Properties and Extensions

In this section we investigate the applicability and numerical stability of the presented HPSC factorization. Pivoting of the applied factorizations as well as iterative refinement of the ob- tained solution are mentioned. A dynamic programming interpretation of the HPSC factoriza- tion as given in [201] is stated.

7.4.1 Applicability

The following theorem shows that, given a KKT system with direct multiple shooting block structure, the HPSC factorization is as widely applicable as the popular null space method for solving the KKT system of a dense QP.

Theorem 7.1 (Applicability of the HPSC Factorization)

The HPSC factorization is applicable to a KKT system with 1. direct multiple shooting block structure,

2. linear independent active constraints (LICQ, definition 3.4),

3. positive definite Hessian on the null space of the active set. ₄

Proof Assumption (2.) implies regularity of the RAF

i and thus existence of the TQ decom- positions. Assumption (3.) guarantees the existence of the CHOLESKY decompositions of the projected Hessian blocks ˜H_{i = Z}T

i HiF FZi. It remains to be shown that the block tridiagonal system (7.34) which we denote by K is positive definite. To this end, observe that we have for (7.34) the representation K =         A₀ BT 1 B₁ A₁ BT 2 B₂ A₂ ... ... ...         (7.55) =         G₀ P₁ G₁ P₂ G2 ... ...                 ˜ H₀ ˜ H₁ ˜ H₂ ...                 GT 0 PT 1 G1T PT 2 G2T ... ...         def = M ˜H MT.

By assumption (3.) we have positive definiteness of all diagonal blocks ˜Hi of ˜H and hence of ˜

H itself, i.e., it holds that

∀w 6= 0 : wTH w > 0.˜ (7.56)

By assumption (2.) the matrix M of equality matching conditions has full row rank and for all v 6= 0 it holds that w = MT_{v 6= 0. Hence}

∀v 6= 0 : (MTv)TH(M˜ Tv) = vT(M ˜H MT)v = vTK v > 0 (7.57) which is the condition for positive definiteness of the system K. This completes the proof. _ƒ

7.4.2 Uniqueness

We investigate the uniqueness of the HPSC factorization.

Theorem 7.2 (Uniqueness of the HPSC Factorization)

The HPSC factorization is unique up to the choice of the signs of the reverse diagonal entries of the southeast triangular factors T_i, and up to the choice of the orthonormal null space column

basis vectors Z_i. ₄

Proof The employed CHOLESKYfactorizations are unique. Thus the uniqueness properties of the initial block QR factorizations carry over to the HPSC factorization. The thin TQ factoriza- tions RAF

i T_Y

i = Ti are unique up to the signs of the reverse diagonal elements of the Ti and the choice of Zi is free subject to orthonormality of Qi = [Zi Yi]. For proofs of the uniqueness

properties of CHOLESKYand QR factorizations we refer to e.g. [89]. _ƒ

7.4.3 Stability

In this section, we address the stability of the HPSC factorization. We are interested in the propagation of roundoff errors in the gradient g and right hand side (b, r, h) through the backsolve with a HPSC factorization to the primal–dual step (v, λ, µ, ν).

Like condensing methods and RICCATI iterations, the HPSC factorization fixes parts of the pivoting sequence, which may possibly lead to stability problems for ill–conditioned KKT sys- tems. The Mathematical Program with Vanishing Constraints (MPVC) Lagrangian formalism introduced in chapter 6 has been introduced specifically to eliminate the major source of ill– conditioning in the targeted class of problems, QPs resulting from MIOCPs treated by outer convexification. Furthermore, all employed factorizations of the matrix blocks are stable un- der the assumptions of theorem 7.1. The use of a SCHURcomplement step in section 7.3.3 still mandates caution for problems with ill–conditioned matching condition Jacobians. This may potentially be the case for processes with highly nonlinear dynamics on different time scales. For the numerical results presented in chapter 9, no problems were observed after use of the MPVC Lagrangian formalism. In the following we briefly mention iterative refinement and opportunities for pivoting of the involved factorizations to improve the backsolve’s accuracy, should the need arise.

Pivoting

Pivoting algorithms can be incorporated into the HPSC factorization in several places to help with the issue of ill–conditioning. Possible extensions include a pivoted QR decomposition of the point constraints,

ΠR_iRAF i T_Q i = h 0 Ti i . (7.58)

and a symmetrically pivoted block cholesky decomposition of the null space Hessian

ΠH_i TH˜iΠHi = UiTUi. (7.59)

We refer to [89] and the references found therein for details. The most promising option prob- ably is symmetric block pivoting of the block tridiagonal CHOLESKY decomposition of system (7.34). This last option requires cheap condition estimates of the diagonal blocks A_i and can be shown to produce at most one additional off-diagonal block in system (7.34).

Iterative Refinement

A different possibility to diminish the error in the KKT system’s solution found using the backsolve algorithm 7.2 is to apply iterative refinement, cf. [89]. This allows to increase the number of significant digits of the primal–dual step from n to N · n at the expense of N − 1 additional backsolves with the residuals. The procedure is given in algorithm 7.3. Iterative refinement has been included in our implementationqpHPSC, see appendix B.

Algorithm 7.3: Iterative refinement of a backsolve with the HPSC factorization. input : HPSC factorization H = (T, Y, Z, U, V, D),

KKT system blocks K = (H, G, P, R), KKT right hand side k = (g, b, r, h), N output: KKT solution v, λ, µ, ν. [v, λ, µ, ν] = 0; δk = k; for i = 1 : N do [v, λ, µ, ν] += hpsc_backsolve(H, K, δk); δk = kkt_multiply(H, [v, λ, µ, ν]) − δk; end

7.4.4 A Dynamic Programming Interpretation

In [201] a dynamic programming interpretation of system (7.1) is given as shown in figure 7.1. The KKT factorization determines the unknowns (xi, ui) on the range spaces of the point constraints defined by R_i, e_i. The null space part remains free and are defined as the result of optimization problems on the manifolds

Ni(x )

def

u₀ u₁

x₀ G₀, P₁, h₀ G₁, P₂, h₁ . . .

R₀, e0 R1, e1 Rm, em

x₁ x₂ x_m

Figure 7.1: Dynamic programming interpretation of the HPSC factorization.

We further define the manifolds of feasible states under the mapping defined by Gi, Pi+1, hi,

Si(xi+1, ui+1)

def

=¦ x ∈ Rnx | ∃u ∈ N (x) : Gixx + Guiu + Pi+1x xi+1+ Pi+1u ui+1= hi© , and the manifold of feasible controls for a given feasible initial state x_i alike,

Ui(xi, xi+1, ui+1)

def

=¦u ∈ N (xi) | Gxixi+ Guiu + Pi+1x xi+1+ Pui+1ui+1= hi© .

For a given state x_m−1, the optimal control u_m−1, steering the process to the terminal state xm∈ Sm is now chosen according to BELLMAN’s principle as minimizer of the objective ϕ

u_{m−1(xm−1) = argmin}

u ϕm−1(u, xm−1, xm) | u ∈ Um−1(xm−1, xm)

_{. (7.60)}

As can be seen, this control can be determined locally, i.e., without consideration of the un- knowns 0¶ i ¶ m − 2, once we have found xm−1. In the same spirit, all further values can be found during a backward sweep starting with i = m − 2 as solutions of local optimization problems depending on the preceeding state,

u_{i(xi) = argmin}

u ϕi(u, xi, xi+1) | u ∈ Ui(xi, xi+1, ui+1)

, 0¶ i ¶ m − 2. (7.61) The initial state x0is finally found by minimizing over S0, and determines all other unknowns. In the case of Nonlinear Model Predictive Control (NMPC), S0 only contains one element, the estimated or measured system state embedded by the initial value embedding constraint.