Assessment and Future Directions of Nonlinear Model Predictive ...

210 A.G. Wills and W.P. Heathto the point x(µ c ). A damped Newton method is used to generate the new pointx k+1 , and, we employ a long-step approach, so that aggressive reductions inµ k are allowed. This has the consequence of increasing theoretical complexitybounds but tends to be favoured for practical algorithms since these bounds areusually conservative.With this in mind, it remains to find an initial point x 0 and value µ 0 suchthat x 0 is close to x(µ 0 ). This is the subject of Section 3.1 which discusses aninitialisation algorithm.Further to this, however, is a minor technical issue. The algorithm presentedin this section requires that the objective function f is linear, but we are interestedin a more general choice of f (typically quadratic). Nevertheless, it isstraightforward to embed problem (P) into a slightly larger formulation whichdoes have a linear objective. In fact, we do this now using the epigraphic form [2].(A) :min t s.t. f(x) ≤ t, x ∈ G, (A µ): min(t,x) (t,x)1µt−ln(t−f(x))+F (x),where t ∈ R and −ln(t − f(x)) is assumed to be a ν f -self-concordant barrierfunction for f(x) ≤ t (e.g. in the case where f is quadratic then this assumptionis satisfied with ν f = 2). Note that (A µ ) has a unique minimiser, denoted by(t(µ),x(µ)). The following lemma shows that if (t(µ),x(µ)) minimises (A µ )thenx(µ) minimises (P µ ) and it therefore appears reasonable to solve problem (A µ )and obtain the minimiser for problem (P µ ) directly.Lemma 1. Let (t(µ),x(µ)) denote the unique minimiser of problem (A µ ) forsome µ>0, thenx(µ) also minimises problem (P µ ) for the same value of µ.3.1 Initialisation StageThe purpose of an initialisation stage is to generate a point (t 0 ,x 0 ) and a valueµ 0 such that (t 0 ,x 0 )iscloseto(t(µ 0 ),x(µ 0 )) (the minimiser of (A µ )), whichenables the main stage algorithm to progress as described above. The usualapproach is to minimise the barrier function F A (t, x) −ln(t − f(x)) + F (x),which in turn provides a point at the “centre” of the constraint set and close tothe central-path of (A) forµ large enough.However, by construction F A (t, x) is unbounded below as t →∞.Thissproblemis overcome, in the literature, by including a further barrier on the maximumvalue of t resulting in the combined barrier functionF B (t, x) − ln(R − t)+F A (x), F A (t, x) −ln(t − f(x)) + F (x). (2)The unique minimiser of the barrier function F B and the central path ofproblem (A) are related in the following important way.Lemma 2. Let (t ∗ ,x ∗ ) denote the unique minimiser of F B (t, x). Then(t ∗ ,x ∗ )coincides with a point on the central path of problem (A) identified by µ = R−t ∗ .Therefore, minimising the barrier function F B actually provides a point onthe central path of (A), which is precisely the goal of the initialisation stage.