I have such an optimization problem to minimize the largest singular value of the matrix:
\min_{\left\{x_i\right\}}\sigma_{max}\left(\mathbf{H}\right) =\sigma_{max}\left(\sum_{i=1}^{L}x_i\mathbf{H}_i\right) \\
\text{s.t.} \sum_{i=1}^{L}x_i=1, x_i\geq0
where \mathbf{H}_i \in \mathbb{C}^{M \times N} and \sigma_{max}\left(\mathbf{X}\right) means the largest singular value of \mathbf{X}, respectively.
Introducing \mathbf{W}=\left[\mathbf{0},\mathbf{H};\mathbf{H}^{\mathrm{T}},\mathbf{0}\right] and \mathbf{W}_i=\left[\mathbf{0},\mathbf{H}_i;\mathbf{H}_i^{\mathrm{T}},\mathbf{0}\right], the optimization problem above can be transformed into:
\min_{\left\{x_i\right\}}\lambda_{max}\left(\mathbf{W}\right) =\lambda_{max}\left(\sum_{i=1}^{L}x_i\mathbf{W}_i\right) \\
\text{s.t.} \sum_{i=1}^{L}x_i=1, x_i\geq0
where \lambda_{max}\left(\mathbf{X}\right) denotes the largest eigenvalue of \mathbf{X}. This problem can be transformed as a SDP problem:
\min_{\boldsymbol{x},t}t \\
\text{s.t. }W(\boldsymbol{x})-t\mathbf{I}_{M+N} \preceq 0,\sum_{i=1}^{L}x_i=1, x_i\geq0
where W(\boldsymbol{x})=\sum_{i=1}^{L}x_i\mathbf{W}_i. The CVX code I used is shown as following:
W0=zeros(Nt+Nr);
cvx_begin sdp
variable x(L);
variable t;
minimize t;
subject to
for i=1:L
W0=W0+x(i)*W(:,:,i);
end
x'*ones(L,1)==1;
x>=0;
t*eye(Nt+Nr)-W0==semidefinite(Nt+Nr);
cvx_end
However, the obtained optimal \boldsymbol{x} is NaN. The cvx_status is shown as following:
Calling Mosek 9.1.9: 1090 variables, 11 equality constraints
For improved efficiency, Mosek is solving the dual problem.
MOSEK Version 9.1.9 (Build date: 2019-11-21 11:34:40)
Copyright © MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 11
Cones : 0
Scalar variables : 270
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 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: 0.01
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 11
Cones : 0
Scalar variables : 270
Matrix variables : 1
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 11
Optimizer - Cones : 1
Optimizer - Scalar variables : 267 conic : 257
Optimizer - Semi-definite variables: 1 scalarized : 820
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 : 66 after factor : 66
Factor - dense dim. : 0 flops : 3.57e+05
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 3.9e+01 1.0e+00 1.0e+00 0.00e+00 0.000000000e+00 0.000000000e+00 1.0e+00 0.03
1 2.1e+01 5.5e-01 1.7e-01 3.51e+00 -3.577315806e-02 -1.371427890e-01 5.5e-01 0.08
2 1.5e+01 3.9e-01 6.7e-01 5.53e-01 -4.330990656e+00 -4.108603708e+00 3.9e-01 0.08
3 6.3e+00 1.6e-01 3.2e-02 3.55e+00 -1.129435191e+00 -1.392990285e+00 1.6e-01 0.08
4 1.7e+00 4.3e-02 2.2e-02 -1.03e-01 -7.632153254e+00 -7.783272849e+00 4.3e-02 0.08
5 1.7e-02 4.4e-04 3.0e-03 -7.90e-01 -1.656095769e+03 -1.611546014e+03 4.4e-04 0.08
6 1.7e-08 4.4e-10 4.2e-06 -9.96e-01 -3.376083486e+09 -3.283855737e+09 4.4e-10 0.09
7 3.1e-15 1.0e-15 9.3e-13 -1.00e+00 -9.067269377e-01 -8.819569976e-01 1.9e-19 0.09
Optimizer terminated. Time: 0.13
Interior-point solution summary
Problem status : DUAL_INFEASIBLE
Solution status : DUAL_INFEASIBLE_CER
Primal. obj: -9.0672693765e-01 nrm: 1e+00 Viol. con: 3e-15 var: 0e+00 barvar: 0e+00
Optimizer summary
Optimizer - time: 0.13
Interior-point - iterations : 7 time: 0.09
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: Infeasible
Optimal value (cvx_optval): +Inf
I wonder whether such a optimization problem can not obtain a non-trivial solution or not? If the answer is yes, how can I get it through CVX? Thank you for your attention.