Thank you for reporting this. We are now aware of this behavior and we are working on it. It can be worked around by perturbing the constraint
In[]:=
LinearOptimization[x+y,{x+y>=1/2,x>=0,y>=10^-30.},{x∈Integers,y}]
Out[]=
{x0,y0.5}
Also specifying alternative methods works fine
In[]:=
LinearOptimization[x+y,{x+y>=1/2,x>=0,y>=0},{x∈Integers,y},Method->"ConvexMixedIntegerBranchAndBound"]
Out[]=
{x0,y0.5}
In[]:=
LinearOptimization[x+y,{x+y>=1/2,x>=0,y>=0},{x∈Integers,y},Method->"Xpress"]
Out[]=
{x0,y0.5}