Assessment and Future Directions of Nonlinear Model Predictive ...

Interior-Point Algorithms for Nonlinear Model Predictive Control 213Given a strictly feasible initial point (x 0 ,s 0 ,y 0 ) ∈ S o (PD), let (x, s, y) =(x 0 ,s 0 ,y 0 ) and define the Newton iterates as follows.1. If termination conditions are satisfied then stop.2. Form (d x ,d y ,d s )bysolving(3)with(x, y, s).3. Update (x, y, s) according to x ← x + αd x , s ← s + αd s , y ← y + αd y ,where α is given at each iteration by α =(µ(x, s) σ s (∇F (x)) + ¯σ) −1 and¯σ max{σ x (d x ),σ s (d s )}.the “closest” point on the central path and cease when suitable proximity isrestored. There exist many proximity measures for the central path, but [9] usea so-called functional proximity measure which is a global measure in the sensethat it has meaning everywhere on S o (PD), and is defined asγ(x, s) =F (x)+F ∗ (s)+νln(µ(x, s)) + ν,µ(x, s) = 1 ν〈s, x〉.A region of the central path, denoted F(β), that uses this measure is defined byF(β) ={(x, y, s) ∈ S o (PD):γ(x, s) ≤ β}.4.1 Centring DirectionGiven a strictly feasible point (x, s, y) ∈ S o (PD) and ω ∈ K o such that∇ 2 F (ω)x = s, the centring direction (d x ,d s ,d y ) is defined as the solution tothe following.∇ 2 F (ω)d x + d s = − 1µ(s, x) s −∇F (x), Ad x =0, A ∗ d y + d s =0. (3)Let u ∈ E and v ∈ K o .Defineσ v (u) = 1 α,whereα>0 is the maximum possiblevalue such that v + αu ∈ K. It is convenient to define a centring algorithm [9].4.2 Affine Scaling DirectionGiven a strictly feasible point (x, s, y) ∈ S o (PD) and ω ∈ K o such that∇ 2 F (ω)x = s, the affine scaling direction (p x ,p s ,p y ) is defined as the solutionto the following.∇ 2 F (ω)p x + p s = −s, Ap x =0, A ∗ p y + p s =0. (4)Note that 〈p s ,p x 〉 = 0 from the last two equations of (4). Furthermore, 〈s, p x 〉 +〈p s ,x〉 = 〈s, x〉 since from the first equation in (4),Thus,−〈s, x〉 = 〈∇ 2 F (ω)p x + p s ,x〉 = 〈s, p x 〉 + 〈p s ,x〉.〈s + αp s ,x+ αp x 〉 = 〈s, x〉 + α(〈s, p x 〉 + 〈p s ,x〉)+α 2 〈p s ,p x 〉, =(1− α)〈s, x〉.Therefore, α can be chosen such that 〈s + αp s ,x + αp x 〉 = νµ c , i.e. α =1 − µ c /µ(s, x). In general, it is not always possible to take a step in direction(p x ,p s ,p y ) with step size α calculated here, since this may result in an infeasible