n = 4;
cvx_begin sdp
variable A(n,n) symmetric;
A >= 0;
A(1,1) == 3;
A(2,2) == 2;
A(3,3) == 1;
A(4,4) == 5;
A(1,2) == .5;
A(1,4) == .25;
A(2,3) == .75;
maximize( log_det( A ) )
cvx_end
disp(['Matrix A with maximum determinant (' num2str(det(A)) ') is:'])
A
disp(['Its eigenvalues are:'])
eigs = eig(A)
Calling Mosek 9.1.9: 59 variables, 18 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 : 18
Cones : 4
Scalar variables : 13
Matrix variables : 2
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 1
Eliminator terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 2 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 : 18
Cones : 4
Scalar variables : 13
Matrix variables : 2
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 16
Optimizer - Cones : 4
Optimizer - Scalar variables : 12 conic : 12
Optimizer - Semi-definite variables: 2 scalarized : 46
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 : 101 after factor : 103
Factor - dense dim. : 0 flops : 2.67e+03
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 3.2e+00 4.0e+00 2.7e+01 0.00e+00 2.282783840e+01 -3.220408006e+00 1.0e+00 0.00
1 6.4e-01 8.0e-01 6.3e+00 -4.82e-01 1.131513779e+01 5.009346393e-01 2.0e-01 0.01
2 1.7e-01 2.2e-01 1.0e+00 5.08e-01 4.413294072e+00 8.163553142e-01 5.5e-02 0.01
3 4.8e-02 6.0e-02 1.6e-01 7.45e-01 3.272925849e+00 2.171307260e+00 1.5e-02 0.01
4 7.8e-03 9.8e-03 1.1e-02 8.47e-01 3.099056911e+00 2.905643702e+00 2.5e-03 0.01
5 4.8e-04 6.0e-04 1.7e-04 9.92e-01 3.028576122e+00 3.016657110e+00 1.5e-04 0.01
6 9.7e-06 1.2e-05 4.8e-07 1.00e+00 3.024305294e+00 3.024063282e+00 3.0e-06 0.01
7 4.7e-08 5.8e-08 1.6e-10 1.00e+00 3.024221238e+00 3.024220077e+00 1.5e-08 0.01
8 2.6e-09 2.4e-09 1.1e-12 1.00e+00 3.024220730e+00 3.024220687e+00 5.6e-10 0.01
Optimizer terminated. Time: 0.02
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: 3.0242207305e+00 nrm: 5e+00 Viol. con: 5e-09 var: 0e+00 barvar: 0e+00 cones: 0e+00
Dual. obj: 3.0242206875e+00 nrm: 7e+00 Viol. con: 0e+00 var: 8e-12 barvar: 8e-09 cones: 0e+00
Optimizer summary
Optimizer - time: 0.02
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): +3.02422
Matrix A with maximum determinant (20.578) is:
A =
3.0000 0.5000 0.1875 0.2500
0.5000 2.0000 0.7500 0.0417
0.1875 0.7500 1.0000 0.0156
0.2500 0.0417 0.0156 5.0000
Its eigenvalues are:
eigs =
0.5964
2.0908
3.2773
5.0355