(5.57) donde es la función de que se muestra en la Figura 5.12 El valor de

To set the stage for the contributions of this chapter, Sum of Squares (SOS) program- ming, originally proposed in the thesis of Parrilo [87] is reviewed. These tools will be key in the development of approximate solutions to the Hamilton Jacobi Bellman equation (1.27), which specifies a set of partial differential equality constraints that the optimal solution must satisfy. Instead of satisfying these constraints exactly, as in Galerkin or collocation techniques, the equality constraints are relaxed directly. The optimization problem is then to find the best approximate solution that lies in the set of polynomials that satisfy these inequality constraints. This is done by reducing these inequalities to a semialgebraic set, allowing for the tools of algebraic geome- try to be employed. Specifically, SOS programming provides a method to perform optimization over such a set.

Definition 6. Let x = (x1, . . . , xn), x ∈ Rn and α = (α1, . . . , αn), α ∈ Nn. The

function zα = xα11x

α2

2 . . . xαnn is a monomial in (x1, . . . , xn) of degree |α| = Pn

i=1αi.

monomials p(x) = X α cαxα = X α cαxα11x α2 2 . . . x αn n , cα ∈R. (2.1)

For brevity of notation, define the ring of polynomials in (x1, . . . , xn) with real

coefficients as _R[x] _{, R}[x1, . . . , xn]. A semialgebraic set is a subset of Rn that is

specified by a finite number of polynomial equations and inequalities.

S ={x∈_Rn _|p i(x)≥0, i= 1, . . . , n} An example is S =(x1, x2)∈R2 |x21+x 2 2 ≤1, x 3 1−x2 ≤0 .

Such a set is not necessarily convex, and testing membership in the set is NP-Hard in general [88, 87]. However, there exists a class of semialgebraic sets that are in fact semidefinite-representable. Key to this development will be the ability to test for non-negativity of a polynomial.

A multivariate polynomialp(x)is asum of squares(SOS) if there exist polynomials p1(x), . . . , pm(x) such that p(x) = m X i=1 p2_i(x).

A seemingly unremarkable observation is that a sum of squares is always positive. Thus, a sufficient condition for non-negativity of a polynomial is that the polynomial is SOS. Perhaps less obvious is that membership in the set of SOS polynomials may be tested as a convex problem and therefore polynomial time-solvable. Denote the functionp(x)being SOS asp(x)∈Σ(x), whereΣ(x)is the set of all SOS polynomials. The key to this reduction in complexity is the following result.

Theorem 7. ([89]) A polynomial p(x) of degree 2d is a sum of squares if and only if there exists a positive semidefinite matrix Q and a vector of monomialsZ(x) con-

taining monomials in x of degree less than or equal to d such that

p=Z(x)TQZ(x). (2.2)

The Q matrix in (2.2) is referred to as the Gram Matrix. The monomials of Z(x)

are not in general algebraicaly independent, meaning that if the equation is expanded and coefficients are matched, there will be free parameters in Q. The result is that optimizations may take place over Q with the constraint Q 0, i.e. Q is positive semidefinite, placing the problem of SOS non-negativity in the realm of semidefinite programming.

Theorem 8. ([87]) Given a finite set of polynomials {pi}m_i=0 ∈ Rn the existence of

ai ∈R for i= 1, . . . , m such that

p0+

m X

i=1

aipi ∈Σ(x)

is a semidefinite programming feasibility problem.

Thus, while the problem of testing non-negativity of a polynomial is intractable in general, by constraining the feasible set to a SOS, the problem becomes tractable. The converse question, is a non-negative polynomial necessarily a sum of squares is, unfortunately, false. This indicates that this test is conservative [87]. Nonetheless, SOS feasibility will be sufficiently powerful for the purposes of this work. Details of how SOS feasibility are reducible to semidefinite programs are given in [36], and have become well known in the control community.

2.1.1.1 The Positivstellensatz

Using SOS theory, it is possible to determine whether a particular polynomial, possi- bly parameterized, is a sum of squares. The next step is to determine how to combine multiple polynomial inequalities, i.e. semialgebraic sets of the form

P ={x∈_Rn _|p

for polynomial functions pi(x) where x ∈ Rn. The answer is given by Stengle’s

Positivstellensatz.

Theorem 9. (Positivstellensatz [90]) The set

X ={x|pi(x)≥0, hj(x) = 0 for all i= 1, . . . m, j = 1, . . . p} (2.3)

is empty if and only if there exists ti ∈R[x], si, rij, . . . ∈Σ(x) such that

−1 =s0+ X i hiti+ X i sipi+ X i6=j rijpipj+· · · (2.4)

Although this theorem is presented in terms of feasibility, it is easily adapted for the purposes of optimization. Given the problem

min p0(x)

s.t. pi(x)≤0 ∀i∈1, . . . , k

a slack factor γ may be introduced to frame the equivalent infeasibility problem

max γ s.t. p0(x)≤γ pi(x)≤0 ∀i∈1, . . . , k    infeasible

which is in a form directly applicable to the Positivestellensatz.

By setting some of the Positivestellensatz multipliers, such as rij or rijk, to zero,

a sufficient condition for infeasibility may be created. Alternatively, it is possible to limit the degree of the multipliershi, si, rij. The search for infeasibility may therefore

begin with a limited polynomial degree, increasing the degree if additional precision is required. This creates ahierarchy of semidefinite relaxations of increasing complexity but also with a decrease in the suboptimality of the solution. This construction is known more broadly as a Lasserre Hierarchy [46], or Theta Body relaxation [91].