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 ), ']' ] );
Calling SeDuMi: 50 variables (10 free), 25 equality constraints
------------------------------------------------------------------------
SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
Split 10 free variables
eqs m = 25, order n = 61, dim = 61, blocks = 1
nnz(A) = 540 + 0, nnz(ADA) = 625, nnz(L) = 325
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 5.04E+001 0.000
1 : 1.25E+001 1.21E+001 0.000 0.2398 0.9000 0.9000 2.40 1 1 9.7E-001
2 : 1.81E+001 2.67E+000 0.000 0.2208 0.9000 0.9000 1.09 1 1 2.5E-001
3 : 1.93E+001 6.94E-001 0.000 0.2598 0.9000 0.9000 1.01 1 1 6.7E-002
4 : 1.95E+001 1.70E-001 0.000 0.2452 0.9000 0.9000 1.00 1 1 1.7E-002
5 : 1.96E+001 6.75E-003 0.000 0.0397 0.9900 0.9900 1.00 1 1
iter seconds digits c*x b*y
5 0.1 15.7 1.9637293286e+001 1.9637293286e+001
|Ax-b| = 5.6e-015, [Ay-c]_+ = 2.8E-015, |x|= 1.1e+001, |y|= 6.7e+000
Detailed timing (sec)
Pre IPM Post
1.001E-002 5.007E-002 4.006E-002
Max-norms: ||b||=2.325211e+000, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 2.6152.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +19.6373
norm(A*x-b,1):
ans = 19.6373
Optimal vector:
x = [ 0.0454 0.7771 -0.4288 -0.2071 -0.6081 0.0065 -0.0013 0.0645 -0.3340 -0.6522 ]
Residual vector:
A*x-b = [ -0.0000 -1.0527 -0.7833 1.6843 0.1257 2.5993 1.2661 0.0000 0.2758 -1.6365 -0.9791 2.6851 0.8774 -0.8686 0.0000 1.6512 -0.0000 1.5824 -0.0000 1.5699 ]
Equality constraints:
C*x = [ -1.0290 0.2431 -1.2566 -0.3472 -0.9414 ]
d = [ -1.0290 0.2431 -1.2566 -0.3472 -0.9414 ]