 0.000 0.3710 0.9000 0.9000 1.61 1 1 5.9E-01
5 : -1.24E+00 1.62E-01 0.000 0.3586 0.9000 0.9000 1.21 1 1 2.0E-01
6 : -1.22E+00 6.27E-02 0.000 0.3860 0.9000 0.9000 1.07 1 1 7.5E-02
7 : -1.21E+00 2.25E-02 0.000 0.3588 0.9000 0.9000 1.03 1 1 2.7E-02
8 : -1.20E+00 9.34E-03 0.000 0.4154 0.9000 0.6609 1.01 1 1 1.1E-02
9 : -1.20E+00 1.44E-03 0.000 0.1545 0.9243 0.9000 1.00 1 1 1.2E-03
10 : -1.20E+00 5.55E-05 0.000 0.0384 0.9904 0.9900 1.00 1 1 3.0E-05
11 : -1.20E+00 8.97E-06 0.000 0.1617 0.9102 0.9000 1.00 1 1 3.8E-06
12 : -1.20E+00 1.56E-06 0.000 0.1737 0.9092 0.9000 1.00 1 1 5.5E-07
13 : -1.20E+00 5.56E-09 0.000 0.0036 0.9990 0.9990 1.00 1 1 2.9E-09
iter seconds digits c*x b*y
13 0.1 8.5 -1.2012704624e+00 -1.2012704662e+00
|Ax-b| = 1.9e-09, [Ay-c]_+ = 0.0E+00, |x|= 3.1e-01, |y|= 1.5e+00
Detailed timing (sec)
Pre IPM Post
1.000E-02 6.000E-02 0.000E+00
Max-norms: ||b||=1, ||c|| = 1.957607e+00,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.35595.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +1.20127