Section 5.2.4: Solves a simple QCQP
randn('state',13);
n = 6;
P0 = randn(n); P0 = P0'*P0 + eps*eye(n);
P1 = randn(n); P1 = P1'*P1;
P2 = randn(n); P2 = P2'*P2;
P3 = randn(n); P3 = P3'*P3;
q0 = randn(n,1); q1 = randn(n,1); q2 = randn(n,1); q3 = randn(n,1);
r0 = randn(1); r1 = randn(1); r2 = randn(1); r3 = randn(1);
fprintf(1,'Computing the optimal value of the QCQP and its dual... ');
cvx_begin
variable x(n)
dual variables lam1 lam2 lam3
minimize( 0.5*quad_form(x,P0) + q0'*x + r0 )
lam1: 0.5*quad_form(x,P1) + q1'*x + r1 <= 0;
lam2: 0.5*quad_form(x,P2) + q2'*x + r2 <= 0;
lam3: 0.5*quad_form(x,P3) + q3'*x + r3 <= 0;
cvx_end
obj1 = cvx_optval;
P_lam = P0 + lam1*P1 + lam2*P2 + lam3*P3;
q_lam = q0 + lam1*q1 + lam2*q2 + lam3*q3;
r_lam = r0 + lam1*r1 + lam2*r2 + lam3*r3;
obj2 = -0.5*q_lam'*inv(P_lam)*q_lam + r_lam;
fprintf(1,'Done! \n');
disp('------------------------------------------------------------------------');
disp('The duality gap is equal to ');
disp(obj1-obj2)
Computing the optimal value of the QCQP and its dual...
Calling sedumi: 35 variables, 10 equality constraints
For improved efficiency, sedumi is solving the dual problem.
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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 = 10, order n = 11, dim = 37, blocks = 6
nnz(A) = 113 + 0, nnz(ADA) = 94, nnz(L) = 52
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 8.45E+00 0.000
1 : 1.19E-02 2.07E+00 0.000 0.2452 0.9000 0.9000 1.63 1 1 1.6E+00
2 : -3.78E+00 3.87E-01 0.000 0.1867 0.9000 0.9000 2.29 1 1 1.6E-01
3 : -4.39E+00 7.69E-02 0.000 0.1990 0.9000 0.9000 1.08 1 1 2.8E-02
4 : -4.66E+00 6.65E-03 0.000 0.0865 0.9900 0.9900 0.84 1 1 2.2E-03
5 : -4.71E+00 2.01E-04 0.000 0.0302 0.9900 0.9900 0.98 1 1 1.1E-04
6 : -4.71E+00 2.62E-07 0.000 0.0013 0.9901 0.9900 1.00 1 1 1.1E-06
7 : -4.71E+00 3.20E-09 0.000 0.0122 0.8709 0.9000 1.00 1 1 2.2E-07
8 : -4.71E+00 2.35E-10 0.089 0.0735 0.9900 0.9900 1.00 1 1 1.6E-08
9 : -4.71E+00 5.45E-11 0.000 0.2318 0.9092 0.9000 1.00 2 2 3.4E-09
iter seconds digits c*x b*y
9 0.0 Inf -4.7144665661e+00 -4.7144665032e+00
|Ax-b| = 7.5e-09, [Ay-c]_+ = 6.6E-09, |x|= 1.6e+01, |y|= 1.4e+00
Detailed timing (sec)
Pre IPM Post
0.000E+00 4.000E-02 1.000E-02
Max-norms: ||b||=9.588449e+00, ||c|| = 6.667862e+00,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 31.0251.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -0.895296
Done!
------------------------------------------------------------------------
The duality gap is equal to
-1.2235e-07