Is it possible to reformuate my problem to get rid of the Invalid constraint with: {convex} <= {convex} error?

Nonconvex
1

Here is my code for the problem formulation

cvx_begin
variable f(J,1) complex

minimize( (w * Hm * f)’ * (w’ * Hm * f))
subject to
(1) a.’ * f == 1;
(2) f’ * f <= 1/(J/gamma);
(3a) for i = 1:K
(3b) (Hm(i , : ) * f)’ * (Hm(i , : ) * f) <= tau. * (f’ * f);
(3c) end
cvx_end

where \mathbf{H}_\mathrm{m} is known complex square matrix, \mathbf{w} and \mathbf{a} are known complex vectors, and tau, eps, gamma, J, K are all known real scalars. And the error invalid constraint {convex} <= {convex} will occur in constraint (3b).

Please let me first explain why I wrote the constraint (3), although I knew it would be an invalid constraint ({convex} <= {convex}). The actual meaning of constraint (3) is that, for the i th receiving antenna of all K receiving antennas, the received power from the transmit signal (unit power transmit signal) after the transmitting weight \mathbf{f} and the channel \mathbf{H}_\mathrm{m}(i, : ) (\mathbf{H}_\mathrm{m} is the channel matrix between transmitting and receiving antennas) is less a certain level, which is proportional to the transmitting power. The transmit power level is calculated by \mathbf{f}^{\prime}\mathbf{f} and the relative loss is tau.

Is that possible that I further formulate this constraint to get rid of the invalid constraint error and make it be accepted by CVX?

2

I don’t believe it would be possible for constraint (3b) to be both convex and for the problem to be feasible, because (3b) convex would require f = vector of zeros, which makes the problem infeasible due to constraint (1).

You need to move all quadratic terms of constraint (3b) to the LHS, and find the matrix A, such that the constraint is of the form f'*A'f <= 0, where A must be positive semidefinite. if that can 't be done, the constraint is not convex. If it can be done, f = vector of zeros would be the only feasible solution of that constraint, but that would violate constraint (1).

Perhaps the model, as is, is a good model for your purposes. But if so, it would be non-convex, and you would need a tool other than CVX, for example YALMIP.

3

Really thanks Mark! I am considering that whether it’s just a good model for my purpose but not gonna have a feasible solution. I will check the LHS formulation first! Thanks again!

4

If your model is always infeasible regardless of the data, that’s not much of a model.

My point is that your model is either non-convex or infeasible. A non-convex model can be a perfectly fine model, but it can’t be implemented in CVX .

5

Yes, I think you are right. Btw, could you enlighten me a little bit more about why the \mathbf{A} has to be PSD to make the inequality constraint convex?

6

LHS <= 0 convex requires LHS to be convex. For hermitian A, f'*A*f is convex iff A is psd.

7

Thanks! I might sound stupid, but to make LHS >=0 and LHS <=0 are convex both require A to be psd? Or did I misunderstand anything?

8

LHS <= 0 is convex constraint if LHS is convex.
LHS >= 0 is convex constraint if LHS is concave.

9

Really thanks Mark! But I have one last stupid question. If A is PSD, there might be a f \neq 0, which makes the quadratic form is 0, right? When A is a singular matrix I think? So maybe for some tau, the A matrix can have non-zero vector solution. Anyway, it’s just very special case maybe.

10

I believe you are correct.

11

Hi Mark, sry it’s me again xD. So I spent some time and now I have got some conclusions and some more questions here.
For all K constraints in (3), the LHS formulation can be written as

(3a) for i = 1:K
(3b) f’ * ( (Hm(i , : )’ * Hm(i , : ) + eps .* I - tau.* I) * f <= 0;
(3c) end
Please note that I added a scalar eps here, which is omitted in my former formulation for simplicity and \mathbf{I} denotes the identity matrix. I have checked all rank 1 Hermitian outcome of Hm(i, : )’ * Hm(i, : ) and they are PSD.

So I further formulate (3b) with eigen decomposition as

f’ * ( U * ( \Gamma + (eps - tau) .* \mathbf{I}) * U’ ) * f <= 0;

As you have taught me, this constraint will be convex iff ( U * ( \Gamma + (eps - tau) .* \mathbf{I}) * U’ ) is PSD, which means the eps has to be larger than tau, otherwise it is non-convex (eps actually saves me here to make the PSD possible). Then for my case, it’s not that common to have eps = tau. Therefore, normally the LHS will be convex for eps > tau (positive definite matrix here), and it will be exactly what you mentioned, which is that f = 0 vector is the solution here.

But maybe due to the exsitence of constraint (1) and (2), when I fed the problem to CVX with eps > tau, it actually solved the problem with non-zero solution. So does it mean CVX just finds a solution under this formulation, which tried its best to satisfy all the constraints here, including letting the LHS in (3) to be as close as to 0?

Meanwhile, I tried to get rid of the quadratic constraint in (3) and reformulate the whole problem as below

cvx_begin
variable f(J,1) complex

minimize( (w * Hm * f)’ * (w’ * Hm * f))
subject to
(1) a.’ * f == J ./ sqrt(gamma);
(2) abs( f ) <= 1 ;
(3a) for i = 1:K
(3b) f’ * ( (Hm(i , : )’ * Hm(i , : ) + eps .* I ) * f <= tau;
(3c) end
cvx_end

The major modification here is in constraint (1) and (2). These two constraints are designed to obtain the beamforing gain that not lower than a desired level, which is \frac{|\mathbf{aw}|^2}{\mathbf{w}^H\mathbf{w}} \geq J/gamma. Before, I fix \mathbf{aw}=1 and let \mathbf{w}^H\mathbf{w}=1 to achieve that gain requirement. Now, by constraint (2), I hope that each entry of \mathbf{f} won’t exceed unit magnitude, and along with the constaint (1), maybe I can constrain the \mathbf{f}^H\mathbf{f} to get closer to the value of (J^2 ./ gamma)? If that’s the case, then maybe I can get rid of the f’ * f in (3) to make LHS <= a scalar that is less than (J^2 ./ gamma) with level of tau. But I am still considering it.

12

Your paragraph

But maybe due to the exsitence of constraint (1) and (2), when I fed the problem to CVX with eps > tau, it actually solved the problem with non-zero solution. So does it mean CVX just finds a solution under this formulation, which tried its best to satisfy all the constraints here, including letting the LHS in (3) to be as close as to 0?

is confusing. Please show a complete reproducible problem, complete will all input data, and all solver and CVX output, as well as the optimal variable value (argmin).

13

Sure, but the Hm matrix is quite big, should I email it to you or just paste it here?

14

The Hm matrix will be split in two parts.

Hm = [Hm1;Hm2];
J=36;
K=36;
w=ones(36,1);
w = w./ norm (w);
as_ele_Tx = ones(36,1);
gamma=10^(1./10); % gamma tradeoff for gain
tau = (10^(-80./10)); % per-antenna residual SI power
eps = (10^(-50 ./ 10)) ; % estimation error

cvx_begin
variable f(J,1) complex

minimize( (w’ * Hm * f)’ * (w’ * Hm * f) + eps * (w’ * w) * (f’ * f) )

subject to

as_ele_Tx.'* f == 1;

f’*f <= 1/(J/gamma);

for i = 1:K

f’ * (Hm(i , : )’ * Hm(i , : ) + eps .* diag(ones(J,1)) - tau.* diag(ones(J,1)))* f <= 0;

end

cvx_end

the solution of f is : f= [0.0303824473837404 + 0.000135791359191675i
0.0303401946348155 + 0.000242656577132709i
0.0301963439166084 + 0.000426996230142492i
0.0301943835237665 + 0.000425920887151213i
0.0303372277021624 + 0.000242264201679140i
0.0303791087959261 + 0.000139930993293148i
0.0300153818336888 + 9.65047112712905e-05i
0.0301074203549018 - 0.000180532345718599i
0.0304842274286617 - 0.000423120965044536i
0.0304854451719929 - 0.000419270752199532i
0.0301120983002693 - 0.000181780042980406i
0.0300227457178182 + 8.95821390103649e-05i
0.0310106810202765 + 0.000329829582266032i
0.0307373430462984 + 0.000787303826581605i
0.0298691758668115 + 0.00101606686535783i
0.0298736523543546 + 0.00100715359941722i
0.0307324280058940 + 0.000790406131397751i
0.0309922388133855 + 0.000341483399698313i
0.0291478606687849 - 0.000569796687402456i
0.0292053798267866 - 0.00165019326964605i
0.0310892088079317 - 0.00105615044316890i
0.0310732690798923 - 0.00106820394178530i
0.0292043401897380 - 0.00165098274402403i
0.0291884525085643 - 0.000578524139070226i
0.0329691279861696 - 9.70720118825339e-05i
0.0313760179897504 + 0.00408315431611590i
0.0304633596993217 + 0.000830387786162535i
0.0304583352092505 + 0.000846174976714681i
0.0313658697955821 + 0.00412702202374999i
0.0329317955067508 - 0.000111536017811344i
0.0142399995914634 - 0.00212056298608348i
0.0145035032890048 + 0.000343407071876693i
0.0138482867364024 - 0.00219676611096253i
0.0138916947827682 - 0.00222008264119658i
0.0144660933163188 + 0.000379515224784629i
0.0143049941041704 - 0.00215697319717633i]

Here is the CVX output:

Calling SDPT3 4.0: 2855 variables, 111 equality constraints
For improved efficiency, SDPT3 is solving the dual problem.

num. of constraints = 111
dim. of sdp var = 4, num. of sdp blk = 1
dim. of socp var = 2812, num. of socp blk = 38
dim. of linear var = 37
dim. of free var = 2 *** 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|5.3e+01|1.3e+01|1.3e+04| 1.895554e+01 0.000000e+00| 0:0:00| chol 1 1
1|0.977|0.975|1.2e+00|4.2e-01|6.3e+02| 2.709536e+02 -1.564900e+01| 0:0:00| chol 1 1
2|0.988|0.996|1.4e-02|1.2e-02|2.7e+01| 2.072589e+01 -5.172325e-01| 0:0:00| chol 1 1
3|0.979|0.963|3.0e-04|1.7e-03|1.0e+00| 4.653641e-01 -1.773503e-02| 0:0:00| chol 1 1
4|0.937|0.669|1.9e-05|6.9e-04|2.4e-01| 3.339385e-02 -5.990972e-03| 0:0:01| chol 1 1
5|0.934|0.977|1.2e-06|3.1e-05|1.0e-02| 2.218612e-03 -1.356540e-04| 0:0:01| chol 1 1
6|0.914|0.975|1.0e-07|1.9e-06|6.1e-04| 9.060307e-05 -1.211056e-05| 0:0:01| chol 1 1
7|0.894|0.778|1.9e-08|5.1e-07|7.2e-05|-1.279028e-04 -1.068150e-05| 0:0:01| chol 1 2
8|0.295|0.147|5.8e-08|4.4e-07|1.3e-04|-7.542869e-04 -1.063566e-05| 0:0:01| chol 2 2
9|0.034|0.055|1.1e-07|4.1e-07|2.2e-04|-2.449515e-03 -1.063154e-05| 0:0:01| chol 1 2
10|0.016|0.009|1.2e-07|4.1e-07|7.8e-04|-1.079578e-02 -1.063059e-05| 0:0:01|
lack of progress in infeas

number of iterations = 10
primal objective value = -1.27902754e-04
dual objective value = -1.06815023e-05
gap := trace(XZ) = 7.15e-05
relative gap = 7.15e-05
actual relative gap = -1.17e-04
rel. primal infeas (scaled problem) = 1.90e-08
rel. dual " " " = 5.07e-07
rel. primal infeas (unscaled problem) = 0.00e+00
rel. dual " " " = 0.00e+00
norm(X), norm(y), norm(Z) = 7.6e+01, 3.1e+00, 4.5e+00
norm(A), norm(b), norm(C) = 5.5e+01, 2.0e+00, 7.3e+00
Total CPU time (secs) = 0.79
CPU time per iteration = 0.08
termination code = 0
DIMACS: 1.9e-08 0.0e+00 1.9e-06 0.0e+00 -1.2e-04 7.2e-05

Status: Solved
Optimal value (cvx_optval): +6.81502e-07

15

Here is Hm1:

Hm1 =[0.00364260800000000 + 0.00150379800000000i 0.00225635200000000 - 0.00107919200000000i 0.000282878700000000 - 0.00137477700000000i -0.000468563000000000 - 0.000511359800000000i -0.000484533200000000 + 0.000236321400000000i 8.19933400000000e-05 + 0.000136496900000000i -0.00717287500000000 - 0.00219247800000000i -0.00359895100000000 + 0.00205271400000000i -0.000265417100000000 + 0.00200448600000000i 0.000872365100000000 + 0.000791984000000000i 0.000665827800000000 - 0.000640995600000000i -0.000318967100000000 - 0.000385946700000000i 0.0141082400000000 + 0.00283914000000000i 0.00526788900000000 - 0.00339907900000000i 0.000739604200000000 - 0.00302681400000000i -0.00244806700000000 - 0.00122470300000000i -0.000248050200000000 + 0.00211988800000000i 0.00121491000000000 - 0.000264584100000000i -0.0259726800000000 - 0.00312138100000000i -0.00874467600000000 + 0.00356050800000000i 0.000173563600000000 + 0.00764228400000000i 0.00588517100000000 - 0.00269312300000000i -0.00360189300000000 - 0.00204636100000000i 0.000216860900000000 + 0.00201329300000000i 0.0460344500000000 + 0.0114869100000000i 0.0177188600000000 - 0.0104541900000000i -0.0140808600000000 - 0.00747839800000000i 0.00139051400000000 + 0.0104116200000000i 0.00310979200000000 - 0.00395207600000000i -0.00206735900000000 + 0.000406579300000000i -0.0923567800000000 - 0.0285803300000000i 0.00829458700000000 + 0.0127615200000000i 0.0137567700000000 - 0.00919952000000000i -0.00879294900000000 - 0.00407963700000000i 0.00231483200000000 + 0.00328226300000000i -0.000524192200000000 - 0.00189450700000000i
0.00196750500000000 - 0.00122703000000000i 0.000443850700000000 - 0.000106160500000000i 0.00101659600000000 + 0.000495489800000000i 0.00106078800000000 - 0.000393351600000000i -0.000171070700000000 - 0.00118374300000000i -0.000333794300000000 + 0.000206129300000000i -0.00289549400000000 + 0.00229140000000000i -0.000890466900000000 - 0.00100889400000000i -0.00259096300000000 - 0.00116274700000000i -0.00207892800000000 + 0.00149774100000000i 0.00117056800000000 + 0.00158450000000000i 0.000551496200000000 - 0.000441811000000000i 0.00377707700000000 - 0.00377415500000000i 0.00289919800000000 + 0.00495399500000000i 0.00657273200000000 + 0.00163654300000000i 0.00279384900000000 - 0.00425464300000000i -0.00328015800000000 - 0.00136281900000000i -0.000468123200000000 + 0.00178529100000000i -0.00540663700000000 + 0.00328739500000000i -0.0108042900000000 - 0.0139455200000000i -0.0147349100000000 - 0.000696758000000000i -2.79436900000000e-05 + 0.0106574000000000i 0.00647044400000000 - 0.00349046500000000i -0.00315443900000000 - 0.00227519200000000i 0.0133852600000000 - 0.00651255100000000i 0.0358952800000000 + 0.0360712400000000i 0.0265995800000000 - 0.0132940000000000i -0.0172008600000000 - 0.00915987800000000i 0.00190364900000000 + 0.0113389700000000i 0.00349223400000000 - 0.00360015500000000i 0.00788561700000000 + 0.00796576800000000i -0.0994357900000000 - 0.0462843300000000i 0.00555236900000000 + 0.0204992500000000i 0.0164968400000000 - 0.0107441200000000i -0.00970411800000000 - 0.00381031700000000i 0.00225073400000000 + 0.00355730300000000i
8.09910600000000e-05 - 0.00123590000000000i 0.00105392400000000 + 0.000475853300000000i 0.00152447100000000 + 0.000460218400000000i 0.00106437000000000 - 0.000400855800000000i 0.00108955700000000 - 0.000426450400000000i -0.000453469700000000 - 0.000324559300000000i 3.44129100000000e-05 + 0.00156289400000000i -0.00269591200000000 - 0.00111550000000000i -0.00298071300000000 - 0.000703778000000000i -0.00189252400000000 - 0.000214561400000000i -0.00204182900000000 + 0.00158293200000000i 0.000613987300000000 + 0.000520801300000000i 0.000367458500000000 - 0.00189128600000000i 0.00678314700000000 + 0.00147478700000000i 0.00495937100000000 + 0.00230493000000000i 0.00520970100000000 + 0.00165498200000000i 0.00262848500000000 - 0.00436098100000000i -0.00166742800000000 - 0.00114191400000000i -0.000765997300000000 + 0.00553021200000000i -0.0149036200000000 - 0.000286246500000000i -0.00986280600000000 - 0.00934975400000000i -0.0141118000000000 - 0.00218835900000000i 0.000163713600000000 + 0.0105604700000000i 0.00526053100000000 - 0.00171055000000000i -0.0111481200000000 - 0.00723174800000000i 0.0262982700000000 - 0.0135719700000000i 0.0313661900000000 + 0.0348314000000000i 0.0277879000000000 - 0.0123466000000000i -0.0171059700000000 - 0.00900544300000000i 0.000402681500000000 + 0.00979005900000000i 0.0133417000000000 - 0.00787425400000000i 0.00533262300000000 + 0.0205458000000000i -0.0973891900000000 - 0.0490334900000000i 0.00429192300000000 + 0.0210471600000000i 0.0165443000000000 - 0.0106015200000000i -0.00872039600000000 - 0.00471149200000000i
-0.000450353300000000 - 0.000327059500000000i 0.00109344300000000 - 0.000421452600000000i 0.00106886900000000 - 0.000390037200000000i 0.00152070200000000 + 0.000463319800000000i 0.00105320200000000 + 0.000481939400000000i 8.59408300000000e-05 - 0.00123509000000000i 0.000607225700000000 + 0.000520362000000000i -0.00205843000000000 + 0.00156534300000000i -0.00189941000000000 - 0.000231118100000000i -0.00296943800000000 - 0.000720289500000000i -0.00269744600000000 - 0.00111103300000000i 2.90406100000000e-05 + 0.00156069900000000i -0.00166027600000000 - 0.00114932000000000i 0.00266334000000000 - 0.00434921500000000i 0.00522270100000000 + 0.00171310300000000i 0.00493582900000000 + 0.00231960500000000i 0.00676322100000000 + 0.00149098400000000i 0.000388581600000000 - 0.00188701900000000i 0.00530627000000000 - 0.00166506400000000i 0.000124399100000000 + 0.0105876900000000i -0.0141041300000000 - 0.00220535700000000i -0.00981099800000000 - 0.00949336700000000i -0.0148262300000000 - 0.000380945400000000i -0.000747518900000000 + 0.00551039900000000i 0.000352306300000000 + 0.00978063800000000i -0.0169441700000000 - 0.00908249100000000i 0.0279177900000000 - 0.0122798000000000i 0.0311241100000000 + 0.0350119600000000i 0.0263449500000000 - 0.0135689000000000i -0.0112161300000000 - 0.00721055400000000i -0.00874386800000000 - 0.00465642900000000i 0.0165415200000000 - 0.0105206700000000i 0.00411247300000000 + 0.0209197600000000i -0.0972910000000000 - 0.0493487000000000i 0.00566943500000000 + 0.0204740400000000i 0.0133201800000000 - 0.00773214300000000i
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16

Here is Hm2, sorry it’s big

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17

I think the nominally non-zero optimal value of f is due to the terrible numerical scaling of the input data, which ends up in the model as small numbers multiplying other small numbers,which makes the LHS positive, but within solver tolerance of the RHS of 0.

You should change units or whatever so that the non-zero input data is within s small number of orders of magnitude of one. I don’t know how much this will help. Ultimately, you appear to have what may be a terrible model, whose only solution is the vector of zeros. So what is the point of the optimization problem? i don’t know anything about the application area, but unless the RHS of the modified version of (3b) is non-trivially greater than 0, it doesn’t seem like a good model.

18

Really thanks for your answer and patience. I get your point. Although the model seems reasonable for my application, it doesn’t seem to be a good model to be solved with bad scaling for the actual requirement. Btw, in your opinion, do you think the new constraint (1) (2) could help me with regulating value of \mathbf{f}^H\mathbf{f}? Or it just doesn’t seem to make sense maybe?

After a second thought, I think maybe it doesn’t make sense. Again, thank you!

19

A constraint
{convex quadratic with no linear or constant term} <= 0
doesn’t seem like a good constraint for an optimization model, for the reasons previously provided.

20

Yes, now I totally get it. Thanks again!