Model Predictive Control

46 4 Linear and Quadratic Optimization
c ′ z = k4
(a) Linear Program -
Case 1
P
c ′ z = k1
z ∗
c ′ z = k1
c ′ c
z = k4
′ z = k3
P
(b) Linear Program -
Case 2
c ′ z = k4
z ∗
c ′ z = k1
P
(c) Linear Program -
Case 3
Figure 4.1 Graphical Interpretation of the Linear Program Solution, ki <
ki−1
not empty. Denote by J ∗ the optimal value and by Z ∗ the set of optimizers of
problem (4.1)
Z ∗ = argmin z∈P c ′ z
Three cases can occur.
Case 1. The LP solution is unbounded, i.e., J ∗ = −∞.
Case 2. The LP solution is bounded, i.e., J ∗ > −∞ and the optimizer is unique.
Z ∗ is a singleton.
Case 3. The LP solution is bounded and there are multiple optima. Z ∗ is an
uncountable subset of R s which can be bounded or unbounded.
The two dimensional geometric interpretation of the three cases discussed above
is depicted in Figure 4.1. The level curves of the cost function c ′ z are represented
by the parallel lines. All points z belonging both to the line c ′ z = ki and to
the polyhedron P are feasible points with an associated cost ki, with ki < ki−1.
Solving (4.1) amounts to finding a feasible z which belongs to the level curve with
the smallest cost ki. Since the gradient of the cost is c ′ , the direction of steepest
descent is −c ′ .
Case 1 is depicted in Figure 4.1(a). The feasible set P is unbounded. One can
move in the direction of steepest descent −c and be always feasible, thus decreasing
the cost to −∞. Case 2 is depicted in Figure 4.1(b). The optimizer is unique and
it coincides with one of the vertices of the feasible polyhedron. Case 3 is depicted
in Figure 4.1(c). The whole bold facet of the feasible polyhedron P is optimal,
i.e., the cost for any point z belonging to the facet equals the optimal value J ∗ .
In general, the optimal facet will be a facet of the polyhedron P parallel to the
hyperplane c ′ z = 0.
From the analysis above we can conclude that the optimizers of any bounded
LP always lie on the boundary of the feasible polyhedron P.