Why is the result of CVX running different from what I theoretically derived?

1

Hi, wish to ask you something.
I’ve converted the equation constraint of X=B*W into the form of LMI as follows
\begin{bmatrix} U & X & B \\ X^H & V & W^H \\ B^H & W & I_M \end{bmatrix} \succeq 0
Tr(U) - M \leq 0 where U \in C^{MN*MN} and V \in C^{K*K} and U \succeq 0,V \succeq 0

The above equivalent forms of proof are as follows

QQ图片20231126173559
QQ图片20231126173559614×765 94 KB

Replace the FH in the image with B to get the result

But when I use LMI as a constraint in cvx, the X I got is not equal to B*W, why is that？

here is my code

numM=4;
numN=16;
numK=4;
gamak = 10^(4/10);
sigmak = 1.0e-4; 

Hhat = [-0.0652405201837634 + 1.67262729732120i,-1.17955547159453 + 1.18751713760266i,-1.67191760052190 + 0.0764800249883790i,-1.28286695049747 - 1.07471836870477i,-0.482353009345158 + 1.60281755490566i,-1.43947387936296 + 0.853939918776870i,-1.63764897082401 - 0.344984216936885i,-0.972516792257723 - 1.36188439697510i,-0.868638242346595 + 1.43070133945419i,-1.60748313133005 + 0.465882848453379i,-1.49884726659665 - 0.744389533521191i,-0.600115527527786 - 1.56209162760993i,-1.19944121666771 + 1.16727447733334i,-1.67286688213672 + 0.0481216424873483i,-1.26438193493919 - 1.09624263443367i,-0.189440188150751 - 1.66256796317943i,-0.0652405201837634 + 1.67262729732120i,-1.17955547159453 + 1.18751713760266i,-1.67191760052190 + 0.0764800249883790i,-1.28286695049747 - 1.07471836870477i,-0.482353009345158 + 1.60281755490566i,-1.43947387936296 + 0.853939918776870i,-1.63764897082401 - 0.344984216936885i,-0.972516792257723 - 1.36188439697510i,-0.868638242346595 + 1.43070133945419i,-1.60748313133005 + 0.465882848453379i,-1.49884726659665 - 0.744389533521191i,-0.600115527527786 - 1.56209162760993i,-1.19944121666771 + 1.16727447733334i,-1.67286688213672 + 0.0481216424873483i,-1.26438193493919 - 1.09624263443367i,-0.189440188150751 - 1.66256796317943i,-0.0652405201837634 + 1.67262729732120i,-1.17955547159453 + 1.18751713760266i,-1.67191760052190 + 0.0764800249883790i,-1.28286695049747 - 1.07471836870477i,-0.482353009345158 + 1.60281755490566i,-1.43947387936296 + 0.853939918776870i,-1.63764897082401 - 0.344984216936885i,-0.972516792257723 - 1.36188439697510i,-0.868638242346595 + 1.43070133945419i,-1.60748313133005 + 0.465882848453379i,-1.49884726659665 - 0.744389533521191i,-0.600115527527786 - 1.56209162760993i,-1.19944121666771 + 1.16727447733334i,-1.67286688213672 + 0.0481216424873483i,-1.26438193493919 - 1.09624263443367i,-0.189440188150751 - 1.66256796317943i,-0.0652405201837634 + 1.67262729732120i,-1.17955547159453 + 1.18751713760266i,-1.67191760052190 + 0.0764800249883790i,-1.28286695049747 - 1.07471836870477i,-0.482353009345158 + 1.60281755490566i,-1.43947387936296 + 0.853939918776870i,-1.63764897082401 - 0.344984216936885i,-0.972516792257723 - 1.36188439697510i,-0.868638242346595 + 1.43070133945419i,-1.60748313133005 + 0.465882848453379i,-1.49884726659665 - 0.744389533521191i,-0.600115527527786 - 1.56209162760993i,-1.19944121666771 + 1.16727447733334i,-1.67286688213672 + 0.0481216424873483i,-1.26438193493919 - 1.09624263443367i,-0.189440188150751 - 1.66256796317943i;-0.343804065086222 + 1.25035870569987i,-1.06795829625980 + 0.735246840784658i,-1.29035582664988 - 0.125018060481049i,-0.906692521323401 - 0.926326781420968i,-1.07896619404889 + 0.719611186651815i,-1.28871167484363 - 0.144066446052317i,-0.893172045483233 - 0.939841133857426i,-0.0782781711894032 - 1.29400736073525i,-1.28678633003226 - 0.163093533043658i,-0.879450156536759 - 0.953158170652693i,-0.0592073577237805 - 1.29536180483867i,0.788609540458150 - 1.02911975684605i,-0.865529733721704 - 0.966274890298573i,-0.0401136141564769 - 1.29643518946567i,0.803908041314568 - 1.01764910784814i,1.27014673574948 - 0.261016824009327i,-0.343804065086222 + 1.25035870569987i,-1.06795829625980 + 0.735246840784658i,-1.29035582664988 - 0.125018060481049i,-0.906692521323401 - 0.926326781420968i,-1.07896619404889 + 0.719611186651815i,-1.28871167484363 - 0.144066446052317i,-0.893172045483233 - 0.939841133857426i,-0.0782781711894032 - 1.29400736073525i,-1.28678633003226 - 0.163093533043658i,-0.879450156536759 - 0.953158170652693i,-0.0592073577237805 - 1.29536180483867i,0.788609540458150 - 1.02911975684605i,-0.865529733721704 - 0.966274890298573i,-0.0401136141564769 - 1.29643518946567i,0.803908041314568 - 1.01764910784814i,1.27014673574948 - 0.261016824009327i,-0.343804065086222 + 1.25035870569987i,-1.06795829625980 + 0.735246840784658i,-1.29035582664988 - 0.125018060481049i,-0.906692521323401 - 0.926326781420968i,-1.07896619404889 + 0.719611186651815i,-1.28871167484363 - 0.144066446052317i,-0.893172045483233 - 0.939841133857426i,-0.0782781711894032 - 1.29400736073525i,-1.28678633003226 - 0.163093533043658i,-0.879450156536759 - 0.953158170652693i,-0.0592073577237805 - 1.29536180483867i,0.788609540458150 - 1.02911975684605i,-0.865529733721704 - 0.966274890298573i,-0.0401136141564769 - 1.29643518946567i,0.803908041314568 - 1.01764910784814i,1.27014673574948 - 0.261016824009327i,-0.343804065086222 + 1.25035870569987i,-1.06795829625980 + 0.735246840784658i,-1.29035582664988 - 0.125018060481049i,-0.906692521323401 - 0.926326781420968i,-1.07896619404889 + 0.719611186651815i,-1.28871167484363 - 0.144066446052317i,-0.893172045483233 - 0.939841133857426i,-0.0782781711894032 - 1.29400736073525i,-1.28678633003226 - 0.163093533043658i,-0.879450156536759 - 0.953158170652693i,-0.0592073577237805 - 1.29536180483867i,0.788609540458150 - 1.02911975684605i,-0.865529733721704 - 0.966274890298573i,-0.0401136141564769 - 1.29643518946567i,0.803908041314568 - 1.01764910784814i,1.27014673574948 - 0.261016824009327i;1.24071728293608 + 0.179700700227814i,0.565630468269524 + 1.11880312846160i,-0.590860743445215 + 1.10568006280365i,-1.24445830743253 + 0.151508734102732i,1.09904450999016 + 0.603062835730714i,0.137709106594817 + 1.24603626150278i,-0.940822794842869 + 0.828498146916922i,-1.21860772769445 - 0.294174213602628i,0.818007754776487 + 0.949925700276277i,-0.307651423969747 + 1.21525004163040i,-1.17145771003041 + 0.446268348504055i,-1.03822464696225 - 0.702527390284345i,0.433253630806475 + 1.17630723947605i,-0.713973131056792 + 1.03035738120908i,-1.25352492371160 + 0.00746783946372853i,-0.726192444501054 - 1.02176884769965i,1.24071728293608 + 0.179700700227814i,0.565630468269524 + 1.11880312846160i,-0.590860743445215 + 1.10568006280365i,-1.24445830743253 + 0.151508734102732i,1.09904450999016 + 0.603062835730714i,0.137709106594817 + 1.24603626150278i,-0.940822794842869 + 0.828498146916922i,-1.21860772769445 - 0.294174213602628i,0.818007754776487 + 0.949925700276277i,-0.307651423969747 + 1.21525004163040i,-1.17145771003041 + 0.446268348504055i,-1.03822464696225 - 0.702527390284345i,0.433253630806475 + 1.17630723947605i,-0.713973131056792 + 1.03035738120908i,-1.25352492371160 + 0.00746783946372853i,-0.726192444501054 - 1.02176884769965i,1.24071728293608 + 0.179700700227814i,0.565630468269524 + 1.11880312846160i,-0.590860743445215 + 1.10568006280365i,-1.24445830743253 + 0.151508734102732i,1.09904450999016 + 0.603062835730714i,0.137709106594817 + 1.24603626150278i,-0.940822794842869 + 0.828498146916922i,-1.21860772769445 - 0.294174213602628i,0.818007754776487 + 0.949925700276277i,-0.307651423969747 + 1.21525004163040i,-1.17145771003041 + 0.446268348504055i,-1.03822464696225 - 0.702527390284345i,0.433253630806475 + 1.17630723947605i,-0.713973131056792 + 1.03035738120908i,-1.25352492371160 + 0.00746783946372853i,-0.726192444501054 - 1.02176884769965i,1.24071728293608 + 0.179700700227814i,0.565630468269524 + 1.11880312846160i,-0.590860743445215 + 1.10568006280365i,-1.24445830743253 + 0.151508734102732i,1.09904450999016 + 0.603062835730714i,0.137709106594817 + 1.24603626150278i,-0.940822794842869 + 0.828498146916922i,-1.21860772769445 - 0.294174213602628i,0.818007754776487 + 0.949925700276277i,-0.307651423969747 + 1.21525004163040i,-1.17145771003041 + 0.446268348504055i,-1.03822464696225 - 0.702527390284345i,0.433253630806475 + 1.17630723947605i,-0.713973131056792 + 1.03035738120908i,-1.25352492371160 + 0.00746783946372853i,-0.726192444501054 - 1.02176884769965i;0.744052980415410 + 0.422486488019040i,0.0782469583530654 + 0.852038966354416i,-0.654643750659819 + 0.550924552138733i,-0.826134857216619 - 0.222622584576821i,0.655475492798011 + 0.550391351049500i,-0.0773324313357631 + 0.852397663636049i,-0.743808051133347 + 0.423432385642987i,-0.772423670857399 - 0.368633143340257i,0.545169516184055 + 0.660180545205781i,-0.230454496282484 + 0.824573953154341i,-0.808437014207177 + 0.281858826121668i,-0.693139653731680 - 0.502549050586954i,0.416768220359671 + 0.748212602618902i,-0.376055042570886 + 0.769469806496849i,-0.846378199753204 + 0.130875705770559i,-0.590887855771216 - 0.619934978684891i,0.744052980415410 + 0.422486488019040i,0.0782469583530654 + 0.852038966354416i,-0.654643750659819 + 0.550924552138733i,-0.826134857216619 - 0.222622584576821i,0.655475492798011 + 0.550391351049500i,-0.0773324313357631 + 0.852397663636049i,-0.743808051133347 + 0.423432385642987i,-0.772423670857399 - 0.368633143340257i,0.545169516184055 + 0.660180545205781i,-0.230454496282484 + 0.824573953154341i,-0.808437014207177 + 0.281858826121668i,-0.693139653731680 - 0.502549050586954i,0.416768220359671 + 0.748212602618902i,-0.376055042570886 + 0.769469806496849i,-0.846378199753204 + 0.130875705770559i,-0.590887855771216 - 0.619934978684891i,0.744052980415410 + 0.422486488019040i,0.0782469583530654 + 0.852038966354416i,-0.654643750659819 + 0.550924552138733i,-0.826134857216619 - 0.222622584576821i,0.655475492798011 + 0.550391351049500i,-0.0773324313357631 + 0.852397663636049i,-0.743808051133347 + 0.423432385642987i,-0.772423670857399 - 0.368633143340257i,0.545169516184055 + 0.660180545205781i,-0.230454496282484 + 0.824573953154341i,-0.808437014207177 + 0.281858826121668i,-0.693139653731680 - 0.502549050586954i,0.416768220359671 + 0.748212602618902i,-0.376055042570886 + 0.769469806496849i,-0.846378199753204 + 0.130875705770559i,-0.590887855771216 - 0.619934978684891i,0.744052980415410 + 0.422486488019040i,0.0782469583530654 + 0.852038966354416i,-0.654643750659819 + 0.550924552138733i,-0.826134857216619 - 0.222622584576821i,0.655475492798011 + 0.550391351049500i,-0.0773324313357631 + 0.852397663636049i,-0.743808051133347 + 0.423432385642987i,-0.772423670857399 - 0.368633143340257i,0.545169516184055 + 0.660180545205781i,-0.230454496282484 + 0.824573953154341i,-0.808437014207177 + 0.281858826121668i,-0.693139653731680 - 0.502549050586954i,0.416768220359671 + 0.748212602618902i,-0.376055042570886 + 0.769469806496849i,-0.846378199753204 + 0.130875705770559i,-0.590887855771216 - 0.619934978684891i]

cvx_begin 
        variable W(numM, numK) complex     
        variable X(numM*numN, numK) complex
        variable U(numM*numN, numM*numN) complex semidefinite  
        variable V(numK, numK) complex semidefinite
        variable I(numM, numM) 
        variable genhaoxia(numK,numK-1) complex
        variable cal(numK,1)
        variable bigmatrix(numM*numN+numK+numM,numM*numN+numK+numM) complex  semidefinite 
        dual variables d1 d2 d3 d4 d5 d6
        minimize( sum(sum_square_abs(W)))   % conj( x ) .* x  square_abs( x )
        subject to
            %X == B*W;
            d5:bigmatrix == [ U  X B ;
                              X' V W';
                              B' W I];                             
            d6:trace(U) - numM <=0            ;                 
            I == eye(numM);        
            tem = conj(Hhat)*X;
            for i=1:numK
                imag(tem(i,i)) == 0;       
            end
            for i=1:numK
                cal(i) == tem(i,i);
            end
            for i=1:numK
                genhaoxia(i,:) == real(tem(i,setdiff(1:numK,i))) + imag(tem(i,setdiff(1:numK,i)))*1i;
            end

            d1:norm([tem(1,:) sigmak]) - real(tem(1,1))*sqrt((1+1/gamak)) <= 0 ;          
            d2:norm([tem(2,:) sigmak]) - real(tem(2,2))*sqrt((1+1/gamak)) <= 0 ;           
            d3:norm([tem(3,:) sigmak]) - real(tem(3,3))*sqrt((1+1/gamak)) <= 0 ;            
            d4:norm([tem(4,:) sigmak]) - real(tem(4,4))*sqrt((1+1/gamak)) <= 0 ;          

    cvx_end
2

Your problem is not reproducible because you have not provided the value of B.

You have not shown us the CVX and solver output, nor shown us the amount by which X differs from B*W.

You have not shown us Lemma 1. So I don;'t even know that the author claims that X must equal B*W. And even if it does, that would only be within solver tolerance, as propagated through the relationships.

There might be other issues, but you first need to address these.

3

sorry for that, the definition for B is

B = zeros(numK*numN,numK);   
B(1,1) = 1;
B(numN+4,2) = 1;
B(numN*2+13,3) = 1;
B(numN*3+16,4) =1;

and Lemma 1 is

The equality constraint X=B*W is equivalent to the following equality constraints in additional variables U and V
\begin{bmatrix} U & X & B \\ X^H & V & W^H \\ B^H & W & I_M \end{bmatrix} \succeq 0 Trace(U - B*B^H) \leq 0

and the result of cvx is

Calling SDPT3 4.0: 9385 variables, 4664 equality constraints
   For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------

 num. of constraints = 4664
 dim. of sdp    var  = 280,   num. of sdp  blk  =  3
 dim. of socp   var  = 80,   num. of socp blk  =  8
 dim. of linear var  =  5
 dim. of free   var  =  4 *** convert ublk to lblk
*******************************************************************
   SDPT3: Infeasible path-following algorithms
*******************************************************************
 version  predcorr  gam  expon  scale_data
   HKM      1      0.000   1        0    
it pstep dstep pinfeas dinfeas  gap      prim-obj      dual-obj    cputime
-------------------------------------------------------------------
 0|0.000|0.000|1.3e+03|3.2e+01|1.3e+06| 1.768000e+03  0.000000e+00| 0:0:00| spchol  1  1 
 1|0.555|0.067|5.8e+02|3.0e+01|8.1e+06| 2.520916e+04 -1.628662e+02| 0:0:01| chol  1  1 
 2|0.815|0.773|1.1e+02|7.1e+00|1.9e+06| 2.379172e+04 -5.927467e+02| 0:0:02| chol  1  1 
 3|0.530|0.653|5.0e+01|2.5e+00|7.1e+05| 2.102215e+04 -8.476864e+02| 0:0:04| chol  1  1 
 4|0.385|0.504|3.1e+01|1.3e+00|4.1e+05| 1.811369e+04 -1.286846e+03| 0:0:05| chol  1  1 
 5|0.608|0.742|1.2e+01|3.6e-01|1.4e+05| 1.157310e+04 -1.085361e+03| 0:0:07| chol  1  1 
 6|0.765|0.397|2.8e+00|2.2e-01|9.4e+04| 4.299680e+03 -8.223373e+02| 0:0:08| chol  1  1 
 7|0.820|0.624|5.1e-01|8.8e-02|3.6e+04| 1.046516e+03 -3.879649e+02| 0:0:09| chol  1  1 
 8|0.671|0.244|1.7e-01|6.7e-02|2.8e+04| 5.360222e+02 -2.926378e+02| 0:0:10| chol  1  1 
 9|0.601|0.361|6.7e-02|4.4e-02|1.8e+04| 2.909562e+02 -1.878144e+02| 0:0:12| chol  1  1 
10|0.516|0.196|3.3e-02|3.9e-02|1.6e+04| 2.003444e+02 -1.516490e+02| 0:0:13| chol  1  1 
11|0.617|0.530|1.2e-02|2.1e-02|8.7e+03| 1.052296e+02 -7.369333e+01| 0:0:14| chol  1  1 
12|0.610|0.204|4.9e-03|1.8e-02|7.8e+03| 6.424645e+01 -5.984993e+01| 0:0:16| chol  1  1 
13|0.627|0.381|1.8e-03|1.2e-02|5.3e+03| 3.286659e+01 -3.897093e+01| 0:0:17| chol  1  1 
14|0.492|0.161|9.2e-04|1.1e-02|4.8e+03| 2.636876e+01 -3.364248e+01| 0:0:18| chol  1  1 
15|0.637|0.554|3.3e-04|5.1e-03|2.2e+03| 1.197037e+01 -1.760534e+01| 0:0:20| chol  1  1 
16|0.595|0.164|1.4e-04|4.4e-03|2.0e+03| 7.322241e+00 -1.550445e+01| 0:0:21| chol  1  1 
17|0.673|0.577|4.5e-05|1.9e-03|8.5e+02| 1.569284e+00 -9.023005e+00| 0:0:22| chol  1  1 
18|0.691|0.167|1.4e-05|1.7e-03|7.3e+02|-2.884482e-01 -8.235530e+00| 0:0:24| chol  1  1 
19|0.803|0.687|2.9e-06|5.3e-04|2.3e+02|-2.726027e+00 -5.374448e+00| 0:0:25| chol  1  1 
20|1.000|0.300|5.9e-07|3.7e-04|1.7e+02|-3.656781e+00 -4.973919e+00| 0:0:26| chol  1  1 
21|1.000|0.702|1.0e-06|1.1e-04|5.0e+01|-3.951924e+00 -4.293589e+00| 0:0:28| chol  1  1 
22|1.000|0.566|5.1e-07|4.9e-05|2.2e+01|-3.990044e+00 -4.128108e+00| 0:0:29| chol  1  1 
23|1.000|0.191|3.0e-07|3.9e-05|1.8e+01|-3.987768e+00 -4.103845e+00| 0:0:30| chol  1  1 
24|1.000|0.479|6.6e-07|2.1e-05|9.3e+00|-3.995225e+00 -4.054313e+00| 0:0:31| chol  2  1 
25|1.000|0.604|4.8e-07|8.2e-06|3.7e+00|-3.999052e+00 -4.021549e+00| 0:0:33| chol  2  2 
26|1.000|0.290|1.8e-09|5.9e-06|2.7e+00|-3.999338e+00 -4.015322e+00| 0:0:34| chol  1  1 
27|1.000|0.295|1.0e-07|4.2e-06|1.9e+00|-3.999454e+00 -4.010823e+00| 0:0:35| chol  2  2 
28|1.000|0.423|1.1e-09|2.4e-06|1.1e+00|-3.999747e+00 -4.006262e+00| 0:0:37| chol  2  2 
29|1.000|0.461|8.4e-10|1.3e-06|5.9e-01|-3.999907e+00 -4.003378e+00| 0:0:38| chol  1  1 
30|1.000|0.354|2.4e-07|8.4e-07|3.8e-01|-3.999948e+00 -4.002185e+00| 0:0:39| chol  2  1 
31|1.000|0.284|8.4e-07|6.0e-07|2.8e-01|-3.999952e+00 -4.001566e+00| 0:0:40| chol  1  2 
32|1.000|0.319|1.2e-09|4.1e-07|1.9e-01|-3.999968e+00 -4.001067e+00| 0:0:42| chol  2  2 
33|1.000|0.375|1.3e-09|2.6e-07|1.2e-01|-3.999982e+00 -4.000668e+00| 0:0:43| chol  2  2 
34|1.000|0.385|1.3e-09|1.6e-07|7.3e-02|-3.999991e+00 -4.000412e+00| 0:0:45| chol  2  2 
35|1.000|0.343|1.3e-09|1.0e-07|4.8e-02|-3.999994e+00 -4.000271e+00| 0:0:46| chol  2  2 
36|1.000|0.307|1.7e-09|7.3e-08|3.3e-02|-3.999996e+00 -4.000188e+00| 0:0:47| chol  2  2 
37|1.000|0.311|4.0e-09|2.1e-05|2.3e-02|-3.999997e+00 -4.000129e+00| 0:0:49| chol  1  1 
38|1.000|0.979|1.3e-08|1.5e-05|5.8e-04|-3.999998e+00 -4.000003e+00| 0:0:50| chol  1  1 
39|0.957|0.975|1.7e-09|3.6e-07|1.5e-05|-4.000000e+00 -4.000000e+00| 0:0:51| chol  1  1 
  stop: primal infeas has deteriorated too much, 3.4e-07
40|1.000|0.897|1.7e-09|3.6e-07|1.5e-05|-4.000000e+00 -4.000000e+00| 0:0:53|
-------------------------------------------------------------------
 number of iterations   = 40
 primal objective value = -3.99999992e+00
 dual   objective value = -4.00000007e+00
 gap := trace(XZ)       = 1.55e-05
 relative gap           = 1.72e-06
 actual relative gap    = 1.74e-08
 rel. primal infeas (scaled problem)   = 1.68e-09
 rel. dual     "        "       "      = 3.60e-07
 rel. primal infeas (unscaled problem) = 0.00e+00
 rel. dual     "        "       "      = 0.00e+00
 norm(X), norm(y), norm(Z) = 1.0e+05, 1.1e+05, 1.1e+05
 norm(A), norm(b), norm(C) = 1.2e+02, 5.0e+00, 6.1e+00
 Total CPU time (secs)  = 52.62  
 CPU time per iteration = 1.32  
 termination code       = -7
 DIMACS: 2.8e-09  0.0e+00  4.4e-07  0.0e+00  1.7e-08  1.7e-06
-------------------------------------------------------------------
 
------------------------------------------------------------
Status: Inaccurate/Solved
Optimal value (cvx_optval): +7.425e-08

I just check whether X==B*W,and the resut is

>> X == B*W

ans =

  64×4 logical 数组

   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0

and the result of X-B*W is

>> X-B*W

ans =

   1.0e-04 *

  列 1 至 3

  -0.3601 - 0.0048i  -0.0431 + 0.1822i  -0.0397 - 0.3263i
  -0.3247 - 0.1759i   0.0279 + 0.2508i   0.3882 - 0.0911i
  -0.2862 + 0.0331i   0.0543 + 0.5207i  -0.0395 + 0.4849i
  -0.6579 - 0.0897i  -0.4393 + 0.7966i  -0.8379 + 0.1995i
  -0.5735 - 0.2432i  -0.4006 + 0.1828i  -0.0085 - 0.5720i
  -0.0437 - 0.2732i  -0.3380 - 0.0235i   0.1887 - 0.2730i
   0.0171 + 0.1110i  -0.2603 - 0.0873i   0.0669 - 0.1187i
  -0.4315 + 0.1651i  -0.2829 - 0.1386i  -0.0114 - 0.2749i
  -0.4424 - 0.1321i  -0.1201 - 0.2917i   0.2497 - 0.1162i
   0.0088 - 0.1119i  -0.0702 - 0.2656i   0.0760 - 0.1140i
  -0.0231 + 0.2757i  -0.0015 - 0.3390i   0.1575 - 0.2914i
  -0.5535 + 0.2857i   0.2086 - 0.3879i   0.5121 - 0.2520i
  -0.6491 + 0.1393i   0.8237 - 0.3867i   0.2017 + 0.8359i
  -0.2879 - 0.0114i   0.5158 + 0.0880i  -0.4141 + 0.2544i
  -0.3102 + 0.1997i   0.2481 + 0.0438i  -0.0950 - 0.3877i
  -0.7451 + 0.0670i   0.3819 - 0.0671i   0.6404 - 0.2338i
  -0.7487 - 0.0106i  -0.0910 + 0.3768i  -0.0832 - 0.6769i
  -0.3247 - 0.1759i   0.0279 + 0.2508i   0.3882 - 0.0911i
  -0.2862 + 0.0331i   0.0543 + 0.5207i  -0.0395 + 0.4849i
  -0.3173 - 0.0429i  -0.2131 + 0.3818i  -0.4031 + 0.0975i
  -0.5735 - 0.2432i  -0.4006 + 0.1828i  -0.0085 - 0.5720i
  -0.0437 - 0.2732i  -0.3380 - 0.0235i   0.1887 - 0.2730i
   0.0171 + 0.1110i  -0.2603 - 0.0873i   0.0669 - 0.1187i
  -0.4315 + 0.1651i  -0.2829 - 0.1386i  -0.0114 - 0.2749i
  -0.4424 - 0.1321i  -0.1201 - 0.2917i   0.2497 - 0.1162i
   0.0088 - 0.1119i  -0.0702 - 0.2656i   0.0760 - 0.1140i
  -0.0231 + 0.2757i  -0.0015 - 0.3390i   0.1575 - 0.2914i
  -0.5535 + 0.2857i   0.2086 - 0.3879i   0.5121 - 0.2520i
  -0.6491 + 0.1393i   0.8237 - 0.3867i   0.2017 + 0.8359i
  -0.2879 - 0.0114i   0.5158 + 0.0880i  -0.4141 + 0.2544i
  -0.3102 + 0.1997i   0.2481 + 0.0438i  -0.0950 - 0.3877i
  -0.7451 + 0.0670i   0.3819 - 0.0671i   0.6404 - 0.2338i
  -0.7487 - 0.0106i  -0.0910 + 0.3768i  -0.0832 - 0.6769i
  -0.3247 - 0.1759i   0.0279 + 0.2508i   0.3882 - 0.0911i
  -0.2862 + 0.0331i   0.0543 + 0.5207i  -0.0395 + 0.4849i
  -0.6579 - 0.0897i  -0.4393 + 0.7966i  -0.8379 + 0.1995i
  -0.5735 - 0.2432i  -0.4006 + 0.1828i  -0.0085 - 0.5720i
  -0.0437 - 0.2732i  -0.3380 - 0.0235i   0.1887 - 0.2730i
   0.0171 + 0.1110i  -0.2603 - 0.0873i   0.0669 - 0.1187i
  -0.4315 + 0.1651i  -0.2829 - 0.1386i  -0.0114 - 0.2749i
  -0.4424 - 0.1321i  -0.1201 - 0.2917i   0.2497 - 0.1162i
   0.0088 - 0.1119i  -0.0702 - 0.2656i   0.0760 - 0.1140i
  -0.0231 + 0.2757i  -0.0015 - 0.3390i   0.1575 - 0.2914i
  -0.5535 + 0.2857i   0.2086 - 0.3879i   0.5121 - 0.2520i
  -0.3131 + 0.0669i   0.3950 - 0.1879i   0.0956 + 0.4028i
  -0.2879 - 0.0114i   0.5158 + 0.0880i  -0.4141 + 0.2544i
  -0.3102 + 0.1997i   0.2481 + 0.0438i  -0.0950 - 0.3877i
  -0.7451 + 0.0670i   0.3819 - 0.0671i   0.6404 - 0.2338i
  -0.7487 - 0.0106i  -0.0910 + 0.3768i  -0.0832 - 0.6769i
  -0.3247 - 0.1759i   0.0279 + 0.2508i   0.3882 - 0.0911i
  -0.2862 + 0.0331i   0.0543 + 0.5207i  -0.0395 + 0.4849i
  -0.6579 - 0.0897i  -0.4393 + 0.7966i  -0.8379 + 0.1995i
  -0.5735 - 0.2432i  -0.4006 + 0.1828i  -0.0085 - 0.5720i
  -0.0437 - 0.2732i  -0.3380 - 0.0235i   0.1887 - 0.2730i
   0.0171 + 0.1110i  -0.2603 - 0.0873i   0.0669 - 0.1187i
  -0.4315 + 0.1651i  -0.2829 - 0.1386i  -0.0114 - 0.2749i
  -0.4424 - 0.1321i  -0.1201 - 0.2917i   0.2497 - 0.1162i
   0.0088 - 0.1119i  -0.0702 - 0.2656i   0.0760 - 0.1140i
  -0.0231 + 0.2757i  -0.0015 - 0.3390i   0.1575 - 0.2914i
  -0.5535 + 0.2857i   0.2086 - 0.3879i   0.5121 - 0.2520i
  -0.6491 + 0.1393i   0.8237 - 0.3867i   0.2017 + 0.8359i
  -0.2879 - 0.0114i   0.5158 + 0.0880i  -0.4141 + 0.2544i
  -0.3102 + 0.1997i   0.2481 + 0.0438i  -0.0950 - 0.3877i
  -0.3584 + 0.0320i   0.1846 - 0.0316i   0.3085 - 0.1131i

  列 4

   0.1446 + 0.1676i
  -0.2470 + 0.0751i
  -0.0475 - 0.5536i
   0.8328 - 0.5596i
   0.3224 + 0.3228i
  -0.0788 + 0.3370i
  -0.2206 + 0.0244i
   0.0308 - 0.1752i
   0.1623 - 0.0717i
   0.0287 + 0.2199i
  -0.3087 + 0.1569i
  -0.3907 - 0.2363i
   0.3443 - 0.9428i
   0.5491 - 0.0860i
  -0.0143 + 0.2584i
  -0.4082 - 0.2080i
   0.2993 + 0.3456i
  -0.2470 + 0.0751i
  -0.0475 - 0.5536i
   0.4007 - 0.2724i
   0.3224 + 0.3228i
  -0.0788 + 0.3370i
  -0.2206 + 0.0244i
   0.0308 - 0.1752i
   0.1623 - 0.0717i
   0.0287 + 0.2199i
  -0.3087 + 0.1569i
  -0.3907 - 0.2363i
   0.3443 - 0.9428i
   0.5491 - 0.0860i
  -0.0143 + 0.2584i
  -0.4082 - 0.2080i
   0.2993 + 0.3456i
  -0.2470 + 0.0751i
  -0.0475 - 0.5536i
   0.8328 - 0.5596i
   0.3224 + 0.3228i
  -0.0788 + 0.3370i
  -0.2206 + 0.0244i
   0.0308 - 0.1752i
   0.1623 - 0.0717i
   0.0287 + 0.2199i
  -0.3087 + 0.1569i
  -0.3907 - 0.2363i
   0.1688 - 0.4544i
   0.5491 - 0.0860i
  -0.0143 + 0.2584i
  -0.4082 - 0.2080i
   0.2993 + 0.3456i
  -0.2470 + 0.0751i
  -0.0475 - 0.5536i
   0.8328 - 0.5596i
   0.3224 + 0.3228i
  -0.0788 + 0.3370i
  -0.2206 + 0.0244i
   0.0308 - 0.1752i
   0.1623 - 0.0717i
   0.0287 + 0.2199i
  -0.3087 + 0.1569i
  -0.3907 - 0.2363i
   0.3443 - 0.9428i
   0.5491 - 0.0860i
  -0.0143 + 0.2584i
  -0.1978 - 0.1003i

is it just within solver tolerance?

4

I ran it with Mosek, which solved to optimality, not Inaccurare/Solved as with SDPT3. The resulting maximum magnitude difference across all elements of X-B*W is 4e-5. That seems like it could be within solver tolerance, but I haven’t done an error propagation analysis (which perhaps would not be so easy) to determine for sure.

Using MATLAB logical equality, == to check for satisfaction of an equality relationship on the output of double precision optimization is meaningless nonsense. In the unlikely event it is satisfied, the relationship holds. If it is not satisfied, it tells you nothing.

1 Like
5

I see, thank you!!!

6

What actually happened on your problem is that the optimal X and W are both within solver tolerance of all zeros. So X= B*W also holds within solver tolerance.

The exact theoretical solution in this case is for X and W to be exactly all zeros. That solution is feasible, and achieves objective value of zero, which has to be optimal; ;so that is the optimal solution. Whether that is a satisfying solution for you is for you to determine.

7

Yes, what you said is indeed correct. The optimal X and W are both within solver tolerance of all zeros in this situation.
But I have another question. I think the reason why the optimal X and W are both within solver tolerance of all zeros is that the value of sigmak is 1.0e-4, which results in the constraint
norm([tem(1, : ) sigmak]) -real(tem(1,1))*sqrt((1+1/gamak)) <= 0 still being satisfied when X is within solver tolerance of all zeros.
In order to make X and W not all 0，I set sigmak=1 to solve the problem. When I use the equality constraint X == B*W, the Mosek solves it . However,when I use the LMI constraints, the Mosek failed. According to Lemma 1，the equality constraint is equivalent to the inequality constraints. Why does this happen, can you give me any suggesstion?

My code is following

clc;clear all;
cvx_solver Mosek
numM=4;
numN=16;
numK=4;
B = zeros(numK*numN,numK);   
B(1,1) = 1;
B(numN+4,2) = 1;
B(numN*2+13,3) = 1;
B(numN*3+16,4) =1;
gamak = 10^(4/10);
sigmak = 1.0%e-4; 

Hhat = [-0.0652405201837634 + 1.67262729732120i,-1.17955547159453 + 1.18751713760266i,-1.67191760052190 + 0.0764800249883790i,-1.28286695049747 - 1.07471836870477i,-0.482353009345158 + 1.60281755490566i,-1.43947387936296 + 0.853939918776870i,-1.63764897082401 - 0.344984216936885i,-0.972516792257723 - 1.36188439697510i,-0.868638242346595 + 1.43070133945419i,-1.60748313133005 + 0.465882848453379i,-1.49884726659665 - 0.744389533521191i,-0.600115527527786 - 1.56209162760993i,-1.19944121666771 + 1.16727447733334i,-1.67286688213672 + 0.0481216424873483i,-1.26438193493919 - 1.09624263443367i,-0.189440188150751 - 1.66256796317943i,-0.0652405201837634 + 1.67262729732120i,-1.17955547159453 + 1.18751713760266i,-1.67191760052190 + 0.0764800249883790i,-1.28286695049747 - 1.07471836870477i,-0.482353009345158 + 1.60281755490566i,-1.43947387936296 + 0.853939918776870i,-1.63764897082401 - 0.344984216936885i,-0.972516792257723 - 1.36188439697510i,-0.868638242346595 + 1.43070133945419i,-1.60748313133005 + 0.465882848453379i,-1.49884726659665 - 0.744389533521191i,-0.600115527527786 - 1.56209162760993i,-1.19944121666771 + 1.16727447733334i,-1.67286688213672 + 0.0481216424873483i,-1.26438193493919 - 1.09624263443367i,-0.189440188150751 - 1.66256796317943i,-0.0652405201837634 + 1.67262729732120i,-1.17955547159453 + 1.18751713760266i,-1.67191760052190 + 0.0764800249883790i,-1.28286695049747 - 1.07471836870477i,-0.482353009345158 + 1.60281755490566i,-1.43947387936296 + 0.853939918776870i,-1.63764897082401 - 0.344984216936885i,-0.972516792257723 - 1.36188439697510i,-0.868638242346595 + 1.43070133945419i,-1.60748313133005 + 0.465882848453379i,-1.49884726659665 - 0.744389533521191i,-0.600115527527786 - 1.56209162760993i,-1.19944121666771 + 1.16727447733334i,-1.67286688213672 + 0.0481216424873483i,-1.26438193493919 - 1.09624263443367i,-0.189440188150751 - 1.66256796317943i,-0.0652405201837634 + 1.67262729732120i,-1.17955547159453 + 1.18751713760266i,-1.67191760052190 + 0.0764800249883790i,-1.28286695049747 - 1.07471836870477i,-0.482353009345158 + 1.60281755490566i,-1.43947387936296 + 0.853939918776870i,-1.63764897082401 - 0.344984216936885i,-0.972516792257723 - 1.36188439697510i,-0.868638242346595 + 1.43070133945419i,-1.60748313133005 + 0.465882848453379i,-1.49884726659665 - 0.744389533521191i,-0.600115527527786 - 1.56209162760993i,-1.19944121666771 + 1.16727447733334i,-1.67286688213672 + 0.0481216424873483i,-1.26438193493919 - 1.09624263443367i,-0.189440188150751 - 1.66256796317943i;-0.343804065086222 + 1.25035870569987i,-1.06795829625980 + 0.735246840784658i,-1.29035582664988 - 0.125018060481049i,-0.906692521323401 - 0.926326781420968i,-1.07896619404889 + 0.719611186651815i,-1.28871167484363 - 0.144066446052317i,-0.893172045483233 - 0.939841133857426i,-0.0782781711894032 - 1.29400736073525i,-1.28678633003226 - 0.163093533043658i,-0.879450156536759 - 0.953158170652693i,-0.0592073577237805 - 1.29536180483867i,0.788609540458150 - 1.02911975684605i,-0.865529733721704 - 0.966274890298573i,-0.0401136141564769 - 1.29643518946567i,0.803908041314568 - 1.01764910784814i,1.27014673574948 - 0.261016824009327i,-0.343804065086222 + 1.25035870569987i,-1.06795829625980 + 0.735246840784658i,-1.29035582664988 - 0.125018060481049i,-0.906692521323401 - 0.926326781420968i,-1.07896619404889 + 0.719611186651815i,-1.28871167484363 - 0.144066446052317i,-0.893172045483233 - 0.939841133857426i,-0.0782781711894032 - 1.29400736073525i,-1.28678633003226 - 0.163093533043658i,-0.879450156536759 - 0.953158170652693i,-0.0592073577237805 - 1.29536180483867i,0.788609540458150 - 1.02911975684605i,-0.865529733721704 - 0.966274890298573i,-0.0401136141564769 - 1.29643518946567i,0.803908041314568 - 1.01764910784814i,1.27014673574948 - 0.261016824009327i,-0.343804065086222 + 1.25035870569987i,-1.06795829625980 + 0.735246840784658i,-1.29035582664988 - 0.125018060481049i,-0.906692521323401 - 0.926326781420968i,-1.07896619404889 + 0.719611186651815i,-1.28871167484363 - 0.144066446052317i,-0.893172045483233 - 0.939841133857426i,-0.0782781711894032 - 1.29400736073525i,-1.28678633003226 - 0.163093533043658i,-0.879450156536759 - 0.953158170652693i,-0.0592073577237805 - 1.29536180483867i,0.788609540458150 - 1.02911975684605i,-0.865529733721704 - 0.966274890298573i,-0.0401136141564769 - 1.29643518946567i,0.803908041314568 - 1.01764910784814i,1.27014673574948 - 0.261016824009327i,-0.343804065086222 + 1.25035870569987i,-1.06795829625980 + 0.735246840784658i,-1.29035582664988 - 0.125018060481049i,-0.906692521323401 - 0.926326781420968i,-1.07896619404889 + 0.719611186651815i,-1.28871167484363 - 0.144066446052317i,-0.893172045483233 - 0.939841133857426i,-0.0782781711894032 - 1.29400736073525i,-1.28678633003226 - 0.163093533043658i,-0.879450156536759 - 0.953158170652693i,-0.0592073577237805 - 1.29536180483867i,0.788609540458150 - 1.02911975684605i,-0.865529733721704 - 0.966274890298573i,-0.0401136141564769 - 1.29643518946567i,0.803908041314568 - 1.01764910784814i,1.27014673574948 - 0.261016824009327i;1.24071728293608 + 0.179700700227814i,0.565630468269524 + 1.11880312846160i,-0.590860743445215 + 1.10568006280365i,-1.24445830743253 + 0.151508734102732i,1.09904450999016 + 0.603062835730714i,0.137709106594817 + 1.24603626150278i,-0.940822794842869 + 0.828498146916922i,-1.21860772769445 - 0.294174213602628i,0.818007754776487 + 0.949925700276277i,-0.307651423969747 + 1.21525004163040i,-1.17145771003041 + 0.446268348504055i,-1.03822464696225 - 0.702527390284345i,0.433253630806475 + 1.17630723947605i,-0.713973131056792 + 1.03035738120908i,-1.25352492371160 + 0.00746783946372853i,-0.726192444501054 - 1.02176884769965i,1.24071728293608 + 0.179700700227814i,0.565630468269524 + 1.11880312846160i,-0.590860743445215 + 1.10568006280365i,-1.24445830743253 + 0.151508734102732i,1.09904450999016 + 0.603062835730714i,0.137709106594817 + 1.24603626150278i,-0.940822794842869 + 0.828498146916922i,-1.21860772769445 - 0.294174213602628i,0.818007754776487 + 0.949925700276277i,-0.307651423969747 + 1.21525004163040i,-1.17145771003041 + 0.446268348504055i,-1.03822464696225 - 0.702527390284345i,0.433253630806475 + 1.17630723947605i,-0.713973131056792 + 1.03035738120908i,-1.25352492371160 + 0.00746783946372853i,-0.726192444501054 - 1.02176884769965i,1.24071728293608 + 0.179700700227814i,0.565630468269524 + 1.11880312846160i,-0.590860743445215 + 1.10568006280365i,-1.24445830743253 + 0.151508734102732i,1.09904450999016 + 0.603062835730714i,0.137709106594817 + 1.24603626150278i,-0.940822794842869 + 0.828498146916922i,-1.21860772769445 - 0.294174213602628i,0.818007754776487 + 0.949925700276277i,-0.307651423969747 + 1.21525004163040i,-1.17145771003041 + 0.446268348504055i,-1.03822464696225 - 0.702527390284345i,0.433253630806475 + 1.17630723947605i,-0.713973131056792 + 1.03035738120908i,-1.25352492371160 + 0.00746783946372853i,-0.726192444501054 - 1.02176884769965i,1.24071728293608 + 0.179700700227814i,0.565630468269524 + 1.11880312846160i,-0.590860743445215 + 1.10568006280365i,-1.24445830743253 + 0.151508734102732i,1.09904450999016 + 0.603062835730714i,0.137709106594817 + 1.24603626150278i,-0.940822794842869 + 0.828498146916922i,-1.21860772769445 - 0.294174213602628i,0.818007754776487 + 0.949925700276277i,-0.307651423969747 + 1.21525004163040i,-1.17145771003041 + 0.446268348504055i,-1.03822464696225 - 0.702527390284345i,0.433253630806475 + 1.17630723947605i,-0.713973131056792 + 1.03035738120908i,-1.25352492371160 + 0.00746783946372853i,-0.726192444501054 - 1.02176884769965i;0.744052980415410 + 0.422486488019040i,0.0782469583530654 + 0.852038966354416i,-0.654643750659819 + 0.550924552138733i,-0.826134857216619 - 0.222622584576821i,0.655475492798011 + 0.550391351049500i,-0.0773324313357631 + 0.852397663636049i,-0.743808051133347 + 0.423432385642987i,-0.772423670857399 - 0.368633143340257i,0.545169516184055 + 0.660180545205781i,-0.230454496282484 + 0.824573953154341i,-0.808437014207177 + 0.281858826121668i,-0.693139653731680 - 0.502549050586954i,0.416768220359671 + 0.748212602618902i,-0.376055042570886 + 0.769469806496849i,-0.846378199753204 + 0.130875705770559i,-0.590887855771216 - 0.619934978684891i,0.744052980415410 + 0.422486488019040i,0.0782469583530654 + 0.852038966354416i,-0.654643750659819 + 0.550924552138733i,-0.826134857216619 - 0.222622584576821i,0.655475492798011 + 0.550391351049500i,-0.0773324313357631 + 0.852397663636049i,-0.743808051133347 + 0.423432385642987i,-0.772423670857399 - 0.368633143340257i,0.545169516184055 + 0.660180545205781i,-0.230454496282484 + 0.824573953154341i,-0.808437014207177 + 0.281858826121668i,-0.693139653731680 - 0.502549050586954i,0.416768220359671 + 0.748212602618902i,-0.376055042570886 + 0.769469806496849i,-0.846378199753204 + 0.130875705770559i,-0.590887855771216 - 0.619934978684891i,0.744052980415410 + 0.422486488019040i,0.0782469583530654 + 0.852038966354416i,-0.654643750659819 + 0.550924552138733i,-0.826134857216619 - 0.222622584576821i,0.655475492798011 + 0.550391351049500i,-0.0773324313357631 + 0.852397663636049i,-0.743808051133347 + 0.423432385642987i,-0.772423670857399 - 0.368633143340257i,0.545169516184055 + 0.660180545205781i,-0.230454496282484 + 0.824573953154341i,-0.808437014207177 + 0.281858826121668i,-0.693139653731680 - 0.502549050586954i,0.416768220359671 + 0.748212602618902i,-0.376055042570886 + 0.769469806496849i,-0.846378199753204 + 0.130875705770559i,-0.590887855771216 - 0.619934978684891i,0.744052980415410 + 0.422486488019040i,0.0782469583530654 + 0.852038966354416i,-0.654643750659819 + 0.550924552138733i,-0.826134857216619 - 0.222622584576821i,0.655475492798011 + 0.550391351049500i,-0.0773324313357631 + 0.852397663636049i,-0.743808051133347 + 0.423432385642987i,-0.772423670857399 - 0.368633143340257i,0.545169516184055 + 0.660180545205781i,-0.230454496282484 + 0.824573953154341i,-0.808437014207177 + 0.281858826121668i,-0.693139653731680 - 0.502549050586954i,0.416768220359671 + 0.748212602618902i,-0.376055042570886 + 0.769469806496849i,-0.846378199753204 + 0.130875705770559i,-0.590887855771216 - 0.619934978684891i];

cvx_begin 
        variable W(numM, numK) complex     
        variable X(numM*numN, numK) complex
        variable U(numM*numN, numM*numN) complex semidefinite  
        variable V(numK, numK) complex semidefinite
        variable I(numM, numM) 
        variable genhaoxia(numK,numK) complex
        variable cal(numK,1)
        variable bigmatrix(numM*numN+numK+numM,numM*numN+numK+numM) complex semidefinite 

        dual variables d1 d2 d3 d4 d5 d6
        minimize( sum(sum_square_abs(W)))   % conj( x ) .* x  square_abs( x )
        subject to
            X == B*W;
            %d5:bigmatrix == [ U  X B ;
             %                 X' V W';
              %                B' W I];                             
            %d6:trace(U) - numM <=0;                 
            %I == eye(numM);        
            tem = conj(Hhat)*X;
            for i=1:numK
                imag(tem(i,i)) == 0;        % 约束C1b 
            end
            for i=1:numK
                cal(i) == tem(i,i);
            end
            for i=1:numK
                genhaoxia(i,:) == tem(i,:);
            end
            
            d1:norm([tem(1,:) sigmak]) - real(tem(1,1))*sqrt((1+1/gamak)) <= 0 ;            % 约束C1a
            d2:norm([tem(2,:) sigmak]) - real(tem(2,2))*sqrt((1+1/gamak)) <= 0 ;            % 约束C1a
            d3:norm([tem(3,:) sigmak]) - real(tem(3,3))*sqrt((1+1/gamak)) <= 0 ;            % 约束C1a
            d4:norm([tem(4,:) sigmak]) - real(tem(4,4))*sqrt((1+1/gamak)) <= 0 ;           % 约束C1a       

    cvx_end

And the results obtained using equality constraints and inequality constraints are as follows, respectively

Calling Mosek 9.1.9: 9396 variables, 48 equality constraints
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:34:40)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 48              
  Cones                  : 8               
  Scalar variables       : 100             
  Matrix variables       : 3               
  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                   : 0.01            
Lin. dep.  - number                 : 0               
Presolve terminated. Time: 0.03    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 48              
  Cones                  : 8               
  Scalar variables       : 100             
  Matrix variables       : 3               
  Integer variables      : 0               

Optimizer  - threads                : 16              
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 44
Optimizer  - Cones                  : 8
Optimizer  - Scalar variables       : 84                conic                  : 80              
Optimizer  - Semi-definite variables: 3                 scalarized             : 18732           
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 : 384               after factor           : 552             
Factor     - dense dim.             : 0                 flops                  : 1.30e+04        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   1.0e+00  1.0e+00  5.0e+00  0.00e+00   4.000000000e+00   0.000000000e+00   1.0e+00  0.06  
1   2.1e-01  2.1e-01  1.0e+00  -6.67e-02  2.819614989e+00   1.755801359e+00   2.1e-01  0.16  
2   4.3e-02  4.3e-02  2.6e-01  -1.35e-01  5.944933638e+00   6.386443505e+00   4.3e-02  0.17  
3   6.8e-03  6.8e-03  6.1e-02  -3.63e-01  1.736659681e+01   2.002187630e+01   6.8e-03  0.19  
4   1.8e-03  1.8e-03  1.7e-02  -1.68e-01  3.074203601e+01   3.413228911e+01   1.8e-03  0.20  
5   5.4e-04  5.4e-04  3.5e-03  3.07e-01   4.156880970e+01   4.311811352e+01   5.4e-04  0.23  
6   8.6e-05  8.6e-05  2.5e-04  6.31e-01   4.792339051e+01   4.824423606e+01   8.6e-05  0.25  
7   3.7e-07  3.7e-07  4.2e-08  9.62e-01   4.927567462e+01   4.927608370e+01   3.7e-07  0.27  
8   2.8e-08  2.8e-08  8.8e-10  1.00e+00   4.928116503e+01   4.928119681e+01   2.8e-08  0.28  
9   3.4e-09  4.0e-09  3.7e-11  1.00e+00   4.928164369e+01   4.928164758e+01   3.4e-09  0.30  
10  2.6e-10  3.7e-10  8.6e-13  1.00e+00   4.928170987e+01   4.928171024e+01   2.6e-10  0.31  
Optimizer terminated. Time: 0.34    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: 4.9281709874e+01    nrm: 6e+01    Viol.  con: 1e-08    var: 2e-08    barvar: 0e+00    cones: 3e-09  
  Dual.    obj: 4.9281710236e+01    nrm: 1e+02    Viol.  con: 0e+00    var: 5e-09    barvar: 1e-08    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.34    
    Interior-point          - iterations : 10        time: 0.31    
      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): +49.2817

Calling Mosek 9.1.9: 9385 variables, 4664 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 (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 4664            
  Cones                  : 8               
  Scalar variables       : 89              
  Matrix variables       : 3               
  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: 0.02    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 4664            
  Cones                  : 8               
  Scalar variables       : 89              
  Matrix variables       : 3               
  Integer variables      : 0               

Optimizer  - threads                : 16              
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 4660
Optimizer  - Cones                  : 9
Optimizer  - Scalar variables       : 86                conic                  : 85              
Optimizer  - Semi-definite variables: 3                 scalarized             : 18732           
Factor     - setup time             : 1.42              dense det. time        : 0.00            
Factor     - ML order time          : 0.67              GP order time          : 0.00            
Factor     - nonzeros before factor : 1.08e+07          after factor           : 1.08e+07        
Factor     - dense dim.             : 10                flops                  : 3.41e+10        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   4.0e+00  4.0e+00  9.0e+00  0.00e+00   8.000000000e+00   0.000000000e+00   1.0e+00  1.50  
1   1.6e+00  1.6e+00  2.2e+00  5.17e-01   9.523308938e-01   -2.616128432e+00  4.1e-01  3.69  
2   8.1e-01  8.1e-01  1.3e+00  1.71e-01   -8.408154725e-01  -3.713291212e+00  2.0e-01  5.77  
3   9.0e-02  9.0e-02  1.3e-01  8.29e-02   -5.118136201e+00  -5.497891750e+00  2.3e-02  7.91  
4   1.1e-02  1.1e-02  1.5e-02  9.06e-02   -5.924596277e+00  -5.801729364e+00  2.8e-03  9.83  
5   1.7e-03  1.7e-03  2.2e-03  5.26e-02   -5.985998399e+00  -5.748189349e+00  4.3e-04  11.81 
6   1.1e-03  1.1e-03  1.4e-03  3.76e-02   -5.874238733e+00  -5.635131122e+00  2.8e-04  13.61 
7   2.8e-04  2.8e-04  3.7e-04  7.87e-02   -6.194472261e+00  -5.888225594e+00  7.0e-05  15.44 
8   4.1e-05  4.1e-05  5.1e-05  -5.32e-02  -5.809294085e+00  -5.514026763e+00  1.0e-05  17.23 
9   3.7e-05  3.7e-05  4.7e-05  -5.18e-03  -5.817432666e+00  -5.519390655e+00  9.3e-06  18.92 
10  9.5e-06  9.5e-06  1.3e-05  1.64e-04   -6.036481401e+00  -5.693598956e+00  2.4e-06  20.66 
11  1.5e-06  1.5e-06  1.9e-06  2.46e-02   -5.859332619e+00  -5.528289562e+00  3.6e-07  22.48 
12  3.5e-07  3.5e-07  4.5e-07  -7.19e-03  -5.723601135e+00  -5.403250015e+00  8.8e-08  24.22 
13  7.2e-08  7.2e-08  9.2e-08  5.49e-03   -5.678358098e+00  -5.361243874e+00  1.8e-08  25.98 
14  1.3e-08  1.3e-08  1.6e-08  -7.15e-03  -5.457268235e+00  -5.166595333e+00  3.3e-09  28.00 
15  4.3e-09  2.0e-09  2.3e-09  7.96e-03   -5.238488932e+00  -4.978429846e+00  5.1e-10  29.86 
16  1.3e-09  6.3e-10  7.3e-10  -1.47e-02  -5.275197566e+00  -5.007927973e+00  1.6e-10  31.91 
17  3.9e-10  1.9e-10  2.3e-10  2.46e-02   -5.508623621e+00  -5.202557205e+00  4.6e-11  33.78 
18  1.5e-10  7.0e-11  8.9e-11  -4.10e-03  -5.572612544e+00  -5.254992063e+00  1.8e-11  35.64 
19  6.3e-11  3.5e-11  3.9e-11  3.90e-02   -5.722748726e+00  -5.381365928e+00  7.5e-12  37.56 
20  1.6e-11  1.6e-11  1.1e-11  -1.05e-02  -5.869920602e+00  -5.506135405e+00  2.0e-12  39.48 
21  7.0e-12  1.4e-10  4.6e-12  -2.75e-03  -5.910066469e+00  -5.538052756e+00  8.3e-13  41.41 
22  1.9e-12  9.6e-11  1.2e-12  2.65e-02   -5.956778759e+00  -5.575789673e+00  2.2e-13  43.25 
23  6.3e-13  1.4e-09  4.3e-13  -6.37e-03  -6.067196784e+00  -5.669165342e+00  7.6e-14  45.11 
24  5.5e-13  1.4e-09  4.3e-13  4.65e-03   -6.067431914e+00  -5.669348256e+00  7.5e-14  47.06 
25  5.4e-13  1.4e-09  4.2e-13  5.15e-03   -6.067488123e+00  -5.669391827e+00  7.5e-14  49.00 
26  5.4e-13  1.4e-09  4.2e-13  5.25e-03   -6.067488123e+00  -5.669391827e+00  7.5e-14  51.00 
27  5.2e-13  1.4e-09  4.2e-13  -9.56e-03  -6.067892944e+00  -5.669721335e+00  7.4e-14  52.91 
28  4.7e-13  1.4e-09  4.1e-13  -7.93e-03  -6.068241883e+00  -5.670002497e+00  7.4e-14  54.89 
29  4.3e-13  1.6e-09  4.1e-13  -1.40e-02  -6.068487404e+00  -5.670205504e+00  7.3e-14  56.94 
30  4.3e-13  1.6e-09  4.1e-13  -1.62e-02  -6.068487404e+00  -5.670205504e+00  7.3e-14  58.97 
31  4.3e-13  1.6e-09  4.1e-13  -1.41e-02  -6.068487404e+00  -5.670205504e+00  7.3e-14  60.91 
32  4.0e-13  1.4e-09  4.0e-13  -1.41e-02  -6.069511852e+00  -5.671012474e+00  7.1e-14  62.80 
33  4.0e-13  1.7e-09  3.9e-13  -1.00e-02  -6.069514517e+00  -5.671014174e+00  7.1e-14  64.75 
34  4.0e-13  1.7e-09  3.9e-13  -1.00e-02  -6.069514517e+00  -5.671014174e+00  7.1e-14  66.69 
35  4.0e-13  1.7e-09  3.9e-13  -1.02e-02  -6.069514517e+00  -5.671014174e+00  7.1e-14  68.64 
Optimizer terminated. Time: 70.64   


Interior-point solution summary
  Problem status  : ILL_POSED
  Solution status : DUAL_ILLPOSED_CER
  Primal.  obj: -3.3188101067e-06   nrm: 4e+00    Viol.  con: 1e-06    var: 0e+00    barvar: 0e+00    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 70.64   
    Interior-point          - iterations : 36        time: 70.61   
      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: Failed
Optimal value (cvx_optval): NaN
8

Mosek reports that the problem is dual ill-posed. CVX provided Mosek the dual of the original problem, so that means Mosek concluded that your problem is primal ill-posed,.

I ran the problem using Mosek 10.1.17, and it solved to claimed optimality, with cvx_optval = 49.2817. The Mosek solver has become more robust since Mosek 9.1.9, so perhaps that’s why my run succeeded and yours didn’t.

You should also use the latest version of Mosek 10.x, rather than 9.1.9 which is bundled with CVX. In order to do so, follow the directions in

9

Thanks for your suggestion! But when I use the Mosek 10.1.17 to solve the following code, it still failed.
My code is

clc;clear all;
cvx_solver Mosek_5 
numM=4;
numN=16;
numK=4;
B = zeros(numK*numN,numK);   
B(1,1) = 1;
B(numN+4,2) = 1;
B(numN*2+13,3) = 1;
B(numN*3+16,4) =1;
gamak = 10^(4/10);
sigmak = 1.0%e-4; 

Hhat = [-0.0652405201837634 + 1.67262729732120i,-1.17955547159453 + 1.18751713760266i,-1.67191760052190 + 0.0764800249883790i,-1.28286695049747 - 1.07471836870477i,-0.482353009345158 + 1.60281755490566i,-1.43947387936296 + 0.853939918776870i,-1.63764897082401 - 0.344984216936885i,-0.972516792257723 - 1.36188439697510i,-0.868638242346595 + 1.43070133945419i,-1.60748313133005 + 0.465882848453379i,-1.49884726659665 - 0.744389533521191i,-0.600115527527786 - 1.56209162760993i,-1.19944121666771 + 1.16727447733334i,-1.67286688213672 + 0.0481216424873483i,-1.26438193493919 - 1.09624263443367i,-0.189440188150751 - 1.66256796317943i,-0.0652405201837634 + 1.67262729732120i,-1.17955547159453 + 1.18751713760266i,-1.67191760052190 + 0.0764800249883790i,-1.28286695049747 - 1.07471836870477i,-0.482353009345158 + 1.60281755490566i,-1.43947387936296 + 0.853939918776870i,-1.63764897082401 - 0.344984216936885i,-0.972516792257723 - 1.36188439697510i,-0.868638242346595 + 1.43070133945419i,-1.60748313133005 + 0.465882848453379i,-1.49884726659665 - 0.744389533521191i,-0.600115527527786 - 1.56209162760993i,-1.19944121666771 + 1.16727447733334i,-1.67286688213672 + 0.0481216424873483i,-1.26438193493919 - 1.09624263443367i,-0.189440188150751 - 1.66256796317943i,-0.0652405201837634 + 1.67262729732120i,-1.17955547159453 + 1.18751713760266i,-1.67191760052190 + 0.0764800249883790i,-1.28286695049747 - 1.07471836870477i,-0.482353009345158 + 1.60281755490566i,-1.43947387936296 + 0.853939918776870i,-1.63764897082401 - 0.344984216936885i,-0.972516792257723 - 1.36188439697510i,-0.868638242346595 + 1.43070133945419i,-1.60748313133005 + 0.465882848453379i,-1.49884726659665 - 0.744389533521191i,-0.600115527527786 - 1.56209162760993i,-1.19944121666771 + 1.16727447733334i,-1.67286688213672 + 0.0481216424873483i,-1.26438193493919 - 1.09624263443367i,-0.189440188150751 - 1.66256796317943i,-0.0652405201837634 + 1.67262729732120i,-1.17955547159453 + 1.18751713760266i,-1.67191760052190 + 0.0764800249883790i,-1.28286695049747 - 1.07471836870477i,-0.482353009345158 + 1.60281755490566i,-1.43947387936296 + 0.853939918776870i,-1.63764897082401 - 0.344984216936885i,-0.972516792257723 - 1.36188439697510i,-0.868638242346595 + 1.43070133945419i,-1.60748313133005 + 0.465882848453379i,-1.49884726659665 - 0.744389533521191i,-0.600115527527786 - 1.56209162760993i,-1.19944121666771 + 1.16727447733334i,-1.67286688213672 + 0.0481216424873483i,-1.26438193493919 - 1.09624263443367i,-0.189440188150751 - 1.66256796317943i;-0.343804065086222 + 1.25035870569987i,-1.06795829625980 + 0.735246840784658i,-1.29035582664988 - 0.125018060481049i,-0.906692521323401 - 0.926326781420968i,-1.07896619404889 + 0.719611186651815i,-1.28871167484363 - 0.144066446052317i,-0.893172045483233 - 0.939841133857426i,-0.0782781711894032 - 1.29400736073525i,-1.28678633003226 - 0.163093533043658i,-0.879450156536759 - 0.953158170652693i,-0.0592073577237805 - 1.29536180483867i,0.788609540458150 - 1.02911975684605i,-0.865529733721704 - 0.966274890298573i,-0.0401136141564769 - 1.29643518946567i,0.803908041314568 - 1.01764910784814i,1.27014673574948 - 0.261016824009327i,-0.343804065086222 + 1.25035870569987i,-1.06795829625980 + 0.735246840784658i,-1.29035582664988 - 0.125018060481049i,-0.906692521323401 - 0.926326781420968i,-1.07896619404889 + 0.719611186651815i,-1.28871167484363 - 0.144066446052317i,-0.893172045483233 - 0.939841133857426i,-0.0782781711894032 - 1.29400736073525i,-1.28678633003226 - 0.163093533043658i,-0.879450156536759 - 0.953158170652693i,-0.0592073577237805 - 1.29536180483867i,0.788609540458150 - 1.02911975684605i,-0.865529733721704 - 0.966274890298573i,-0.0401136141564769 - 1.29643518946567i,0.803908041314568 - 1.01764910784814i,1.27014673574948 - 0.261016824009327i,-0.343804065086222 + 1.25035870569987i,-1.06795829625980 + 0.735246840784658i,-1.29035582664988 - 0.125018060481049i,-0.906692521323401 - 0.926326781420968i,-1.07896619404889 + 0.719611186651815i,-1.28871167484363 - 0.144066446052317i,-0.893172045483233 - 0.939841133857426i,-0.0782781711894032 - 1.29400736073525i,-1.28678633003226 - 0.163093533043658i,-0.879450156536759 - 0.953158170652693i,-0.0592073577237805 - 1.29536180483867i,0.788609540458150 - 1.02911975684605i,-0.865529733721704 - 0.966274890298573i,-0.0401136141564769 - 1.29643518946567i,0.803908041314568 - 1.01764910784814i,1.27014673574948 - 0.261016824009327i,-0.343804065086222 + 1.25035870569987i,-1.06795829625980 + 0.735246840784658i,-1.29035582664988 - 0.125018060481049i,-0.906692521323401 - 0.926326781420968i,-1.07896619404889 + 0.719611186651815i,-1.28871167484363 - 0.144066446052317i,-0.893172045483233 - 0.939841133857426i,-0.0782781711894032 - 1.29400736073525i,-1.28678633003226 - 0.163093533043658i,-0.879450156536759 - 0.953158170652693i,-0.0592073577237805 - 1.29536180483867i,0.788609540458150 - 1.02911975684605i,-0.865529733721704 - 0.966274890298573i,-0.0401136141564769 - 1.29643518946567i,0.803908041314568 - 1.01764910784814i,1.27014673574948 - 0.261016824009327i;1.24071728293608 + 0.179700700227814i,0.565630468269524 + 1.11880312846160i,-0.590860743445215 + 1.10568006280365i,-1.24445830743253 + 0.151508734102732i,1.09904450999016 + 0.603062835730714i,0.137709106594817 + 1.24603626150278i,-0.940822794842869 + 0.828498146916922i,-1.21860772769445 - 0.294174213602628i,0.818007754776487 + 0.949925700276277i,-0.307651423969747 + 1.21525004163040i,-1.17145771003041 + 0.446268348504055i,-1.03822464696225 - 0.702527390284345i,0.433253630806475 + 1.17630723947605i,-0.713973131056792 + 1.03035738120908i,-1.25352492371160 + 0.00746783946372853i,-0.726192444501054 - 1.02176884769965i,1.24071728293608 + 0.179700700227814i,0.565630468269524 + 1.11880312846160i,-0.590860743445215 + 1.10568006280365i,-1.24445830743253 + 0.151508734102732i,1.09904450999016 + 0.603062835730714i,0.137709106594817 + 1.24603626150278i,-0.940822794842869 + 0.828498146916922i,-1.21860772769445 - 0.294174213602628i,0.818007754776487 + 0.949925700276277i,-0.307651423969747 + 1.21525004163040i,-1.17145771003041 + 0.446268348504055i,-1.03822464696225 - 0.702527390284345i,0.433253630806475 + 1.17630723947605i,-0.713973131056792 + 1.03035738120908i,-1.25352492371160 + 0.00746783946372853i,-0.726192444501054 - 1.02176884769965i,1.24071728293608 + 0.179700700227814i,0.565630468269524 + 1.11880312846160i,-0.590860743445215 + 1.10568006280365i,-1.24445830743253 + 0.151508734102732i,1.09904450999016 + 0.603062835730714i,0.137709106594817 + 1.24603626150278i,-0.940822794842869 + 0.828498146916922i,-1.21860772769445 - 0.294174213602628i,0.818007754776487 + 0.949925700276277i,-0.307651423969747 + 1.21525004163040i,-1.17145771003041 + 0.446268348504055i,-1.03822464696225 - 0.702527390284345i,0.433253630806475 + 1.17630723947605i,-0.713973131056792 + 1.03035738120908i,-1.25352492371160 + 0.00746783946372853i,-0.726192444501054 - 1.02176884769965i,1.24071728293608 + 0.179700700227814i,0.565630468269524 + 1.11880312846160i,-0.590860743445215 + 1.10568006280365i,-1.24445830743253 + 0.151508734102732i,1.09904450999016 + 0.603062835730714i,0.137709106594817 + 1.24603626150278i,-0.940822794842869 + 0.828498146916922i,-1.21860772769445 - 0.294174213602628i,0.818007754776487 + 0.949925700276277i,-0.307651423969747 + 1.21525004163040i,-1.17145771003041 + 0.446268348504055i,-1.03822464696225 - 0.702527390284345i,0.433253630806475 + 1.17630723947605i,-0.713973131056792 + 1.03035738120908i,-1.25352492371160 + 0.00746783946372853i,-0.726192444501054 - 1.02176884769965i;0.744052980415410 + 0.422486488019040i,0.0782469583530654 + 0.852038966354416i,-0.654643750659819 + 0.550924552138733i,-0.826134857216619 - 0.222622584576821i,0.655475492798011 + 0.550391351049500i,-0.0773324313357631 + 0.852397663636049i,-0.743808051133347 + 0.423432385642987i,-0.772423670857399 - 0.368633143340257i,0.545169516184055 + 0.660180545205781i,-0.230454496282484 + 0.824573953154341i,-0.808437014207177 + 0.281858826121668i,-0.693139653731680 - 0.502549050586954i,0.416768220359671 + 0.748212602618902i,-0.376055042570886 + 0.769469806496849i,-0.846378199753204 + 0.130875705770559i,-0.590887855771216 - 0.619934978684891i,0.744052980415410 + 0.422486488019040i,0.0782469583530654 + 0.852038966354416i,-0.654643750659819 + 0.550924552138733i,-0.826134857216619 - 0.222622584576821i,0.655475492798011 + 0.550391351049500i,-0.0773324313357631 + 0.852397663636049i,-0.743808051133347 + 0.423432385642987i,-0.772423670857399 - 0.368633143340257i,0.545169516184055 + 0.660180545205781i,-0.230454496282484 + 0.824573953154341i,-0.808437014207177 + 0.281858826121668i,-0.693139653731680 - 0.502549050586954i,0.416768220359671 + 0.748212602618902i,-0.376055042570886 + 0.769469806496849i,-0.846378199753204 + 0.130875705770559i,-0.590887855771216 - 0.619934978684891i,0.744052980415410 + 0.422486488019040i,0.0782469583530654 + 0.852038966354416i,-0.654643750659819 + 0.550924552138733i,-0.826134857216619 - 0.222622584576821i,0.655475492798011 + 0.550391351049500i,-0.0773324313357631 + 0.852397663636049i,-0.743808051133347 + 0.423432385642987i,-0.772423670857399 - 0.368633143340257i,0.545169516184055 + 0.660180545205781i,-0.230454496282484 + 0.824573953154341i,-0.808437014207177 + 0.281858826121668i,-0.693139653731680 - 0.502549050586954i,0.416768220359671 + 0.748212602618902i,-0.376055042570886 + 0.769469806496849i,-0.846378199753204 + 0.130875705770559i,-0.590887855771216 - 0.619934978684891i,0.744052980415410 + 0.422486488019040i,0.0782469583530654 + 0.852038966354416i,-0.654643750659819 + 0.550924552138733i,-0.826134857216619 - 0.222622584576821i,0.655475492798011 + 0.550391351049500i,-0.0773324313357631 + 0.852397663636049i,-0.743808051133347 + 0.423432385642987i,-0.772423670857399 - 0.368633143340257i,0.545169516184055 + 0.660180545205781i,-0.230454496282484 + 0.824573953154341i,-0.808437014207177 + 0.281858826121668i,-0.693139653731680 - 0.502549050586954i,0.416768220359671 + 0.748212602618902i,-0.376055042570886 + 0.769469806496849i,-0.846378199753204 + 0.130875705770559i,-0.590887855771216 - 0.619934978684891i];

cvx_begin 
        variable W(numM, numK) complex     
        variable X(numM*numN, numK) complex
        variable U(numM*numN, numM*numN) complex semidefinite  
        variable V(numK, numK) complex semidefinite
        variable I(numM, numM) 
        variable genhaoxia(numK,numK) complex
        variable cal(numK,1)
        variable bigmatrix(numM*numN+numK+numM,numM*numN+numK+numM) complex semidefinite 

        dual variables d1 d2 d3 d4 d5 d6
        minimize( sum(sum_square_abs(W)))   % conj( x ) .* x  square_abs( x )
        subject to
            %X == B*W;
            d5:bigmatrix == [ U  X B ;
                              X' V W';
                              B' W I];                             
            d6:trace(U) - numM <=0;                 
            I == eye(numM);        
            tem = conj(Hhat)*X;
            for i=1:numK
                imag(tem(i,i)) == 0;        % 约束C1b 这里注意只有对角线上的虚部是0
            end
            for i=1:numK
                cal(i) == tem(i,i);
            end
            for i=1:numK
                genhaoxia(i,:) == tem(i,:);
            end
            
            d1:norm([tem(1,:) sigmak]) - real(tem(1,1))*sqrt((1+1/gamak)) <= 0 ;            % 约束C1a
            d2:norm([tem(2,:) sigmak]) - real(tem(2,2))*sqrt((1+1/gamak)) <= 0 ;            % 约束C1a
            d3:norm([tem(3,:) sigmak]) - real(tem(3,3))*sqrt((1+1/gamak)) <= 0 ;            % 约束C1a
            d4:norm([tem(4,:) sigmak]) - real(tem(4,4))*sqrt((1+1/gamak)) <= 0 ;           % 约束C1a       

    cvx_end

and my result is

Calling Mosek_5 10.1.17: 9385 variables, 4664 equality constraints
   For improved efficiency, Mosek_5 is solving the dual problem.
------------------------------------------------------------

MOSEK Version 10.1.16 (Build date: 2023-10-26 10:16:52)
Copyright (c) MOSEK ApS, Denmark WWW: mosek.com
Platform: Windows/64-X86

Problem
  Name                   :                 
  Objective sense        : minimize        
  Type                   : CONIC (conic optimization problem)
  Constraints            : 4664            
  Affine conic cons.     : 0               
  Disjunctive cons.      : 0               
  Cones                  : 8               
  Scalar variables       : 89              
  Matrix variables       : 3 (scalarized: 18732)
  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.  - primal attempts        : 1                 successes              : 1               
Lin. dep.  - dual attempts          : 0                 successes              : 0               
Lin. dep.  - primal deps.           : 0                 dual deps.             : 0               
Presolve terminated. Time: 0.01    
GP based matrix reordering started.
GP based matrix reordering terminated.
Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 4660            
Optimizer  - Cones                  : 9               
Optimizer  - Scalar variables       : 86                conic                  : 85              
Optimizer  - Semi-definite variables: 3                 scalarized             : 18732           
Factor     - setup time             : 1.49            
Factor     - dense det. time        : 0.55              GP order time          : 0.00            
Factor     - nonzeros before factor : 1.08e+07          after factor           : 1.09e+07        
Factor     - dense dim.             : 0                 flops                  : 3.42e+10        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   4.0e+00  4.0e+00  9.0e+00  0.00e+00   8.000000000e+00   0.000000000e+00   1.0e+00  1.53  
1   1.6e+00  1.6e+00  2.2e+00  5.17e-01   9.523308938e-01   -2.616128432e+00  4.1e-01  2.08  
2   8.1e-01  8.1e-01  1.3e+00  1.71e-01   -8.408154725e-01  -3.713291212e+00  2.0e-01  2.55  
3   9.0e-02  9.0e-02  1.3e-01  8.29e-02   -5.118136201e+00  -5.497891750e+00  2.3e-02  3.14  
4   1.1e-02  1.1e-02  1.5e-02  9.06e-02   -5.924596277e+00  -5.801729364e+00  2.8e-03  3.73  
5   1.7e-03  1.7e-03  2.2e-03  5.26e-02   -5.985998834e+00  -5.748188960e+00  4.3e-04  4.31  
6   1.1e-03  1.1e-03  1.4e-03  3.76e-02   -5.874184137e+00  -5.635087383e+00  2.8e-04  4.78  
7   2.8e-04  2.8e-04  3.7e-04  7.87e-02   -6.195855261e+00  -5.889307945e+00  7.0e-05  5.26  
8   4.1e-05  4.1e-05  5.1e-05  -5.34e-02  -5.783991777e+00  -5.491291099e+00  1.0e-05  5.81  
9   3.7e-05  3.7e-05  4.6e-05  -6.17e-03  -5.788450728e+00  -5.493266136e+00  9.3e-06  6.31  
10  1.1e-05  1.1e-05  1.5e-05  1.11e-03   -6.030221058e+00  -5.687989772e+00  2.8e-06  6.78  
11  1.1e-06  1.1e-06  1.5e-06  -1.11e-03  -5.864316378e+00  -5.527839477e+00  2.8e-07  7.38  
12  1.0e-06  1.0e-06  1.3e-06  8.16e-03   -5.878232834e+00  -5.539279596e+00  2.6e-07  7.92  
13  7.1e-07  7.1e-07  9.4e-07  3.29e-02   -5.897573909e+00  -5.554151509e+00  1.8e-07  8.39  
14  1.7e-07  1.7e-07  2.2e-07  2.31e-03   -5.803470463e+00  -5.465562394e+00  4.1e-08  8.94  
15  2.3e-08  2.1e-08  2.7e-08  -1.00e-02  -5.507099071e+00  -5.203979079e+00  5.4e-09  9.59  
16  9.7e-09  6.2e-09  7.7e-09  1.42e-03   -5.491524769e+00  -5.189377769e+00  1.6e-09  10.08 
17  2.7e-09  1.8e-09  2.2e-09  8.31e-05   -5.581575762e+00  -5.264661612e+00  4.4e-10  10.67 
18  1.5e-09  1.0e-09  1.3e-09  1.45e-02   -5.655570035e+00  -5.326042577e+00  2.5e-10  11.23 
19  3.9e-10  2.5e-10  3.3e-10  4.83e-03   -5.742627223e+00  -5.399025416e+00  6.2e-11  11.80 
20  1.4e-10  9.0e-11  1.2e-10  -4.60e-05  -5.870299550e+00  -5.505944250e+00  2.3e-11  12.44 
21  5.2e-11  3.4e-11  4.6e-11  4.10e-03   -5.931942981e+00  -5.557207480e+00  8.4e-12  13.00 
22  2.5e-11  1.1e-10  2.2e-11  1.51e-02   -5.960086751e+00  -5.578809713e+00  4.0e-12  13.58 
23  6.8e-12  5.7e-11  6.1e-12  1.75e-02   -5.994520129e+00  -5.606992266e+00  1.1e-12  14.16 
24  2.7e-12  9.4e-11  2.5e-12  1.01e-02   -6.048996917e+00  -5.652183105e+00  4.4e-13  14.72 
25  8.8e-13  1.8e-10  8.1e-13  1.31e-03   -6.138470296e+00  -5.728137202e+00  1.4e-13  15.30 
26  4.4e-13  1.8e-10  3.8e-13  7.40e-03   -6.176417884e+00  -5.759105288e+00  6.8e-14  15.95 
27  4.0e-13  3.4e-10  3.8e-13  6.50e-03   -6.177246542e+00  -5.759799610e+00  6.6e-14  16.64 
28  3.9e-13  4.4e-10  3.8e-13  6.41e-03   -6.177248060e+00  -5.759800873e+00  6.6e-14  17.36 
29  3.5e-13  5.9e-10  3.7e-13  6.28e-03   -6.177347475e+00  -5.759884143e+00  6.6e-14  17.95 
30  3.5e-13  1.0e-09  3.6e-13  5.53e-03   -6.177348327e+00  -5.759884832e+00  6.6e-14  18.58 
31  3.5e-13  1.0e-09  3.6e-13  5.53e-03   -6.177348327e+00  -5.759884832e+00  6.6e-14  19.22 
32  3.5e-13  1.0e-09  3.6e-13  5.53e-03   -6.177348327e+00  -5.759884832e+00  6.6e-14  19.84 
33  3.1e-13  9.4e-10  3.5e-13  1.55e-02   -6.177337680e+00  -5.759875205e+00  6.6e-14  20.42 
34  3.1e-13  9.4e-10  3.5e-13  1.55e-02   -6.177337680e+00  -5.759875205e+00  6.6e-14  21.03 
35  3.1e-13  9.4e-10  3.5e-13  1.55e-02   -6.177337680e+00  -5.759875205e+00  6.6e-14  21.64 
Optimizer terminated. Time: 22.30   


Interior-point solution summary
  Problem status  : ILL_POSED
  Solution status : DUAL_ILLPOSED_CER
  Primal.  obj: -2.8886390844e-06   nrm: 4e+00    Viol.  con: 9e-07    var: 0e+00    barvar: 0e+00    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 22.30   
    Interior-point          - iterations : 36        time: 22.25   
      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: Failed
Optimal value (cvx_optval): NaN
10

This one also was dual ill-posed on the dual, i.e., primal ill-posed, when I ran it. I will defer to the Mosek forum members to further comment.

11

Thank you very much!!!