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 Mosek 9.1.9: 35 variables, 10 equality constraints
For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------
MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 10
Cones : 4
Scalar variables : 35
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 1 time : 0.00
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.00
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 10
Cones : 4
Scalar variables : 35
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 10
Optimizer - Cones : 4
Optimizer - Scalar variables : 35 conic : 32
Optimizer - Semi-definite variables: 0 scalarized : 0
Factor - setup time : 0.00 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 49 after factor : 55
Factor - dense dim. : 0 flops : 1.25e+03
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 5.6e+00 5.0e+00 1.0e+00 0.00e+00 0.000000000e+00 0.000000000e+00 1.0e+00 0.00
1 1.8e+00 1.6e+00 3.6e-01 -5.80e-01 9.950141487e-02 7.175750246e-01 3.3e-01 0.01
2 4.3e-01 3.8e-01 3.8e-02 7.42e-01 -2.392908578e+00 -2.297090534e+00 7.7e-02 0.01
3 5.3e-02 4.7e-02 1.8e-03 8.74e-01 -3.035369662e+00 -3.018245201e+00 9.5e-03 0.01
4 1.4e-02 1.2e-02 2.5e-04 9.53e-01 -3.163983578e+00 -3.158419311e+00 2.4e-03 0.01
5 2.2e-03 2.0e-03 1.9e-05 8.95e-01 -3.217054679e+00 -3.215643247e+00 3.9e-04 0.01
6 2.0e-05 1.8e-05 1.5e-08 1.00e+00 -3.225854306e+00 -3.225843854e+00 3.6e-06 0.01
7 2.0e-06 1.8e-06 4.9e-10 1.00e+00 -3.225929324e+00 -3.225928277e+00 3.6e-07 0.01
8 1.6e-07 1.4e-07 1.1e-11 1.00e+00 -3.225937738e+00 -3.225937655e+00 2.8e-08 0.01
9 1.9e-08 1.7e-08 4.4e-13 1.00e+00 -3.225938379e+00 -3.225938370e+00 3.3e-09 0.01
Optimizer terminated. Time: 0.01
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -3.2259383795e+00 nrm: 7e+00 Viol. con: 4e-08 var: 0e+00 cones: 0e+00
Dual. obj: -3.2259383698e+00 nrm: 7e-01 Viol. con: 0e+00 var: 4e-08 cones: 0e+00
Optimizer summary
Optimizer - time: 0.01
Interior-point - iterations : 9 time: 0.01
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -0.895296
Done!
------------------------------------------------------------------------
The duality gap is equal to
-5.8546e-08