Assessment and Future Directions of Nonlinear Model Predictive ...

214 A.G. Wills and W.P. HeathChoose ɛ, β and ∆ such that ɛ>0, 0 βiterate the following steps.a) Compute the affine scaling direction (p x ,p s ,p y ) by solving (4).b) Update λ with λ ← 1 − µ c /µ(x, s). If λ>0 then find η>0 such thatγ(x + ηp x ,s+ ηp s )=∆ and update α with α ← min{η, λ}.Otherwise find η < 0 such that γ(x + ηp x ,s + ηp s ) = ∆ and α ←max{η, λ}.c) Update the predictor point (x + ,s + ,y + )usingx + ← x + αp x , s + ←s + αp s , y + ← y + αp y .d) Update the iterate (x, s, y) using the Newton process 4.1 starting from(x + ,s + ,y + ) and stopping as soon as a point in F(β) is found.2. Update the iterate (x, s, y) using Algorithm 4.1 starting from (x, s, y) andstopping as soon as a point in F(ɛ) is found.point. However, if it is possible to take the full step of size α then the dualitygap is equal to νµ c .4.3 Path-Following AlgorithmThe structure of the algorithm is described as follows: starting from a strictlyfeasible initial point (x 0 ,s 0 ,y 0 ) ∈ S o (PD), then firstly a predictor step is computedwhich aims at reducing the distance between µ(x, s) andµ c . The step sizeis computed in order to maintain iterates within a certain region of the centralpath If it is possible to take a step that reduces the gap |µ(x, s) − µ c | to zero,whilst remaining inside the allowed region of the central path, then this step willbe taken. After computing the intermediate predictor point, the algorithm proceedsto correct the iterates towards the central path until they are sufficientlyclose. This process sets the scene for a new predictor step in which further reductionof the gap |µ(x, s) − µ c | may be achieved. From Theorem 7.1 in [9], thenumber of corrector steps n c in each iteration is bounded byn c ≤ ∆ (τ − ln(1 + τ)), τ =0.5(3β/(1 + β)) 1/2 . (5)The number of final corrector steps n fc is given by the same relation but withβ = ɛ.Also from Theorem 7.1 in [9], if µ 0 ≥ µ c , then the number of predictor stepsn p,1 is bounded from above by⌈ ⌉( (1 ln(µ0 /µ c )n p,1 ≤, δ 1 =− 1) ) 2 1/2ln(1/(1 − δ 1 ))2ν c(∆, β)+ 1 2ν c(∆, β) +4 1 ν c(∆, β) ,(6)where c(∆, β) is a positive constant that depends on ∆ and β only. Furthermore,in [10] it is noted that Theorem 7.1 from [9] holds for negative values of α. Hence,if µ 0 ≤ µ c , then the number of predictor steps n p,2 is bounded from above by