Newton’s Method for Global Free Flight Trajectory Optimization

Globally optimal free flight trajectory optimization can be achieved with a combination of discrete and continuous optimization. A key requirement is that Newton’s method for continuous optimization converges in a sufficiently large neighborhood around a minimizer. We show in this paper that, under certain assumptions, this is the case.


Introduction
Around the world countries are implementing Free Flight airspaces that allow aircraft to choose their own route, as opposed to being restricted to a predetermined three-dimensional network.The primary factors that influence costs are time and fuel consumption, which are closely interrelated [1].Based on the relative weights of these factors (cf.cost index) the optimal airspeed can be determined, which typically remains largely constant [2][3][4].Additionally, the vertical flight path can usually be predetermined using aircraft performance data [5].Consequently, the problem can be well approximated in a way proposed by Zermelo in 1931 [6], which involves finding the most efficient trajectory from point A to B for an aircraft flying at a constant airspeed in a given two-dimensional wind field.
The Free Flight Trajectory Optimization Problem is usually solved using direct or indirect methods from Optimal Control [5,[7][8][9][10][11].These are highly efficient, but suffer from one key drawback: They only converge locally and are thus dependent on a sufficiently good starting point.This makes such methods, used as a standalone, incapable of meeting airlines' high expectations regarding the global optimality of routes.
In [12][13][14] a deterministic two-stage algorithm was proposed that combines discrete and continuous optimization in order to find a globally optimal solution to the free flight trajectory optimization problem.With this approach the exponential complexity of other branch and bound based algorithms is circumvented.
The primary objective of the first stage is to obtain a finite sample in a systematic manner that adequately covers the search space.This deterministic approach eliminates the potential for infinite runtime, which may occur when using stochastic global optimization algorithms, such as Particle Swarm Optimization, Simulated Annealing, or Monotonic Basin Hopping [15][16][17][18].One approach is to create a locally dense directed graph with a specific density determined by the node spacing h and connectivity length ℓ, thereby implicitly defining the sample.The instances can then be selected in order of quality by applying Yen's algorithm [19] to calculate the k th shortest paths.Promising paths serve as initial guesses for a subsequent refinement stage in which a continuous solution to the problem is calculated up to the desired accuracy.
Analytical evidence and numerical experiments have demonstrated that the new hybrid algorithm has a time complexity of O(ℓ −1 ), making it superior to the conventional purely discrete approach, which has a time complexity of O(ℓ −6 ) [12].In this context, ℓ refers to the maximum arc length in a graph and the discretization length in a continuous optimization scenario.Thus, ℓ −1 serves as a comparable metric for the precision of the solution.
The present paper is concerned with the second stage.One way to generate a continuous solution is to apply Newton's method to the first order necessary conditions (the KKT-conditions) -an approach commonly referred to as Newton-KKT or Sequential Quadratic Programming (SQP) (see e.g., [20]).It is now shown that there is a quantifiable domain around a global optimum such that Newton-KKT converges if initialized accordingly.
Since the computational effort of the first graph-searching stage depends exclusively on the problem instance, i.e., the wind conditions, the algorithm asymptotically inherits the super fast convergence rates of the Newton-KKT method.
The paper is structured as follows.After defining the problem and introducing a formulation that is convenient for the analytical discussion in Section 2, we formally state the necessary and sufficient conditions as well as the Newton-KKT approach in Section 3. The proof of convergence is provided in Section 4 followed by a conclusion emphasizing the impact on previous and future work. 2 The Free Flight Trajectory Optimization Problem

Notation
Throughout this article lower case subscripts like e.g., x t or ξ τ , denote partial derivatives, while total derivatives are indicated by primes, e.g., T ′ or f ′ .Locally and globally optimal quantities are indicated by single and double superscript stars, respectively, e.g., ξ ⋆ or ξ ⋆⋆ .If not stated otherwise, we assume • to denote the l 2 -norm.Accordingly, we use the following quantitative definition of the L ∞ -norm in terms of the l 2 -norm. (1)

Problem Statement
Neglecting any traffic flow restrictions, we consider Lipschitz-continuous flight paths ξ ∈ C 0,1 ((0, 1), R 2 ) connecting origin ξ(0) = x O and destination ξ(1) = x D .By Rademacher's theorem, such paths are almost everywhere differentiable, and moreover contained in the Sobolev space W 1,∞ ((0, 1), R 2 ).A short calculation reveals that an aircraft travelling along such a path ξ with constant airspeed v through a three times continuously differentiable wind field w ∈ C 3 (R 2 , R 2 ) with bounded magnitude w(x) < v reaches the destination after a flight duration with ξ τ denoting the time derivative of ξ and see [12][13][14].Among these paths ξ, we need to find one with minimal flight duration T (ξ), since that is essentially proportional to fuel consumption [1].This classic of optimal control is known as Zermelo's navigation problem [6].It can easily be shown that in case of bounded wind speed, the optimal trajectory cannot be arbitrarily longer than the straight connection of origin and destination.Hence, every global minimizer is contained in an ellipse Ω ⊂ R 2 with focal points x O and x D .
The flight duration T as defined in (2) is based on a time reparametrization from actual flight time t ∈ [0, T ] to pseudo-time τ ∈ [0, 1] according to the actual flight trajectory x(t) = ξ(τ (t)) such that x t (t) − w(x(t)) = v, where x t denotes the so called ground speed, i.e., the derivative of position x with respect to the unscaled time t.As a consequence, the actual parametrization of ξ in terms of pseudo-time τ is irrelevant for the value of T .Calling two paths ξ, ξ equivalent if there exists a Lipschitz-continuous bijection r : (0, 1) → (0, 1) such that ξ(r(τ )) = ξ(τ ), we can restrict the optimization to equivalence classes.Moreover, every equivalence class contains a representative with constant ground speed ξ τ (τ ) = L for almost all τ , that can be obtained from any ξ with ξτ (τ ) = 0 ∀τ via ξ(τ Hence, we introduce z := (L, ξ) ∈ Z := R × X and the affine space of valid trajectories and subsequently consider the equivalent constrained minimization problem min z∈Z T (ξ), s.t.h(z) = 0 for a.a.τ ∈ (0, 1) with h(z) = 0 expressing the constant ground speed requirement, as for L ≤ L max with an arbitrary continuation for L > L max that is linear in ξ τ .If the constraint is satisfied, L also represents the path length, since Note that T : X → R is Fréchet differentiable with respect to the corresponding linear space δX := W 1,∞ 0 ((0, 1), R 2 ) (9) of directions δξ with zero boundary values, which consequently do not change origin and destination.This space is equipped with the norm Further we define the linear space δZ := R × δX (11) and equip the spaces Z and δZ with the norms 3 Continuous Optimization: Newton-KKT In order to find a continuous solution to the free flight optimization problem (6) we apply Newton's method to the first order necessary conditions (the KKTconditions), which is also known as sequential quadratic programming (SQP).Before we formally introduce Newton's method, we discuss the necessary and sufficient conditions for optimality, which also defines the goal of the presented algorithm.

Necessary Conditions
The goal of the present paper is to find an isolated globally optimal solution ξ ⋆⋆ to (6) that satisfies T (ξ ⋆⋆ ) ≤ T (ξ) ∀ξ ∈ X, contrary to a local optimizer ξ ⋆ that is only superior to trajectories in a certain neighborhood, T (ξ ⋆ ) ≤ T (ξ) ∀ξ ∈ N (ξ ⋆ ) ⊆ X.An isolated global minimizer satisfies the necessary Karush-Kuhn-Tucker (KKT) optimality conditions [21] given that it is a regular point, which is always the case, as confirmed by the following Theorem.

Sufficient Conditions
Now we turn to the second order sufficient conditions for optimality.In general, a stationary point (z ⋆ , λ ⋆ ) is a strict minimizer, if, in addition to the necessary conditions above, the well known Ladyzhenskaya-Babuška-Brezzi (LBB) conditions (e.g., [22]) are satisfied, which comprise a) the so called inf-sup condition and b) the requirement that the Lagrangian's Hessian regarding z, L zz , need be positive definite on the kernel of h ′ .
The inf-sup condition states that for the minimizer z ⋆ there is a κ > 0 such that inf δλ =0∈L 2 (0,1) Formally, the second part of the LBB-conditions requires that there is a B > 0 such that for any δz ∈ δZ that satisfies In the present case, this reads for any δz ∈ δZ such that In case of a global minimizer z ⋆⋆ = (ξ ⋆⋆ , L ⋆⋆ ), this can be reduced using λ ⋆⋆ = 0 from Lemma 2.Moreover, the constraint is equivalent to requiring that δξ T τ ξ ⋆⋆ τ = δL L ⋆⋆ almost everywhere.With this, we conclude that for any isolated global minimizer z ⋆⋆ of (6) that satisfies the inf-sup condition, there exists a B > 0 such that for any δz ∈ δZ such that δξ T τ ξ ⋆⋆ τ = δL L ⋆⋆ almost everywhere.It is important to note that the second order sufficient conditions are formulated in a L 2 -setting, while differentiability only holds in L ∞ .This is known as two-norm-discrepancy [23].

Newton's Method
In order to provide a more compact notation, we use χ = (z, λ) ∈ Z × L 2 (0, 1) =: Y in this context and define F as the total derivative of the Lagrangian, On Y we define the following norms, The problem is now to find a χ ⋆⋆ such that the first order necessary conditions for optimality as stated in ( 14) are satisfied, which translates to Applying Newton's method, we iteratively solve for ∆χ k and proceed with χ k+1 ← χ k + ∆χ k , starting with some initial value χ 0 .In other words, in every iteration we need to find (∆z k , ∆λ k ) such that

Proof of Convergence
On the way to prove the existence of a non-empty domain B(χ ⋆⋆ , R) such that Newton's method as defined in Section 3.2 converges to the corresponding global minimizer χ ⋆⋆ , if initialized with a starting point within this neighborhood, we first prove that the KKT-operator F ′ is invertible and that the Newton step ∆χ k is always well defined.Essentially, this is the case if the LBB-conditions as given in ( 18) and ( 20) are satisfied.Hence, we will show that there is a R > 0 such that the inf-sup condition is satisfied and that the Lagrangian is positive definite on the kernel of the constraints for any χ ∈ B(χ ⋆⋆ , R).Further, we show that an affine covariant Lipschitz condition holds, which finally helps to complete the proof.Before we get there, we recall the following Lemma from [13, Lemma 7] which provides a bound for the path length of a global minimizer.Lemma 3. Let z ⋆⋆ = (L ⋆⋆ , ξ ⋆⋆ ) be a global minimizer of (6), let w L ∞ (Ω) ≤ c 0 , and define As most of the subsequent results hold in a L ∞ -neighborhood of a minimizer, we introduce the following notation. respectively.
Moreover, we provide three simple yet useful bounds that hold in such a L ∞ -neighborhood of a minimizer.Lemma 4. Let χ ⋆⋆ = (z ⋆⋆ , λ ⋆⋆ ) be a global minimizer of (6) and the corresponding Lagrange multipliers.Then for every χ ∈ B(χ ⋆⋆ , R) it holds that Proof The first two inequalities follow immediately, since a global minimizer satisfies the constraint from (6).The latter one is a direct consequence of Lemma 2.

Inf-Sup Condition
We now show that the inf-sup condition, introduced in ( 18), holds in a certain neighborhood around a global minimizer.First, however, we point out that deviations δξ and δξ τ from a trajectory are inherently related and that the former is always bounded by the latter.

Positive Definiteness of the Lagrangian
The next step in order prove invertibility of the KKT-operator F ′ (χ), (24), is to show that the second partial derivative of the Lagrangian L(χ), (13), with respect to the state z is positive definite on the kernel of the linearized constraints.On the way we derive a similar result for the objective T (ξ), ( 2) for which we first derive an upper bound for its third derivative.
Proof From the definition of T in (2), we know that Inserting the bound from Lemmas 4 and 7 above and using Young's inequality yields ∆ξ C 0,1 (0,1) .
Having bounded the third derivative of T , we can estimate the potential decay of T ′′ and thus derive a lower bound for the size of this neighborhood.Similarly, we can bound h ′′ and hence L zz .
Proof Let ∆ξ := ξ − ξ ⋆⋆ and note that ∆ξ . Then we obtain Further, we point out that which together with the bounds from Lemma 4 yields Together, these bounds yield the claim with

Upper Bound for the Lagrangian
As a counterpart to the previous Lemma, we also derive an upper bound for L zz close to a minimizer.Again we start with the underlying function f in order to bound the error in the objective function T .
Then, for any ξ ∈ X, the second directional derivative of f as given in (3) is bounded by with The proof can be found in the appendix.
Proof From the definition of T in (2) we know that which, together with the bounds from Lemmas 4 and 10 as well as Young's inequality, then leads to Theorem 12. Let χ ⋆⋆ = (z ⋆⋆ , λ ⋆⋆ ) be a global minimizer of (6) and the corresponding Lagrange multipliers.Then for every χ ∈ B(χ ⋆⋆ , R) and every δz ∈ δZ it holds that with B(R) from Theorem 11.
Proof Using the bound from Theorem 11 and Young's inequality, we get

Invertibility of the KKT-Operator
Using the previous three results, which together state the existence of a neighborhood around a minimizer such that the LBB-conditions are satisfied, we are now ready to prove that the KKT-operator F ′ is invertible.
Lemma 13.Let χ ⋆⋆ = (z ⋆⋆ , λ ⋆⋆ ) be a global minimizer of (6), that satisfies the first and second order conditions for optimality with some B > 0, and the corresponding Lagrange multipliers.Further, let there be a u with u = 1 such that u T ξ ⋆⋆ τ ≥ c > 0 for almost all τ ∈ (0, 1).Then for F as given in (21) it holds that for every χ = (z, λ) ∈ B(χ ⋆⋆ , R) and and B(R) and κ(R) as given in Theorem 11 and Theorem 6, respectively.
Proof The proof builds on some prerequisites that have been established above and are briefly summarized.
i) In Theorem 6 it was proved that the inf-sup condition is satisfied: ≥ κ > 0.
ii) In Theorem 9 it was proved that Lzz is positive definite on the kernel of the constraints, i.e., for all δz ∈ δZ such that h ′ (z)[δz] = 0.
iii) In Theorem 12 it was proved that Lzz is bounded from above as Under these conditions, it follows from Brezzi's Splitting Theorem [22,Thm. 4.3] that F ′ (x) is isomorphic.Further, it can be shown that for every right hand side F (x) of the saddle point problem (24) there is exactly one solution (∆z, ∆λ) with with . This completes the proof, since

Lipschitz Constant
We are on the verge of presenting a Lipschitz constant for the free flight problem.To accomplish this, we introduce an additional bound in the form of a Lemma.This bound incorporates the constant B, which is derived in the appendix (Lemma 16).It serves to define an upper limit on the second derivative of f as defined in Equation (3).Its value is contingent upon the overall characteristics of the wind field.
Lemma 14.Let χ ⋆⋆ = (z ⋆⋆ , λ ⋆⋆ ) be a global minimizer of (6) and the corresponding Lagrange multipliers.For any χ i∈{1,2} ∈ B(χ ⋆⋆ , R) there is a B such that with Proof From Lemma 4 it directly follows that Using these bounds as well as the Cauchy-Schwarz inequality and Young's inequality, we show that for any δχ ∈ δZ × L 2 (0, 1) with it holds that as well as As shown in Lemma 16 in the appendix, there is a B < ∞ such that which provides the following bound, as Finally, we use the bounds derived above to show that for any δχ it holds that This directly yields the claim, as

Convergence of Newton's Method
We are now ready to connect the results outlined above to prove that the Newton-KKT method applied to the free flight optimization problem (6) converges to a global minimizer as characterized in Section 3.1 provided that there is a u ∈ R 2 with u = 1 such that u T ξ ⋆⋆ τ ≥ c.Roughly speaking, the optimal route needs to head towards the destination, dominating any route that involves flying the opposite direction.It is intuitively clear that this holds even for relatively strong wind conditions.Theorem 15.Let χ ⋆⋆ = (z ⋆⋆ , λ ⋆⋆ ) be a global solution of (6) that satisfies the first and second order conditions for optimality with B > 0. Moreover let there be a c > 0 and a u ∈ R 2 with u = 1 such that u T ξ ⋆⋆ τ ≥ c for almost all τ ∈ (0, 1).Finally, let ω := ω 1 ω 2 , as given in Lemmas 13 and 14.Then there is a R C > 0, such that the ordinary Newton iterates defined in Section 3.2 converge to χ ⋆⋆ at an estimated rate if initialized with χ 0 ∈ B(χ ⋆⋆ , R C ) and provided that the iterates χ k remain in Proof In Theorems 6, 9 and 12 we showed that the inf-sup condition is satisfied, that, Lzz(χ) is positive definite on the kernel of the constraint for all x ∈ B(χ ⋆⋆ , R C ), and that it is bounded from above.Consequently, F ′ (χ) is invertible with as confirmed in Lemma 13.Further, it follows from Lemmas 13 and 14 that We now define e k := χ k − χ ⋆⋆ and proceed for µ ∈ (0, 1) as follows: which yields the claim with µ = 1 as In order to prove uniqueness in B(χ ⋆⋆ , R C ), assume there is a second solution χ ⋆ = χ ⋆⋆ with F (χ ⋆ ) = 0 and χ ⋆ ∈ B(χ ⋆⋆ , R C ). Initialized with χ 0 := χ ⋆ it certainly holds that χ 1 = χ ⋆ .However, from (47) we obtain due to ω < 2, which yields a contradiction.

Conclusion
It has been demonstrated that the Newton-KKT method can be used to solve the free flight trajectory optimization problem under certain conditions.These conditions are i) the requirement for the iterates to remain within a L ∞neighborhood of the solution, and ii) a starting point that is sufficiently close to the solution.Such a suitable starting point can be found efficiently by calculating shortest paths on a specific graph [13].Hence an important tool for efficient deterministic global optimization of the free flight problem has been established.
and in particular which yields and respectively.Finally, the third directional derivative is (64) Newton's Method for Global Free Flight Trajectory Optimization which is bounded by Before we turn to the length term f 2 , we first consider the term We also note that which is bounded by The second derivative is and in particular which yields and respectively.The third derivative is which is bounded by For the length term f 2 = g −1 √ F , we thus obtain which is bounded by The second derivative is and in particular The third derivative is Newton's Method for Global Free Flight Trajectory Optimization which is bounded by Proof We obtain f by adding f 1 and f 2 .The third derivative of f can thus be bounded using (65), (82), and the triangle inequality.
, we note that and obtain Then, for any ξ ∈ X, the second directional derivative of f as given in (3) is bounded by Proof We obtain f by adding f 1 and f 2 .The second derivative of f can thus be bounded using (62), (79), and the triangle inequality.
, we note that