Mosek solved this “as is”.
At best, all of the 1e-14 are meaningless to the solver, and at worst, can screw it up, and should be eliminated. I don’t think it made a difference in this case using Mosek, because the result was the same when I eliminated them.
S/w = 0.15 and is the only use made of either S or w, so that is probably o.k. (with these values). That leaves h(1)/sigma = h(2)/sigma = 5.06e+02 as the only place h or sigma are used, so that is probably o.k. However, it is better not to have these “extreme” numbers, because you can’t be certain that transformations by CVX will always be in terms of these “net” numbers., unless the divisions are in parentheses, whereby the division occurs before CVX ever sees the individual numbers).
Calling Mosek 9.3.6: 17 variables, 6 equality constraints
For improved efficiency, Mosek is solving the dual problem.
MOSEK Version 9.3.10 (Build date: 2021-11-5 08:42:07)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 6
Cones : 2
Scalar variables : 17
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 : 0.01
Lin. dep. - number : 0
Presolve terminated. Time: 0.03
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 6
Cones : 2
Scalar variables : 17
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 4
Optimizer - Cones : 2
Optimizer - Scalar variables : 15 conic : 6
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 : 7 after factor : 7
Factor - dense dim. : 0 flops : 7.90e+01
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.0e+00 4.0e+00 8.3e-01 0.00e+00 -1.674162547e-01 0.000000000e+00 1.0e+00 0.06
1 1.9e-01 7.5e-01 7.0e-02 3.00e-01 -8.253068901e-01 -8.081031594e-01 1.9e-01 0.16
2 3.9e-02 1.5e-01 5.2e-03 1.51e+00 -2.230186035e-01 -2.267921025e-01 3.9e-02 0.16
3 6.9e-03 2.8e-02 3.8e-04 1.13e+00 -6.455245714e-02 -6.518045223e-02 6.9e-03 0.17
4 1.3e-03 5.3e-03 3.4e-05 1.00e+00 -5.879743404e-02 -5.886635862e-02 1.3e-03 0.17
5 1.5e-04 5.9e-04 1.3e-06 9.95e-01 -5.421801150e-02 -5.422412341e-02 1.5e-04 0.17
6 1.8e-05 7.0e-05 5.1e-08 9.98e-01 -5.365307705e-02 -5.365378774e-02 1.8e-05 0.19
7 1.9e-06 7.5e-06 1.8e-09 1.00e+00 -5.359284960e-02 -5.359292302e-02 1.9e-06 0.19
8 2.4e-07 9.7e-07 8.2e-11 1.00e+00 -5.358692995e-02 -5.358693935e-02 2.4e-07 0.19
9 8.9e-09 3.5e-08 5.9e-13 1.00e+00 -5.358629922e-02 -5.358629933e-02 8.9e-09 0.20
Optimizer terminated. Time: 0.27
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -5.3586299223e-02 nrm: 1e+02 Viol. con: 1e-06 var: 2e-08 cones: 0e+00
Dual. obj: -5.3586299332e-02 nrm: 2e+00 Viol. con: 0e+00 var: 8e-09 cones: 0e+00
Optimizer summary
Optimizer - time: 0.27
Interior-point - iterations : 9 time: 0.20
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): +0.0535863
disp(t)
0.465510584602382
0.534487451108615
disp(y)
1.0e-03 *
0.230370156931148
0.305492836586113
If you have available CVX 2.2 plus Mosek 9.x, use that, because it is the easiest and most reliable and robust solution option. Otherwise, use CVXQUAD as discussed in CVXQUAD: How to use CVXQUAD's Pade Approximant instead of CVX's unreliable Successive Approximation for GP mode, log, exp, entr, rel_entr, kl_div, log_det, det_rootn, exponential cone. CVXQUAD's Quantum (Matrix) Entropy & Matrix Log related functions .