Section 5.2.4: Solves a simple QCQP

% Boyd & Vandenberghe, "Convex Optimization"
% Joëlle Skaf - 08/23/05
%
% Solved a QCQP with 3 inequalities:
%           minimize    1/2 x'*P0*x + q0'*r + r0
%               s.t.    1/2 x'*Pi*x + qi'*r + ri <= 0   for i=1,2,3
% and verifies that strong duality holds.

% Input data
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');

% Displaying results
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.
------------------------------------------------------------
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