x is binary, 0=< y <=K, Z is their product. The transformation is z≥0, z≤y, z≤Kx, y−z≤K(1−x)

Is the transformation of the multiplication of two variables correct? I have tried the method ,but it seems that it is not right. The optimization problem becomes unbounded.

if x equals 0, then

```
z>=0 z<=y z<=0 y-z <= K
-> Z = 0, 0<= y <= K
```

if x equals 1, then

```
z>=0 z<=y z<=K y-z <=0
-> z = y
```

It is correct, but when adding the constraints to the problem it gets unbounded. In the following, T = X * S.

```
K = 10;
cvx_begin
variable X(3);
variable Y(3);
variable Z(3);
variable T(3);
variable S(3) binary;
minimize(sum(Z));
subject to
{T,1000*Y,Z} == exponential(3);
X>=0;
X<=K;
Y>=1e-10;
Y<=1;
sum(Y) <= 1;
% transformation
T >= 0; T <= X;
T <= K.*S;
X - T <= K.*(1-S);
sum(T) >= 5;
sum(S) <= 2;
cvx_end
```

Using Mosek to solve this problem, it gives the result of unbounded.