Model Predictive Control

50 4 Linear and Quadratic Optimization
x2
2
c = 1
optimal primal solutions
max
(a) Case 2 - Primal LP
1
x1
y2
dual degenerate
′
1
1
min
c = 1
(b) Case 2 - Dual LP
Figure 4.4 LP with Primal and Dual Degeneracy
y = 1
active constraints, exactly as many as the number of decision variables. The
same holds true in the dual space, where the optimal dual variable is (0, 1).
Case 2. The situation changes when c = 1, as portrayed in Figure 4.4. Consider
the two solutions for the primal LP denoted with 1 and 2 in Figure 4.4(a) and
referred to as “basic” solutions. Basic solution 1 is primal non-degenerate, since
it is defined by exactly as many active constraints as there are variables. Basic
solution 2 is primal degenerate, since it is defined by three active constraints,
i.e., more than two. Any convex combination of optimal solutions 1 and 2
is also optimal. This continuum of optimal solutions in the primal problem
corresponds to a degenerate solution in the dual space, hence the primal problem
is dual-degenerate. That is, the dual problem is primal-degenerate. Both basic
solutions correspond to a degenerate solution point in the dual space, as seen
on Figure 4.4(b). In conclusion, Figures 4.4(a) and 4.4(b) show an example
of a primal problem with multiple optima and the corresponding dual problem
being primal degenerate.
Case 3. We want to show that the statement “if the dual problem is primal degenerate
then the primal problem has multiple optima” is, in general, not true.
Consider case 2 and switch dual and primal problems, i.e.,call the “dual problem”
primal problem and the “primal problem” dual problem (this can be done
since the dual of the dual problem is the primal problem). Then, we have a
dual problem which is primal degenerate in solution 2 while the primal problem
does not present multiple optima.
4.1.5 Convex Piecewise Linear Optimization
Consider a continuous and convex piecewise affine function J : R ⊆ R s → R:
J(z) = c ′ iz + di for [ z ] ∈ Ri, (4.10)
y1