Concept guide
Optimization problem: feasibility first, then objective values, over nine integer candidates
The candidate set C={0,1,2}² has nine points, the constraint is x+2y≥3 and the objective is f(x,y)=x+y. Minimize with a scan algorithm that first decides feasibility and then compares values one by one, and prove the result with an invariant: optimal value m=2, argmin={(0,2),(1,1)} (a tie of two); explain why (0,0) and (2,0) are excluded although their objective values do not exceed 2; find the maximum; and show that under the same constraint over the real nonnegative quadrant, the minimum is 3/2, attained at (0,3/2), which is not in the declared finite candidate set.