8. Computational Issues
8.2 On the S-procedure in Piecewise Quadratic Analysis
Chapter 8. Computational Issues PROPOSITION8.4—SIMPLEXBOUNDING
Let Xibe a simplex with non-empty interior, and let ¯Gibe the correspond-ing cell identification matrix, as computed in(8.5). Then, the ellipsoid of minimum volume that contains Xi is given by
¯xTG¯iTG¯i¯x≤1.
Proof: See Appendix A.
PROPOSITION8.5—PARALLELEPIPED BOUNDING
Let Xi⊂Rnbe a parallelepiped with non-empty interior, Xi {x∈Rnt tciTx− exit ≤di, i1, . . . ,n} and let
T¯
c1T/d1 −ex1/d1
... ... cnT/dn −exn/dn
Then, the ellipsoid of minimal volume that contains Xi is given by
¯xTT¯TT ¯x¯ ≤n
Proof: See Appendix A.
8.2 On the S-procedure in Piecewise Quadratic Analysis be a quadratic set. Then the quadratic form
V(x) xTPix (8.6) is positive for all x∈Eiwith x60 if and only if there exists a non-negative scalar ui≥0 such that the following LMI condition is satisfied
Pi−uiSi>0.
Clearly, if we consider a set given as the intersection of several quad-ratic sets,E ∩mi1Ei and there exists ui≥0 such that
Pi− Xm
i1
uiSi>0 (8.7)
then V(x)defined in(8.6)is positive for all x∈E. However, as was shown by a simple example in [153]this condition is only sufficient. In other words, there are quadratic functions V(x)that are positive on sets on the form Ewith m>1, but where no solution to the LMI condition(8.7)can be found.
Let eik denote the columns of ¯Ei. Then, the polyhedral S-procedure-relaxation can be seen as a special case of(8.7)via
Pi−E¯iTUiE¯iPi−X
j,k
ujkeijeTik>0
This indicates that the S-procedure is only a sufficient condition for veri-fying positivity of a piecewise quadratic function on a polyhedral domain.
It is then easy to come to the premature conclusion that it is more conservative to use polytopic relaxations than to fix a quadratic set ap-proximation and use this in the LMI computations. In this section, we will show how this is not the case. For some important classes of par-titions we will be able to prove that the polytopic relaxation is always stronger than quadratic relaxations. Moreover, based on separation re-sults for so-called copositive matrices, we will prove that the polytopic S-procedure-relaxation is non-conservative for simplex partitions in Rn with n≤3.
Polyhedral Relaxation is Stronger than Ellipsoidal
By a comparative example in Chapter 4, we illustrated how the use of ellipsoidal cell description in the S-procedure allows significant computa-tional savings in the analysis computations compared to the use of poly-topic relaxations. However, as the same example indicated, these savings
Chapter 8. Computational Issues
come at the price of increased conservatism in the analysis. The develop-ments in Section 8.1 will now allow us to be more precise in this issue.
More specifically, we will show that if the piecewise quadratic computa-tions with minimum volume ellipsoids as cell boundings have a solution, then so have the computations in Theorem 4.1, while the opposite is not always true. This is contrary to a statement in[44](Section 4.1)where it was indicated that computations using ellipsoidal cell boundings would be less conservative than those using polyhedral relaxations, since the S-procedure may be lossy when several quadratic terms are used.
PROPOSITION8.6—POLYHEDRALRELAXATION ISSTRONGER THANELLIPSOIDAL
Let Xi be a simplex cell. Let ¯Ei be the associated cell bounding satisfy-ing(4.18), and let ¯Si describe the minimal volume ellipsoidal boundings computed as in Proposition 8.4. Then, the polytopic S-procedure relax-ation ¯EiTUiE¯i is stronger than the S-procedure using minimum volume ellipsoids, uiS¯i. More precisely, if the LMI
P¯i−τiS¯i>0 (8.8) has a solution, then so has the LMI
P¯i−E¯iTUiE¯i >0, (8.9) but there are cases when(8.9)admits a solution while(8.8)does not.
Proof: See Appendix A.
A similar result can be established also for hyper-rectangular cells.
To understand the use of the S-procedure in the piecewise quadratic analysis, it is fruitful to consider the problem of verifying the constraint
¯xTP¯i¯x>0 x∈Xi
using LMI computations. In this case, the role of the S-procedure is to separate the set Vi− {xt¯xTP¯i¯x<0}from the set Xi. The volume of the covering ellipsoid may have very little to do with this separation. This is illustrated in Figure 8.4. The minimum volume ellipsoid of Xi intersects the set V−, hence it cannot be used to verify the desired inequality. By using the polyhedral relaxation, there is a lot of freedom in optimizing over the quadratic bounding, and separation can easily be accomplished, see Figure 8.4(right).
Another point is that although the S-procedure is only sufficient when there is more than one quadratic constraint, adding new constraints can never make the inequalities harder to satisfy since the associated multi-pliers can always be set to zero. On the contrary, adding new terms may allow separations that would otherwise not be possible.
8.2 On the S-procedure in Piecewise Quadratic Analysis
0 1 2 3
0 1 2 3
Vi−
Xi
0 1 2 3
0 1 2 3
Vi−
Xi
Figure 8.4 The counter example in Proposition 8.6. The minimal volume ellipsoid fails to separate Xi from Vi− (left), while optimizing over the covering ellipsoids using the polyhedral formulation easily finds a separating supset.
Copositivity and Non-Conservatism of the S-procedure
There is a lot of structure in the way computations are made for simplex partitions. In this section, we will use this structure further to prove that the polytopic relaxation is both necessary and sufficient for the piecewise quadratic computations on simplices inRnwith n1,2,3.
In what follows, we let Xibe a simplex inRnwith associated continuity matrix ¯Fi and cell bounding ¯Ei computed as in Section 8.1. Consider verification of the inequality
¯xTF¯iTT ¯Fi¯x≥0 x∈Xi. (8.10) Recall that in the simplex case, the matrices ¯Fi and ¯Ei are used to map the state vector x into partition coordinates z. It can be verified that
zE¯i¯x x∈ Xi
z
E
iTF¯i¯x x∈ Xiwhere
E
i is the vertex extraction matrix for Xi. Moreover, ¯FiE
iE¯i andXi (
z :E¯ixtz0, Xn+1
i1
zi1 )
Hence, verification of the inequality(8.10), is equivalent to verification of
zTTiz≥0 z0. (8.11)
with Ti
E
iTTE
i. The constraint Pizi 1 can be disregarded due to homogeneity. Problems of the type (8.11)occur, for example, in the so-lution to some non-standard LQG problems[52]and has attracted some
Chapter 8. Computational Issues
attention in the linear algebra literature. Matrices that satisfy(8.11)are called copositive matrices. If the inequality in (8.11)is strict for z 6 0, Tiis called a strictly copositive matrix. For some time it was conjectured that if Ti is copositive, then it can be written as the sum of two matrices
TiPi+Ui (8.12)
where Pi is positive semidefinite and Ui 0, i.e., Ui has non-negative entries. The following result was proved in[36], see also[52].
PROPOSITION8.7—DECOMPOSITION OFCOPOSITIVEMATRICES [36]
For dimensions n≤4, every copositive matrix Ti can be decomposed in the form (8.12). However, indecomposable copositive matrices exist for n≥5.
Proposition 8.7 implies that for n≤4, the inequality(8.11)holds if and only if there exists a matrix Ui 0 such that
Ti−Ui≥0. (8.13)
Since ¯Ei is invertible, this is equivalent to E¯iTTiE¯i−E¯iTUiE¯i>0
and hence
F¯iTT ¯Fi−E¯iTUiE¯i≥0.
This is a non-strict version of the LMI condition used in Theorem 4.1.
Moreover, the matrix Ui used in this decomposition has non-negative en-tries(uij0 is allowed).
As this cannot be handled in a solver which only treats strict inequal-ities, we will extend the result to treat strict inequalities and allow the entries of Ui to be(strictly)positive.
PROPOSITION8.8—NON-CONSERVATISM OF THES-PROCEDURE
Let{Xi}i∈I be a simplex partition inRnwith n≤3, and with constraint matrices ¯Ei and ¯Fi computed as in Section 8.1. Then
V(x) ¯xTF¯iTT ¯Fi¯x>0 for x∈Xi\{0},i∈I
if and only if there exists a matrix Ui with positive entries such that F¯iTT ¯Fi−E¯iTUiE¯i>0.