Equality constrained norm minimization.
p = 1;
n = 10; m = 2*n; q=0.5*n;
A = randn(m,n);
b = randn(m,1);
C = randn(q,n);
d = randn(q,1);
cvx_begin
variable x(n)
dual variable y
minimize( norm( A * x - b, p ) )
subject to
y : C * x == d;
cvx_end
disp( sprintf( 'norm(A*x-b,%g):', p ) );
disp( [ ' ans = ', sprintf( '%7.4f', norm(A*x-b,p) ) ] );
disp( 'Optimal vector:' );
disp( [ ' x = [ ', sprintf( '%7.4f ', x ), ']' ] );
disp( 'Residual vector:' );
disp( [ ' A*x-b = [ ', sprintf( '%7.4f ', A*x-b ), ']' ] );
disp( 'Equality constraints:' );
disp( [ ' C*x = [ ', sprintf( '%7.4f ', C*x ), ']' ] );
disp( [ ' d = [ ', sprintf( '%7.4f ', d ), ']' ] );
disp( 'Lagrange multiplier for C*x==d:' );
disp( [ ' y = [ ', sprintf( '%7.4f ', y ), ']' ] );
Calling sedumi: 50 variables, 25 equality constraints
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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 = 25, order n = 43, dim = 52, blocks = 22
nnz(A) = 270 + 0, nnz(ADA) = 625, nnz(L) = 325
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 3.35E+01 0.000
1 : 1.53E+01 1.01E+01 0.000 0.3005 0.9000 0.9000 2.36 1 1 1.2E+00
2 : 1.72E+01 2.89E+00 0.000 0.2873 0.9000 0.9000 1.29 1 1 3.3E-01
3 : 1.78E+01 6.67E-01 0.000 0.2308 0.9000 0.9000 1.07 1 1 7.8E-02
4 : 1.80E+01 1.45E-01 0.000 0.2181 0.9000 0.9000 1.02 1 1 1.7E-02
5 : 1.80E+01 1.42E-02 0.000 0.0974 0.9900 0.9900 1.00 1 1 1.7E-03
6 : 1.80E+01 4.75E-04 0.000 0.0335 0.9907 0.9900 1.00 1 1 9.6E-05
7 : 1.80E+01 1.07E-07 0.000 0.0002 0.9999 0.9999 1.00 1 1 1.3E-08
iter seconds digits c*x b*y
7 0.1 8.1 1.7972427055e+01 1.7972426900e+01
|Ax-b| = 5.3e-09, [Ay-c]_+ = 1.9E-09, |x|= 8.5e+00, |y|= 8.1e+00
Detailed timing (sec)
Pre IPM Post
2.000E-02 1.200E-01 1.000E-02
Max-norms: ||b||=1.858593e+00, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 2.65245.
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Status: Solved
Optimal value (cvx_optval): +17.9724
norm(A*x-b,1):
ans = 17.9724
Optimal vector:
x = [ 0.6963 0.1554 0.5142 0.3923 -0.4455 0.0931 0.4494 -0.3651 -0.4819 -0.5498 ]
Residual vector:
A*x-b = [ 1.1860 -0.0000 0.8197 0.0000 1.4400 2.4199 0.5118 1.9345 0.1147 -1.7785 2.6667 -0.0000 0.0000 -0.4222 -0.9773 0.9364 -0.7506 0.2294 -1.7848 0.0000 ]
Equality constraints:
C*x = [ -0.0638 0.6113 0.1093 1.8140 0.3120 ]
d = [ -0.0638 0.6113 0.1093 1.8140 0.3120 ]
Lagrange multiplier for C*x==d:
y = [ -1.5037 1.2411 -2.9373 5.5888 2.3990 ]