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
------------------------------------------------------------------------
SeDuMi 1.1R3 by AdvOL, 2006 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 : 3.80E+001 0.000
1 : 1.28E+001 1.12E+001 0.000 0.2946 0.9000 0.9000 2.70 1 1 1.1E+000
2 : 1.48E+001 3.02E+000 0.000 0.2699 0.9000 0.9000 1.26 1 1 3.0E-001
3 : 1.52E+001 9.88E-001 0.000 0.3272 0.9000 0.9000 1.07 1 1 1.0E-001
4 : 1.54E+001 3.70E-001 0.000 0.3747 0.9000 0.9000 1.02 1 1 3.8E-002
5 : 1.55E+001 9.39E-002 0.000 0.2537 0.9000 0.9000 1.01 1 1 9.8E-003
6 : 1.55E+001 1.84E-002 0.000 0.1956 0.9000 0.9120 1.00 1 1 1.6E-003
7 : 1.55E+001 2.15E-003 0.000 0.1170 0.9091 0.9000 1.00 1 1 2.1E-004
8 : 1.55E+001 5.50E-005 0.000 0.0256 0.9905 0.9900 1.00 1 1 7.4E-006
9 : 1.55E+001 2.49E-009 0.000 0.0000 1.0000 1.0000 1.00 1 1 2.1E-010
iter seconds digits c*x b*y
9 0.1 9.7 1.5491581695e+001 1.5491581692e+001
|Ax-b| = 1.9e-010, [Ay-c]_+ = 1.7E-011, |x|= 8.4e+000, |y|= 7.8e+000
Detailed timing (sec)
Pre IPM Post
3.004E-002 1.202E-001 1.001E-002
Max-norms: ||b||=2.275599e+000, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 2.32468.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +15.4916
norm(A*x-b,1):
ans = 15.4916
Optimal vector:
x = [ 0.1140 0.1739 -0.8716 -0.4524 -0.1927 0.3188 -1.2567 0.2236 0.5423 1.0767 ]
Residual vector:
A*x-b = [ -0.2762 -0.0000 -0.6846 -0.0000 1.7275 -0.0000 -0.0149 0.0480 0.5530 1.0724 0.7873 0.0000 0.0000 0.6145 -0.5936 -1.4800 0.8559 2.3848 -1.6966 -2.7023 ]
Equality constraints:
C*x = [ -1.4776 0.7381 -1.0904 -1.7868 1.6391 ]
d = [ -1.4776 0.7381 -1.0904 -1.7868 1.6391 ]
Lagrange multiplier for C*x==d:
y = [ 4.3036 2.2856 -3.2557 -1.3588 2.9295 ]