5. Increase k by one and start over in point 1.
As seen in section 4.2, step 2. of the above algorithm essentially involves the efficient solution of a QP or a parametric QP making use of the current iterate as an initializer. We need to make sure, though, that this solution completes as fast as required to maintain local contractivity and thus nominal stability of the scheme. Appropriate techniques are presented in chapters 7 and 8.
4.5 Summary
In this chapter we have presented the real–time iteration scheme for fast NMPC using active– set based NLP methods such as Sequential Quadratic Programming to repeatedly compute feedback controls based on a multiple shooting discretization of the predictive control prob- lem. The underlying algorithmic ideas have been described and NLP theory has been stud- ied as necessary to sketch a proof of local contractivity of the real–time iteration scheme. We have developed a new extension to the real–time iteration scheme that is applicable to mixed–integer NMPC problems. We have further derived a new sufficient condition for local contractivity of the mixed–integer real–time iterations. The proof relies on the existing con- tractivity statement for the continuous case. There, such a statement could be shown under the assumption that the repeatedly fixed controls are indeed fixed onto the exact outcome of the last NEWTON–type step. To make this framework accessible, we showed existence of a the- oretical modification to the approximate inverse KKT matrix that provides the NEWTON–type step including the contribution due to a rounding scheme being applied afterwards. The proof of existence relies on a symmetric rank one or BROYDEN type rank two update. We derived a sufficient condition for the modified KKT matrix to satisfy the κ–condition for local con- tractivity, and showed that this condition can actually be satisfied by a sampling time chosen small enough. A dependency on the granularity of the binary or integer control discretization along the lines of chapter 2 has thus been established also for mixed–integer Model Predictive Control. Vice versa, we showed that, if appropriate bounds and LIPSCHITZconstants on certain model functions are available, an upper bound on the sampling time can be derived.
In this chapter, we take a closer look at combinatorial issues raised by applying outer con- vexification to constraints that directly depend on a discrete control. We describe several un- desirable phenomena that arise when treating such problems with a Sequential Quadratic Programming (SQP) method, and give explanations based on the violation of Linear Inde- pendence Constraint Qualification (LICQ). We show that the Nonlinear Programs (NLPs) can instead be treated favorably as Mathematical Programs with Vanishing Constraints (MPVCs), a class of challenging problems with complementary constraints and nonconvex feasible set. We give a summary of a Lagrangian framework for MPVCs due to [3] and others, which al- lows for the design of derivative based descent methods for the efficient solution of MPVCs in the next chapter.
5.1 Constraints Depending on Integer Controls
We have seen in chapter 2 that for Mixed–Integer Optimal Control Problems (MIOCPs) with integer or binary controls that directly enter one or more of the constraints, rounding the relaxed optimal solution to a binary one is likely to violate these constraints.
5.1.1 The Standard Formulation after Outer Convexification
Having applied outer convexification to the binary control vector w (·) ∈ {0, 1}nw
, a constraint g directly depending on w (t) can be written as constraining the convexified residual
nw X i=1 ˜wi(t) · g (t, x (t), ωi, p) ¾ 0 ∀t ∈ T , (5.1) nw X i=1 ˜ wi(t) = 1 ∀t ∈ T .
This ensures that the constraint function g (·) is evaluated in integer feasible controls ωionly. While in addition it is obvious that any rounded solution derived from the relaxed optimal one will be feasible, this formulation leads to relaxed optimal solutions that show compensation effects. Choices ˜wi(t) > 0 for some 1 ¶ i ¶ nwthat are infeasible with respect to g (ωi) can be compensated for by nonzero choices ˜wj(t) > 0 that are not part of the MIOCP’s solution.
5.1.2 Outer Convexification of Constraints
In chapter 2 we already mentioned that this issue can be resolved by applying the outer convexification technique not only to the process dynamics but also to the affected constraints.
This amounts to the substitution of g (t, x(t), u(t), w(t), p) ¾ 0 ∀t ∈ T , (5.2) w (t) ∈ {0, 1}nw ∀t ∈ T , by ˜ wi(t) · g (t, x (t), ωi, p) ¾ 0, ∀t ∈ T 1¶ i ¶ nw, (5.3) ˜ w (t) ∈ [0, 1]nw, nw X i=1 ˜ wi(t) = 1 ∀t ∈ T .
It is immediately clear that an optimal solution satisfying the constraints g for some relaxed optimal control ˜w ∈ [0, 1]nw
will also satisfy these constraints for any rounded binary control. This convexification of the constraints g does not introduce any additional unknowns into the discretized MIOCP. The increased number of constraints is put in context by the observation that the majority of the convexified constraints can be considered inactive as the majority of the components of ˜w is likely to be active at zero. Active set methods are an appropriate tool to exploit this property. As constraints of type (5.3) violate LICQ in ˜wi(t) = 0, we briefly survey further alternative approaches at maintaining feasibility of g after rounding, before we investigate the structure of (5.3) in more detail.
5.1.3 Relaxation and Homotopy
In the spirit of a Big–M method for feasibility determination [157], the constraints g can be relaxed using a relaxation parameter M > 0,
g (t, x(t), ωi, p) ¾ −M(1 − ˜wi(t)), ∀t ∈ T 1¶ i ¶ nw, (5.4) ˜ w (t) ∈ [0, 1]nw, nw X i=1 ˜ wi(t) = 1 ∀t ∈ T .
Clearly for ˜wi(t) = 1 feasibility of the constraints g (ωi) is enforced while for ˜wi(t) = 0 the constraint g (ωi) does not affect the solution if M is chosen large enough. This approach has been followed e.g. by [178]. The drawback of this formulation is the fact that the choice of the relaxation parameter M is a priori unclear. In addition, the relaxation enlarges the feasible set of the Optimal Control Problem (OCP) unnecessarily, thereby possibly attracting fractional relaxed solutions of the OCP due to compensation effects.
5.1.4 Perspective Cuts
A recently emerging idea based on disjunctive programming [93] is to employ a reformulation based on perspective cuts for a constraint w · g(x) ¾ 0 depending on a continuous variable
x ∈ [0, xup] ⊂ R and a binary variable w ∈ {0, 1},
0¶ λg(x/λ), (5.5)
0¶ x ¶ λxup, λ∈ [0, 1] ⊂ R.
Clearly again for λ = 1 the original constraint on x is enforced, while for λ = 0 is can be seen by TAYLOR’s expansion of λg(x/λ) that the constraint is always satisfied. For simplicity of notation let us assume for a moment that g(x(t), w(t)) be a scalar constraint function depending on a scalar state trajectory x(t). The proposed reformulation then reads
˜ wi(t) · g ˜x i(t) ˜ wi(t), ωi ¾ 0 ∀t ∈ T , 1¶ i ¶ nw, (5.6) ˜ wi(t) · xup¾ ˜xi(t) ∀t ∈ T , 1¶ i ¶ nw, ˜ xi(t) ∈ [0, xup], nw X i=1 ˜ xi(t) = x(t) ∀t ∈ T , 1¶ i ¶ nw, ˜ w (t) ∈ [0, 1]nw, nw X i=1 ˜ wi(t) = 1 ∀t ∈ T .
Here we have introduced slack variables ˜xi(t), 1 ¶ i ¶ nw for the state x(t). Clearly if ˜
wi(t) = 1 the constraint g(ωi
) will be satisfied by ˜xi(t) and thus by x(t), while if ˜wi(t) = 0 this constraint does not affect the solution. This reformulation is promising as LICQ holds for the resulting convexified NLPs on the whole of the feasible set if it held for the original one. As the number of unknowns increased further due to introduction of the slack variables ˜xi(t), structure exploiting methods tailored to the case of many control parameters become even more crucial for the performance of the mixed–integer OCP algorithm. In addition, a connec- tion between the convex combination ˜x(t) of the state trajectory x(t) and the convexification of the dynamics must be established.
5.2 Lack of Constraint Qualification
In this section we investigate numerical properties of an NLP with a constraint depending on integer or binary variables that has been treated by outer convexification as proposed in section 5.1.2. We show that certain constraint qualifications are violated which are commonly assumed to hold by numerical codes for the solution of NLPs. Linearization of constraints treated by outer convexification bears significant potential for severe ill–conditioning of the generated linearized subproblems, as has been briefly noted by [3] in the context of truss op- timization, and we exemplarily investigate this situation. Finally, extensive numerical studies [116] have revealed that SQP methods attempting to solve MPVCs without special consid- eration of the inherent combinatorial structure of the feasible set are prone to very frequent cycling of the active set. We present an explanation of this phenomenon for the case of lin- earizations in NLP–infeasible iterates of the SQP method.
5.2.1 Constraint Qualifications
The local optimality conditions of theorem 3.1 were given under the assumption that LICQ holds in the candidate point x? in order to ensure a certain wellbehavedness of the feasi- ble set in its neighborhood. The KARUSH–KUHN–TUCKER(KKT) theorem (3.1) can however be proven under less restrictive conditions, and various Constraint Qualifications (CQs) differing in strength and applicability have been devised by numerous authors. We present in the fol- lowing four popular CQs in order of decreasing strength which will be of interest during the investigation of MPVCs in section 5.3.
We first require the definitions of the tangent and the linearized cone.
Definition 5.1 (Tangent Cone, Linearized Cone)