Model Predictive Control

58 4 Linear and Quadratic Optimization
Case 1. The MIQP solution is unbounded, i.e., J ∗ = −∞. This cannot happen if
H ≻ 0.
Case 2. The MIQP solution is bounded, i.e., J ∗ > −∞ and the optimizer is
unique. Z ∗ is a singleton.
Case 3. The MIQP solution is bounded and there are infinitely many optima
corresponding to the same integer value. Z∗ is the Cartesian product of an
infinite dimensional subset of Rs and an integer number z∗ b . This cannot
happen if H ≻ 0.
Case 4. The MIQP solution is bounded and there are finitely many optima corresponding
to different integer values. Z ∗ is a finite set of optimizers {(z ∗ 1,c , z∗ 1,b ), . . . , (z∗ N,c , z∗ N,b )}.
Case 5. The union of Case 3 and Case 4.