Thank you very much for your kind help. The advises are very illuminating to my work.

Now I used small number 1e-6 plus eye(3) to replace 0. And sometimes the violation still happen. And if I change the small number to 1e-4 then the problem disappeared. Does the threshold change with the problem and need to be adjusted case by case?

I have wrote a test program and code is listed below. At the end of the test program I used min(eig(sa)) to check whether sa=Y-x*x’ is semi definite positive. Here eig(sa) is the matlab function which return the eigenvalues of the matrix sa. The result is negative, so that sa is not a semi-definite positive matrix and the schur complement equivalent constraint is violated.

The input of the problem are generated randomly to perform Monte carlo simulation. In the test program, beta, z_hat ,and L are computed from some random generated scenario parameters. For some scenarios this violation appears while others not. The code just show a violation case. If the smallnumber parameter is large enough (1e-4) then the violation will never happen.

The result solution x and Y are used for a rounding procedure to produce some solution that is feasible with the constraint Y=x*x' and L=l*l'( rank one constraint which is not convex). In the rounding procedure I used Y-x*x' as a covariance matrix and use x as the mean. That means I generated some random vectors \xi_x(t)\sim N(x,Y-x*x') and used them to construct some feasible solutions to the problem with the nonconvex rank-one constraints. So that Y-x*x’ must be PSD.

May I trouble you for checking this test program and give me some advice ? The solver is SDPT3 4.0

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

```
clear;
clc;
M=4;
smallnumber=1e-6;
z_hat=[349.049118988982,273.191449032727,37.1430167833719,28.2125744474959;343.322423198038,121.879112283012,324.779715601743,112.393700815180];
L=[109.006172583561,90.4708901710594,115.813983123910,114.041111701073];
Xi=5;
beta=[199.620782097572;48.1241426764543;336.614462093126;293.790032116872];
cvx_begin sdp
variable t(1,1)
variable Y(2,2) semidefinite
variable x(2,1)
variable K(M,M) semidefinite
variable k(M,1)
minimize(t)
for i=1:M
trace(Y)-2*z_hat(:,i)'*x+z_hat(:,i)'*z_hat(:,i)+2*Xi*k(i)+Xi^2<=t*beta(i)^2;
end
for i=1:M
[ trace(Y)-2*z_hat(:,i)'*x+z_hat(:,i)'*z_hat(:,i)-2*Xi*k(i)+Xi^2,beta(i);beta(i),t]>=0;
end
for i=1:M
K(i,i)==[1;-z_hat(:,i)]'*[trace(Y),x';x,eye(2)]*[1;-z_hat(:,i)];
end
for i=1:M
for j=1:M
if i~=j
K(i,j)>=[1;-z_hat(:,i)]'*[trace(Y),x';x,eye(2)]*[1;-z_hat(:,j)];
end
end
end
[K,k;k',1]>=smallnumber*eye(M+1);
[Y,x;x',1]>=smallnumber*eye(3);
cvx_end
sa1=Y-x*x';
min(eig(sa1))
```

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The output of this test program is

## Calling SDPT3 4.0: 66 variables, 20 equality constraints

For improved efficiency, SDPT3 is solving the dual problem.

num. of constraints = 20

dim. of sdp var = 22, num. of sdp blk = 8

dim. of linear var = 16

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|1.2e+06|6.0e+00|2.1e+08|-8.375836e+06 0.000000e+00| 0:0:00| chol 1 1

1|0.001|0.419|1.2e+06|3.5e+00|8.8e+07|-8.347404e+06 -5.022124e+00| 0:0:00| chol 1 1

2|0.394|0.634|7.3e+05|1.3e+00|5.2e+07|-2.506617e+06 -7.763444e+00| 0:0:00| chol 1 1

3|0.766|0.434|1.7e+05|7.2e-01|3.0e+07| 3.013552e+06 -9.487711e+00| 0:0:00| chol 1 1

4|0.795|0.919|3.5e+04|5.8e-02|5.7e+06| 2.730554e+06 -1.438932e+01| 0:0:00| chol 1 1

5|0.739|0.537|9.1e+03|2.7e-02|2.3e+06| 1.331237e+06 -2.048347e+01| 0:0:00| chol 1 1

6|0.759|0.334|2.2e+03|1.8e-02|7.3e+05| 3.823449e+05 -2.477292e+01| 0:0:00| chol 1 1

7|0.621|0.614|8.4e+02|6.9e-03|3.2e+05| 1.893795e+05 -3.795200e+01| 0:0:00| chol 1 1

8|0.839|0.798|1.3e+02|1.4e-03|7.0e+04| 4.627635e+04 -5.433469e+01| 0:0:00| chol 1 1

9|0.740|0.331|3.5e+01|9.4e-04|2.4e+04| 1.720520e+04 -6.818425e+01| 0:0:00| chol 1 1

10|0.975|0.952|8.7e-01|4.5e-05|7.4e+02| 5.142954e+02 -7.290390e+01| 0:0:00| chol 1 2

11|0.970|0.978|2.6e-02|1.0e-06|5.1e+01| 1.765441e+01 -3.142451e+01| 0:0:00| chol 2 2

12|1.000|0.094|6.6e-07|5.3e-06|3.6e+01| 3.435643e+00 -2.870878e+01| 0:0:00| chol 2 2

13|1.000|0.726|6.7e-07|1.6e-06|1.4e+01| 2.934840e+00 -9.748479e+00| 0:0:00| chol 2 2

14|0.899|0.827|2.3e-07|4.1e-07|2.4e+00|-1.318272e-01 -2.317013e+00| 0:0:00| chol 2 2

15|1.000|0.110|9.5e-09|4.1e-07|2.0e+00|-3.670402e-01 -2.197370e+00| 0:0:00| chol 2 3

16|1.000|0.499|4.3e-06|2.1e-07|9.8e-01|-7.351979e-01 -1.648034e+00| 0:0:00| chol 2 2

17|1.000|0.442|1.8e-06|1.2e-07|5.3e-01|-8.657176e-01 -1.361772e+00| 0:0:00| chol 2 2

18|1.000|0.595|4.8e-07|5.2e-08|2.0e-01|-9.396068e-01 -1.127610e+00| 0:0:00| chol 2 2

19|1.000|0.726|6.6e-08|2.1e-08|5.3e-02|-9.594094e-01 -1.007562e+00| 0:0:00| chol 2 2

20|1.000|0.267|1.1e-08|2.5e-08|4.1e-02|-9.642802e-01 -9.969930e-01| 0:0:00| chol 2 2

21|1.000|0.618|9.0e-09|1.2e-08|1.6e-02|-9.683979e-01 -9.802903e-01| 0:0:00| chol 2 2

22|1.000|0.777|6.2e-09|3.5e-09|3.6e-03|-9.704330e-01 -9.727611e-01| 0:0:00| chol 2 2

23|0.986|0.865|1.8e-09|5.9e-10|4.8e-04|-9.707177e-01 -9.709803e-01| 0:0:00|# chol 2 2

24|0.978|0.924|5.7e-08|5.4e-11|2.9e-05|-9.786030e-01 -9.707748e-01| 0:0:00|# chol 2 2

stop: progress in duality gap has deteriorated, 3.1e-05

25|0.009|0.008|5.7e-08|5.4e-11|2.9e-05|-9.786030e-01 -9.707748e-01| 0:0:00|

## number of iterations = 25

primal objective value = -9.78602968e-01

dual objective value = -9.70774785e-01

gap := trace(XZ) = 2.92e-05

relative gap = 9.90e-06

actual relative gap = -2.65e-03

rel. primal infeas (scaled problem) = 5.74e-08

rel. dual " " " = 5.39e-11

rel. primal infeas (unscaled problem) = 0.00e+00

rel. dual " " " = 0.00e+00

norm(X), norm(y), norm(Z) = 4.0e+00, 2.3e+05, 4.3e+05

norm(A), norm(b), norm© = 1.5e+05, 2.0e+00, 6.2e+05

Total CPU time (secs) = 0.17

CPU time per iteration = 0.01

termination code = -8

DIMACS: 5.7e-08 0.0e+00 1.4e-10 0.0e+00 -2.7e-03 9.9e-06

Status: Inaccurate/Solved

Optimal value (cvx_optval): +0.970775

ans =

-0.1969