n = 4;
fprintf(1,'Solving the upper bound SDP ...');
cvx_begin sdp
variable C1(n,n) symmetric
maximize ( C1(1,4) )
C1 >= 0;
diag(C1) == ones(n,1);
C1(1,2) >= 0.6;
C1(1,2) <= 0.9;
C1(1,3) >= 0.8;
C1(1,3) <= 0.9;
C1(2,4) >= 0.5;
C1(2,4) <= 0.7;
C1(3,4) >= -0.8;
C1(3,4) <= -0.4;
cvx_end
fprintf(1,'Done! \n');
fprintf(1,'Solving the lower bound SDP ...');
cvx_begin sdp
variable C2(n,n) symmetric
minimize ( C2(1,4) )
C2 >= 0;
diag(C2) == ones(n,1);
C2(1,2) >= 0.6;
C2(1,2) <= 0.9;
C2(1,3) >= 0.8;
C2(1,3) <= 0.9;
C2(2,4) >= 0.5;
C2(2,4) <= 0.7;
C2(3,4) >= -0.8;
C2(3,4) <= -0.4;
cvx_end
fprintf(1,'Done! \n');
disp('--------------------------------------------------------------------------------');
disp(['The minimum and maximum values of rho_14 are: ' num2str(C2(1,4)) ' and ' num2str(C1(1,4))]);
disp('with corresponding correlation matrices: ');
disp(C2)
disp(C1)
Solving the upper bound SDP ...
Calling Mosek 9.1.9: 18 variables, 6 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 : 6
Cones : 0
Scalar variables : 8
Matrix variables : 1
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 : 6
Cones : 0
Scalar variables : 8
Matrix variables : 1
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 6
Optimizer - Cones : 0
Optimizer - Scalar variables : 8 conic : 0
Optimizer - Semi-definite variables: 1 scalarized : 10
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 : 21 after factor : 21
Factor - dense dim. : 0 flops : 5.29e+02
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.0e+00 8.0e-01 5.0e+00 0.00e+00 4.000000000e+00 0.000000000e+00 1.0e+00 0.00
1 2.2e-01 1.8e-01 6.8e-01 5.68e-01 1.601847865e+00 5.752271707e-01 2.2e-01 0.01
2 7.2e-02 5.8e-02 1.0e-01 1.20e+00 6.556609770e-01 3.423728347e-01 7.2e-02 0.01
3 7.0e-03 5.6e-03 2.5e-03 1.50e+00 2.625263863e-01 2.385818161e-01 7.0e-03 0.01
4 8.2e-05 6.5e-05 3.2e-06 1.05e+00 2.303212228e-01 2.300505955e-01 8.2e-05 0.01
5 2.5e-06 2.0e-06 1.7e-08 9.99e-01 2.299221058e-01 2.299139040e-01 2.5e-06 0.01
6 5.3e-08 4.2e-08 5.3e-11 1.00e+00 2.299093602e-01 2.299091858e-01 5.3e-08 0.01
7 5.8e-09 4.7e-09 1.9e-12 1.00e+00 2.299091122e-01 2.299090929e-01 5.8e-09 0.01
8 8.5e-10 5.2e-09 1.1e-13 1.00e+00 2.299090875e-01 2.299090847e-01 8.5e-10 0.01
Optimizer terminated. Time: 0.01
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: 2.2990908751e-01 nrm: 2e+00 Viol. con: 7e-10 var: 0e+00 barvar: 0e+00
Dual. obj: 2.2990908467e-01 nrm: 1e+00 Viol. con: 0e+00 var: 3e-10 barvar: 4e-09
Optimizer summary
Optimizer - time: 0.01
Interior-point - iterations : 8 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): +0.229909
Done!
Solving the lower bound SDP ...
Calling Mosek 9.1.9: 18 variables, 6 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 : 6
Cones : 0
Scalar variables : 8
Matrix variables : 1
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 : 6
Cones : 0
Scalar variables : 8
Matrix variables : 1
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 6
Optimizer - Cones : 0
Optimizer - Scalar variables : 8 conic : 0
Optimizer - Semi-definite variables: 1 scalarized : 10
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 : 21 after factor : 21
Factor - dense dim. : 0 flops : 5.29e+02
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.0e+00 8.0e-01 5.0e+00 0.00e+00 4.000000000e+00 0.000000000e+00 1.0e+00 0.00
1 2.2e-01 1.8e-01 6.7e-01 5.68e-01 1.623017452e+00 5.938125873e-01 2.2e-01 0.01
2 5.2e-02 4.2e-02 4.7e-02 1.36e+00 6.610632060e-01 4.381337029e-01 5.2e-02 0.01
3 5.0e-03 4.0e-03 1.2e-03 1.45e+00 4.138204992e-01 3.969081187e-01 5.0e-03 0.01
4 1.0e-05 8.3e-06 1.0e-07 1.06e+00 3.928604383e-01 3.928251029e-01 1.0e-05 0.01
5 3.3e-07 2.6e-07 5.7e-10 1.00e+00 3.928217350e-01 3.928206256e-01 3.3e-07 0.01
6 3.6e-08 2.9e-08 2.1e-11 1.00e+00 3.928204884e-01 3.928203659e-01 3.6e-08 0.01
7 1.8e-09 1.1e-09 2.3e-13 1.00e+00 3.928203313e-01 3.928203253e-01 1.8e-09 0.01
Optimizer terminated. Time: 0.02
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: 3.9282033134e-01 nrm: 1e+00 Viol. con: 1e-09 var: 9e-11 barvar: 0e+00
Dual. obj: 3.9282032532e-01 nrm: 1e+00 Viol. con: 0e+00 var: 5e-10 barvar: 9e-10
Optimizer summary
Optimizer - time: 0.02
Interior-point - iterations : 7 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): -0.39282
Done!
--------------------------------------------------------------------------------
The minimum and maximum values of rho_14 are: -0.39282 and 0.22991
with corresponding correlation matrices:
1.0000 0.6000 0.8542 -0.3928
0.6000 1.0000 0.3042 0.5000
0.8542 0.3042 1.0000 -0.5751
-0.3928 0.5000 -0.5751 1.0000
1.0000 0.7447 0.8000 0.2299
0.7447 1.0000 0.3306 0.6013
0.8000 0.3306 1.0000 -0.4000
0.2299 0.6013 -0.4000 1.0000