Inexact Restoration for Euler Discretization of Box-constrained Optimal

Inexact Restoration for Euler Discretization of Box-constrained Optimal Control Problems∗ Nahid Banihashemi†

C. Yal¸cın Kaya‡

July 21, 2012

Communicated by Hans J. Oberle Abstract The Inexact Restoration method for Euler discretization of state and control constrained optimal control problems is studied. Convergence of the discretized (finitedimensional optimization) problem to an approximate solution using the Inexact Restoration method and convergence of the approximate solution to a continuous-time solution of the original problem are established. It is proved that a sufficient condition for convergence of the Inexact Restoration method is guaranteed to hold for the constrained optimal control problem. Numerical experiments employing the modelling language AMPL and optimization software Ipopt are carried out to illustrate the robustness of the Inexact Restoration method by means of two computationally challenging optimal control problems, one involving a container crane and the other a free-flying robot. The experiments interestingly demonstrate that one might be better-off using Ipopt as part of the Inexact Restoration method (in its subproblems) rather than using Ipopt directly on its own.

Key words: Optimal control, inexact restoration, Euler discretization, container crane, free-flying robot. AMS subject classifications. 49K15, 49M05, 49M25, 65K10, 65L06

1

Introduction

Finding a solution to an optimal control problem is in general difficult because one has to deal with an infinite-dimensional optimization problem. It is common practice to discretize an optimal control problem by using Euler, midpoint, trepezoidal, or in general, Runge-Kutta schemes ∗

The authors are indebted to Helmut Maurer for passing on to them his (the Ipopt-alone) AMPL code for the container crane example, and for further useful discussions. They also thank the referees and the editor for their comments and suggestions, which improved the paper. † School of Mathematics and Statistics, University of South Australia, Mawson Lakes , SA 5095, Australia. E-mail: [email protected] . ‡ Corresponding Author. School of Mathematics and Statistics, University of South Australia, Mawson Lakes, SA 5095, Australia. E-mail: [email protected] .

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Inexact Restoration for Euler Discretization of Constrained Optimal Control Problems by N. Banihashemi & C. Y. Kaya

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(see, e.g., [1-7]), so as to convert the problem to a finite-dimensional optimization problem. Euler discretization, the simplest of these schemes, is often the chosen approximation, because it is easier to manipulate and it allows the conditions of optimality to be written down easily. Once one has the discretized optimal control problem, it is important that (i) the optimization method employed to get an approximate solution is efficient and (ii) the approximate solution converges to a solution of the continuous-time problem as discretization is progressively made finer. The Inexact Restoration (IR) method, which is an iterative finite-dimensional optimization method developed by Mart´ınez and his coworkers [8-10], has been demonstrated to be particularly suited to discretization of optimal control problems in [1-2]. In particular, it is shown in both [1] and [2] that one of the sufficient conditions for convergence of the IR method is automatically satisfied for the discretized problem. Moreover, it is observed in [1-2], via numerical experiments, that the IR method is more robust compared to the Newton and projected Newton methods, and that it seems to obey the mesh independence principle, just as Newton’s method does. A local convergence analysis and an associated algorithm for the IR method are given by Birgin and Mart´ınez [10]. In a technical report [11], which is an expanded version of [1], it is proved that a special instance of the IR method is a projected Newton method. In [1-2], unconstrained optimal control problems with unspecified free terminal state variables are considered. In the current paper, we consider constraints on the state and control variables. In particular, we focus on the case when state and control variables have simple bounds (box constraints), because then projections on the feasible set as part of the IR method are easy to carry out. For the Euler discretization of these more general and challenging problems we utilize the IR method, with the above-listed advantages in mind. This choice addresses (i). Convergence of the solution of the Euler discretized problem to a continuous-time solution of state and control constrained optimal control problems has been studied by Dontchev, Hager and Malanowski [4]. We will employ their result in this paper. This addresses (ii). It should be noted that there exists another popular discretization scheme, called control parameterization, in which one approximates the control variables in each time subdivision by constants, linear functions or splines, and the state variables are obtained by solving the dynamical system equations with these approximated controls (see [12-17]). Our approach here is markedly different from such schemes. Each IR iteration consists of two phases, namely the feasibility and optimality phases in which separate subproblems are solved. For the constrained optimal control problem considered in this paper, as opposed to the case studied in [1-2], one needs to update not only the states but also the controls in the feasibility phase. We prove in this paper a similar result to those proved in [1-2] that, if feasibility is improved and the magnitude of the update in control variables is kept small, then the magnitude of the update in state variables is also small, in the sense to be explained in Section 4. We also furnish convergence results for the new Euler discretized problem: We establish that the IR method is convergent in the new setting, and that, as one takes finer subdivisions, the aproximate solution converges to a continuous-time solution. Birgin and Mart´ınez also propose a practical algorithm for the IR method in [10], which they use to make comparisons and justify robustness of the IR method compared to the popular general-purpose optimization software Lancelot. We adapt their algorithm to our optimal control setting and code it in the modelling language AMPL [19] which uses the optimization software Ipopt [20]. We test our implementation of IR on two nontrivial optimal control problems from the literature, namely the container crane and free-flying robot problems. We solve the same optimal control problems by also using Ipopt on its own (again interfaced with


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AMPL), and make comparisons with our implementation of IR. We run the AMPL codes for each approach 100 times with randomly generated initial guesses. Our experiments show that one might be better-off using Ipopt as part of the IR method (in its subproblems in the feasibility and optimality phases) rather than using Ipopt directly on its own. Dontchev, Hager and Malanowski [4] provide their results using the so-called indirect adjoining approach, where the time-derivative of the pure state constraints are adjoined to the Hamiltonian function [21]. The adjoint (or costate) variables found this way differ from those found by using the direct adjoining approach. In our paper, we identify a process of converting one set of adjoint variables obtained by one approach to the other, and illustrate this in our numerical experiments. The paper is organized as follows. In Section 2, the problem is stated and the convergence result of Dontchev, Hager and Malanowski [4] is cited in detail for our setting. In Section 3, Euler discretization of the optimal control problem is described and necessary conditions of optimality are derived. The IR method for the optimal control problem and associated results are presented in Section 4. In Section 5, we give the IR algorithm for our particular problem. In Section 6, we carry out numerical experiments with the container crane and free-flying robot problems, make comparisons and provide discussion.

2

The Optimal Control Problem

Consider an optimal control problem subject to box constraints on the state and control variables as follows. ⎧ tf ⎪ ⎪ ⎪ minimize φ(x(tf )) + f0 (x(t), u(t)) ⎪ ⎪ ⎪ t 0 ⎨ (P) subject to x(t) ˙ = f (x(t), u(t)) for a.e. t ∈ [t0 , tf ] , ⎪ ⎪ ⎪ x(t0 ) = x0 , ξ(x(tf )) = 0 , ⎪ ⎪ ⎪ ⎩ (x(t), u(t)) ∈ χ for a.e. t ∈ [t0 , tf ] , where the state variable x ∈ W 2,∞ (t0 , tf ; IRn ), x˙ = dx/dt, the control variable u ∈ L∞ (t0 , tf ; IRm ). The functions f0 : IRn × IRm → IR, f : IRn × IRm → IRn , ξ : IRn → IRn and φ : IRn → IR are twice continuously differentiable. The initial state is specified as x0 . The constraint set χ ⊆ IRn×m is a closed and convex set; in particular we choose χ to be a “box”: χ = {(x(t), u(t)) | ai ≤ xi (t) ≤ ai ,

bj ≤ uj (t) ≤ bj ,

i = 1, . . . , n,

j = 1, . . . , m} ,

where ai , ai , bj and bj are real constants, with ai < ai and bj < bj , and xi (t) and uj (t) are the ith and the jth components of x(t) and u(t), respectively. The functions ξ and φ are also twice continuously differentiable. In the rest of this section, we adopt the following terminology from Dontchev et al. [4]. In this problem, t0 and tf are fixed and Lα (t0 , tf : IRn ) denotes the Lebesgue space of measurable functions x : [t0 , tf ] → IRn with x(·)α integrable, equipped with the norm xLα :=

tf

t0

x(t)

α

1/α ,

where · is the Euclidean norm. The case α = ∞ corresponds to the space of essentially bounded, measurable functions equipped with the essential supremum norm. Furthermore,


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W k,α(t0 , tf ; IRn ) is the Sobolev space consisting of functions x : [t0 , tf ] → IRn , whose jth derivative lies in L∞ for all 0 ≤ j ≤ k with the norm xW k,α

k j

d x

:=

dtj

.

Lα

j=0

We split the constraint (x(t), u(t)) ∈ χ into two sets of inequalities: ai ≤ xi (t) ≤ ai for i = 1, . . . , n, and bj ≤ uj (t) ≤ bj for j = 1, . . . , m. Problem (P) can now be re-written as

(P1)

⎧ ⎪ ⎪ minimize ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ subject to ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

φ(x(tf )) +

tf t0

f0 (x(t), u(t))

x(t) ˙ = f (x(t), u(t)) for a.e. t ∈ [t0 , tf ], x(t0 ) = x0 , ξ(x(tf )) = 0 , ϑ(x(t)) ≤ 0 , for all t ∈ [t0 , tf ] , θ(u(t)) ≤ 0 , for a.e. t ∈ [t0 , tf ] ,

where ϑ(x(t)) := (x1 (t) − a1 , −x1 (t) + a1 , . . . , xn (t) − an , −xn (t) + an ) , θ(u(t)) := (u1 (t) − b1 , −u1 (t) + b1 , . . . , um (t) − bm , −um (t) + bm ) . Next, we pose assumptions as in [4] and introduce the Euler discretization of Problem (P1). The result, which guarantees convergence of the solution of the discretized problem to that of the original one, will be cited from [4] at the end of this section. (A1) Problem (P1) has a local solution (x∗ , u∗ ) which lies in W 2,∞ ×W 1,∞ and ϑ(x∗ (t0 )) < 0. (A2) In a neighbourhood of (x∗ , u∗ ), f0 , ξ and f have first- and second-order partial derivatives which are Lipschitz continuous. (The assumption that ϑ has Lipschitz continuous thirdorder derivative in [4] is automatically satisfied, given the definition of ϑ above.) (A3) There exist Lagrange multipliers ψ ∗ ∈ W 2,∞ , σ ∗ ∈ IRr and κ∗ , ν ∗ ∈ W 1,∞ associated with (x∗ , u∗) for which the following form of the first-order necessary optimality conditions (minimum principle) is satisfied at (x∗ , u∗ , ψ ∗ , σ ∗ , κ∗ , υ ∗ ), namely x˙ = f (x, u) ,

(1)

x(t0 ) − x0 = 0 ,

(2)

ξ(x(tf )) = 0 ,

(3)

∂f0 ∂f (x, u) + (ψ − ν 1 + ν 2 )T (x, u) = 0 , ψ˙ + ∂x ∂x −ψ(tf ) + ∇x (φ(x(tf )) + ξ(x(tf ))T σ) = 0 , ∂f0 ∂f (x, u) + (ψ − ν 1 + ν 2 )T (x, u) + κ1 − κ2 = 0 , ∂u ∂u 1 2 1 κ ≥ 0, (κ − κ )u − (κ + κ2 ) b = 0 , ν˙ ≥ 0 ,

(ν˙ 1 − ν˙ 2 )x − (ν˙ 1 + ν˙ 2 ) a = 0 ,

(4) (5) (6) (7) (8)

where ν(t) := (ν11 (t), ν12 (t), . . . , νn1 (t), νn2 (t)) is the multiplier corresponding to ϑ(x(t)) ≤ 0. By this definition, νi1 (t) corresponds to xi (t) − ai ≤ 0 and νi2 (t) to −xi (t) + ai ≤ 0 for i = 1, . . . , n. We also define ν 1 := (ν11 , . . . , νn1 ) and similarly ν 2 := (ν12 , . . . , νn2 ). The multiplier


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κ(t) := (κ11 (t), κ21 (t), . . . , κ1m (t), κ2m (t)) corresponds to θ(u(t)) ≤ 0, in the same way, and κ1 and κ2 are defined as (κ11 (t), . . . , κ1m (t)) and (κ21 (t), . . . , κ2m (t)), respectively. The Hamiltonian function for Problem (P1) is defined by T

H(x, u, ψ, κ) = f0 (x, u) + f (x, u) ψ +

m

κ1j (uj − bj ) + κ2j (−uj + bj ) .

j=1

Equations (1)-(8) of the minimum principle are the result of applying the indirect adjoining approach with continuous adjoint functions [21]. Note that the variational properties of the multipliers are given in the necessary conditions of optimality (1)-(8). In the sequel, we will provide and discuss the relationship between the multipliers for the cases of direct and indirect adjoining approaches, respectively - see Remark 3.1 and the results and comments in the container crane example along with Figures 6 and 10. The following matrices are defined to pose the remaining assumptions. A(t) = ∇x f (x∗ (t), u∗ (t)) ,

B(t) = ∇u f (x∗ (t), u∗ (t)) ,

Θ(t) = ∇u θ(u∗ (t)) ,

Ξ = ∇x ξ(x∗ (tf )) ,

Υ(t) = ∇x ϑ(x∗ (t)) .

For a scalar α and t ∈ [t0 , tf ], the following functions are also defined. θαi = min{0, θi (u∗ (t)) + α} i = 1, . . . , 2m, ϑjα = min{0, ϑj (x∗ (t)) + α} , j = 1, . . . , 2n . Furthermore, the 2m × 2m matrix Uα (t), 2n × 2n matrix Tα (t) and 2(n + m) × (3m + 2n) matrix Vα (t) are defined as follows. Uα (t) = diag θαi (t) , Tα (t) = diag ϑjα (t) , Θ(t) Uα (t) 0 Vα (t) = . Υ(t)B(t) 0 Tα (t) In [4], a so-called Independence Condition is posed concerning general state and control constraints for Problem (P1), namely that Vα (t)Vα (t)T ζ ≥ β ζ ,

for all ζ ∈ IR2(n+m) and all t ∈ [t0 , tf ] ,

(9)

for some α > 0 and β > 0. In the following proposition, we provide a condition, which is equivalent to Condition (9), but easier to check, for the box-constrained case. Proposition 2.1 Suppose that bi ≤ u∗i (t)) ≤ bi for all i ∈ J u := {i1 , . . . , iq } ⊂ {1, . . . , m} =: Ju , and aj ≤ x∗j (t)) ≤ aj for all j ∈ J x := {j1 , . . . , js } ⊂ {1, . . . , n} =: Jx , are active. Then Condition (9) holds if, and only if, ⎡ ⎤ ei1

⎢ ⎥ .. ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ e

i q ⎢ ⎥ ⎢ ············ ⎥ ⎢ ⎥ ⎥ = q + s ≤ m + n , for all t ∈ [t0 , tf ] , rank ⎢ (10) ⎢ ∂fj1 ∗ ⎥ ∗ (x (t), u (t)) ⎥ ⎢ ⎢ ∂u ⎥ ⎢ ⎥ . .. ⎢ ⎥ ⎢ ⎥ ⎣ ∂fj ⎦ s (x∗ (t), u∗ (t)) ∂u where eik , k = 1, . . . , q, are the ik th standard basis vectors in IRm .


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Proof. See Appendix A.

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So, we pose the following assumption, instead of Condition 9. (A4) (Independence) Condition (10) holds. Remark 2.1 For Assumption (A4) to hold, at any given time, at most as many state (box) constraints as the number of control variables are allowed to be active. For example, for a single-input control system, only one state constraint can become active at any given time. If none of the state constraints is active then Assumption (A4) is not needed, because (10) holds automatically. The rank condition in (10) can be interpreted further: In view of the proof of the proposition, ∂ x˙ jk /∂uk = ∂fjk /∂uk (x∗ , u∗ ) = 0, for some k = 1, . . . , m, whenever either −x∗jk + ajk ≤ 0 or x∗jk − ajk ≤ 0 is active. Therefore, the (box) state constraints considered in this paper are required to be of order one. It should also be noted that the rank condition (10) is reminiscent of the constraint qualification (4.23) in [21] for first-order state constraints. (A5) (Controllability) There exists α > 0 such that for each q ∈ IRn there exists a solution (c, d, x, u) ∈ W 1,∞ (t0 , tf ; IR2n ) × L∞ (t0 , tf ; IR2m ) × W 1,∞ (t0 , tf ; IRn ) × L∞ (t0 , tf ; IRm ) of the following linear system: x(t) ˙ − A(t)x(t) − B(t)u(t) = 0,

x(0) = x0 ,

Ξx(tf ) = q , Θ(t)u(t) + Uα (t)c(t) = 0 , Υ(t)x(t) + Tα (t)d(t) = 0 . The last assumption (A6) below is known as coercivity which is a strong form of a second-order sufficient optimality condition. First, define w ∗ = (x∗ , u∗, ψ ∗ , σ ∗ , κ∗ , ν ∗ ), as well as ∗ ) = H(x∗ , u∗ , ψ ∗ , κ∗ ) − f (x∗ , u∗)T ∇x ϑ(x∗ )T ν ∗ , H(w ∗) , Q∗11 = ∇xx H(w

∗) , Q∗12 = ∇xu H(w

∗) , Q∗22 = ∇uu H(w

R∗ = ∇xx (φ(x∗ (tf )) + ξ(x∗ (tf )T σ ∗ ) . Introduce the quadratic form 1 tf 1 ∗ [x(t)T Q∗11 (t)x(t) + 2x(t)T Q∗12 u(t) + u(t)T Q∗22 (t)u(t)]dt + x(tf )T R∗ (t)x(tf ) . B (x, u) = 2 t0 2 (A6) (Coercivity) There exists a constant γ > 0 such that B∗ (x, u) ≥ γ u2L2

for all (x, u) ∈ M ∗ ,

where M ∗ = {(x, u) ∈ W01,2 (t0 , tf ; IRn ) × L2 (t0 , tf ; IRm ) | x(t) ˙ − A(t)x(t) − B(t)u(t) = 0 , Ξx(tf ) = 0}.


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7

Euler Discretization

In this section, we introduce Euler discretization of Problem (P1) in a way which is reminiscent of that in [1]. First subdivide the time horizon [t0 , tf ] into N pieces. Take the subdivision points ti , i = 0, .., N, equidistantly with the mesh size Δt = (tf − t0 )/N, so that ti = t0 + iΔt. In this case, tN = tf . Define the partition π = {t0 , t1 , . . . , tN } , and the vectors xπ = (xT0 , xT2 , . . . , xTN ) ∈ IRn(N +1) ,

uπ = (uT0 , uT2 , . . . , uTN −1) ∈ IRmN ,

where xi and ui are the approximation of the state and control variables at time ti , respectively, so xi ≈ x(ti ) and ui ≈ u(ti) for i = 0, . . . , N. Introduce the forward difference xi = and approximate the integral

xi+1 − xi , Δt

tf

f0 (x(t), u(t))

t0

by the Riemann sum Δt

N −1

f0 (xi , ui ) .

i=0

Now, the Euler discretization of Problem (P1) can be given by ⎧ N −1 ⎪ ⎪ ⎪ minimize φ(xN ) + Δt f0 (xi , ui) ⎪ ⎪ ⎪ ⎪ i=0 ⎪ ⎪ ⎨ subject to xi+1 = xi + Δtf (xi , ui), (PE) ⎪ x0 = x0 , ξ(xN ) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ θ(ui ) ≤ 0, ⎪ ⎪ ⎪ ⎩ ϑ(xi+1 ) ≤ 0, i = 0, . . . , N − 1, where θ(ui ) := (ui1 − b1 , −ui1 + b1 , . . . , uim − bm , −uim + bm ) and ϑ(xi ) := (xi1 − a1 , −xi1 + a1 , . . . , xin − an , −xin + an ) . We denote by xij the jth component of xi , i = 0, . . . , N, j = 1, . . . , n, and by uik the kth component of ui, k = 1, . . . , m. Define the Lagrangian function for Problem (PE) as π , σ ¯ ) := Δt LP E (xπ , uπ , λπ , κπ , μ

N −1

f0 (xi , ui ) +

i=0

N −1

λTi+1 (−xi+1 + xi + Δtf (xi , ui))

i=0 n N

+ λT0 (x0 − x0 ) + Δt

[(xij − aj ) μ1ij + (−xij + aj ) μ2ij ]

j=1 i=0

+ Δt

m N −1 [(uij − bj )κ1ij + (−uij + bj ) κ2ij ] + φ(xN ) + σ ¯ T ξ(xN ), j=1 i=0


8

where σ ¯ ∈ IRn , λπ := (λ0 , . . . , λN ) ∈ IRn(N +1) , κπ := (κ0 , κ1 , . . . , κN −1 ) ∈ IR2mN and μ0 , μ 1 , . . . , μ N ) ∈ IR2n(N +1) are the Lagrange multipliers. We also define μ π := ( 1 2 1 2 1 2 1 2 κi := (κi1 , κi1 , . . . , κim , κim ) and similarly μ i := ( μi1 , μ i1 , . . . , μ in , μ in ). First-order necessary conditions, namely the Karush-Kuhn-Tucker (KKT) conditions, for Problem (PE) are given by ∇LP E (xπ , uπ , λπ , κπ , μ π , σ ¯) = 0 , κTi ≥ 0,

κi θ(ui ) = 0 ,

μ Ti ≥ 0,

μ i ϑ(xi ) = 0 ,

that is, Δt

∂f0 ∂f μ1i − μ 2i )T (xi , ui ) + λTi+1 (xi , ui ) + ( ∂x ∂x

+ λTi+1 − λTi = 0,

(11)

∂f ∂f0 (xi , ui ) + λTi+1 (xi , ui) + (κ1i − κ2i )T = 0 , ∂u ∂u xi+1 − xi − Δt f (xi , ui) = 0 ,

(13)

ξ(xN ) = 0,

(14)

x0 − x0 = 0,

(15)

(12)

κi ≥ 0,

κTi θ(ui ) = 0 ,

(16)

μ i ≥ 0,

μ Ti ϑ(xi ) = 0 ,

(17)

σ ¯T

∂ξ(xN ) ∂φ(xN ) μ1N − μ 2N ) = 0 , + − λN + Δt ( ∂x ∂x

(18)

for i = 0, . . . , N − 1. In order to relate the first order necessary conditions of optimality (11)(18) for the discretized problem (PE) to those of the original problem (P1) one needs to define the following transformed dual variables: νik

:= −Δt

N

μ kj ,

ψi := λi + νi1 − νi2 ,

i = 0, . . . , N , k = 1, 2 .

(19)

j=i k Note that μ ki = (νi+1 − νik )/Δt =: (νik ) . After substituting (19) into (11)-(18), and carrying out manipulations, we obtain the following optimality conditions for Problem (PE):

∂f0 1 2 (xi , ui) + (ψi+1 − νi+1 + νi+1 )T ∂x ∂f0 1 2 (xi , ui) + (ψi+1 − νi+1 + νi+1 )T ∂u xi+1 − xi − Δf (xi , ui ) = 0 ,

∂f (xi , ui) + ψi = 0 , ∂x ∂f (xi , ui) + (κ1i − κ2i )T = 0 , ∂u

(20) (21) (22)

ξ(xN ) = 0 ,

(23)

x0 − x0 = 0 ,

(24)

κi ≥ 0,

κTi θ(ui )

νi

≥ 0,

(νi )T ϑ(xi )

σ ¯T

∂ξ(xN ) ∂φ(xN ) + − ψN = 0 , ∂x ∂x

= 0, = 0,

(25) (26) (27)

where ψi = (ψi+1 − ψi )/Δt, i = 0, 1, . . . , N − 1. One can easily check that Equations (20)-(27) constitute Euler discretization of (1)-(8).


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Remark 3.1 The transformation defined in (19) has further implications. It is a useful tool in relating the multipliers for the direct and indirect adjoining approaches described in [21]. The multiplier of the state constraints for the direct adjoining approach should be identified here as Δt μ π , and the multiplier for the direct adjoining approach as νπ . While the discretization of the adjoint variables for the direct adjoining approach is λπ , that for the indirect adjoining approach is ψπ . For a given Δt, we say that (x∗π , u∗π , ψπ∗ , σ ¯ ∗ , κ∗π , νπ∗ ) is a critical solution of Problem (PE) if it satisfies (20)-(27). Define the discrete Lp and W 1,p norms for xπ as N 1/p 1/p xπ Lp = Δtxi p , xπ L∞ = sup |xi | , xπ W 1,p = [xπ pLp + xπ pLp ] , i=0

0≤i≤N

where xπ = (x0 , x1 , . . . , xN −1 ). Here xπ is said to be Lipschitz continuous in discrete time if xπ L∞ ≤ η, where η is a Lipschitz constant. Define x∗π − x∗ to be discrete such that (x∗π − x∗ )i = x∗i − x∗ (ti ), for i = 0, . . . , N. Discrete norms for the other variables are defined similarly. Theorem 3.1 (Dontchev, Hager and Malanowski [4]) If Assumptions (A1)-(A6) hold, then for all sufficiently small Δt, there exists a locally unique local solution (x∗π , u∗π ) of the discrete ¯ ∗ , κ∗π , νπ∗ ) and a constant optimal control problem (PE), associated Lagrange multipliers (ψπ∗ , σ c > 0, independent of Δt such that x∗π − x∗ W 1,2 , u∗π − u∗ L2 , ψπ∗ − ψ ∗ W 1,2 , κ∗π − κ∗ L2 , νπ∗ − ν ∗ L2 , ¯ σ ∗ − σ ∗ ≤ c Δt . Moreover, x∗π , u∗π , ψπ∗ , κ∗π and νπ∗ are Lipschitz continuous in time with Lipschitz constants independent of Δt. Corollary 3.1 If (A1)-(A6) hold, as Δt → 0, a local solution (x∗π , u∗π , ψπ∗ , σ ¯ ∗ , κ∗π , νπ∗ ) of Prob∗ ∗ ∗ ∗ ∗ ∗ lem (PE) converges to a local solution (x , u , ψ , σ , κ , ν ) of Problem (P), with respect to the norms indicated in Theorem 3.1.

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IR for Optimal Control

In this section, we will formulate the IR method for solving optimal control problems. For the sake of simplicity in appearance, we define the functions for the equality constraints as follows. h0 (x0 ) = x0 − x0 , hi+1 (xi+1 , xi , ui) = Δt f (xi , ui) − xi+1 + xi , hN +1 (xN ) = ξ(xN ) .

i = 0, 1, . . . , N − 1,

We define the vector function for the equality constraints as h(xπ , uπ ) = (hT0 (x0 ), hT1 (x0 , x1 , u0), hT2 (x1 , x2 , u1 ), . . . , hTN (xN −1 , xN , uN −1 ), hTN +1 (xN ))T . Now Problem (PE) can be re-written as N −1 ⎧ ⎪ ⎨ minimize Δt f0 (xi , ui) + φ(xN ) (PIR) i=0 ⎪ ⎩ subject to h(xπ , uπ ) = 0 , (xπ , uπ ) ∈ Ω ,


10

where Ω := {(xπ , uπ ) | aj ≤ xij ≤ aj , bk ≤ uik ≤ bk , i = 0, . . . , N − 1 , j = 1, . . . , n , k = 1, . . . , m} . Problem (PIR) is a box-constrained optimization problem. Birgin and Mart´ınez [10] present the IR method to solve this class of problems. The idea of their method can be translated into our setting of Euler-discretized state and control constrained optimal control problem as follows. The IR iterations start with a given point (xπ , uπ ) ∈ Ω and first find a “more feasible” point (yπ , vπ ) ∈ Ω. This process is referred to as the feasibility phase. Next, in the so-called optimality phase, a “more optimal ” point (zπ , wπ ) ∈ Ω is found in the plane tangent to the constraints and passing through (yπ , vπ ). In order to formulate these two phases we need to define the Lagrangian for Problem (PIR) in the same way Birgin and Mart´ınez [10] and later Kaya and Mart´ınez [1] did, but this time for our setting of constrained optimal control problems. The Lagrangian function is given, for all (xπ , uπ ) ∈ Ω, by L(xπ , uπ , λπ , σ) = Δt

N −1

f0 (xi , ui) + λT0 h0 (x0 ) +

i=0

N −1

λTi+1 hi+1 (xi , xi+1 , ui )

i=0

+ σ T ξ(xN ) + φ(xN ) , where σ ∈ IRn and λπ = (λT0 , . . . , λTN ) ∈ IR(N +1)n are the Lagrange multipliers. It turns out that λπ converges, as discretization is made finer, to the costate (or adjoint) variable of the direct adjoining approach. The relationship between the costates of the direct and indirect adjoining approaches are furnished by (19) and Remark 3.1. The relationship is then illustrated in Figures 6 and 10 for the crane control example. We say that (x∗π , u∗π , σ ∗ , λ∗π ) is a critical point of Problem (PIR), if h(x∗π , u∗π ) = 0 , G(x∗π , u∗π , λ∗π , σ ∗ ) = P [(x∗π , u∗π ) − ∇L(x∗π , u∗π , λ∗π , σ ∗ )] − (x∗π , u∗π ) = 0 ,

(28) (29)

where ∇ is the gradient operator with respect to (xπ , uπ ), and P is the projection operator on Ω with respect to the Euclidian norm · . Lemma 4.1 The necessary conditions of optimality (11)-(18) and (28)-(29) yield the same solution (x∗π , u∗π , σ ∗ , λ∗π ). Proof. See, for example, Proposition 3.3.14 in [22]. The inequality constraints are denoted by gj , j = 1, . . . , r, in this proposition. Define gi := θ(ui ) and gN +i := ϑ(xi+1 ), i = 0, . . . , N − 1. 2 Conditions (28)-(29) are consistent with the IR setting in [10]. The use of the projection operator prevents us from having to deal with the complementarity conditions (16)-(17). This π . way, we also have fewer Lagrange multipliers: We do not have to compute κπ and μ Define the vectors

∂L ∂f0 T ∂f (xi , ui ) , := xi − = xi − Δt λi + λi+1 (xi , ui) + ∂xi ∂x ∂x ∂L ∂f0 2 T ∂f (xi , ui ) , Gi (xi , ui, λi , λi+1 ) := ui − = ui − Δt λi+1 (xi , ui) + ∂ui ∂u ∂u G1i (xi , ui, λi , λi+1 )

where λi = (λi+1 − λi )/Δt, i = 0, . . . , N − 1, and G1N (xN , σ)

∂φ(xN ) ∂L T ∂ξ(xN ) + := xN − = xN − σ . ∂xN ∂x ∂x


11

We also define the projection vector functions Pi1 (xi , ui, λi , λi+1 ) and Pi2(xi , ui, λi , λi+1 ) as follows. ⎧ 1 G − xij , if aj ≤ G1ij ≤ aj , ⎪ ⎨ ij Pij1 (xi , ui , λi, λi+1 ) := aj − xij , if G1ij > aj , ⎪ ⎩ aj − xij , if G1ij < aj , where j = 1, . . . , n, and ⎧ ⎪ G2 − uik , if bj ≤ G2ik | ≤ bk , ⎪ ⎨ ik Pik2 (xi , ui, λi , λi+1 ) := bk − uik , if G2ik > bk , ⎪ ⎪ ⎩ b − u , if G2 < b , ik k k ik where k = 1, . . . , m. Now, for i = 0, . . . , N − 1, let Gi (xi , ui, λi , λi+1 ) = (Pi1 (xi , ui , λi , λi+1 ), Pi2(xi , ui, λi , λi+1 )) , and

GN (xN , σ) = (PN1 (xN , σ), 0) ,

so that G(xπ , uπ , λπ , σ) = (G0 (x0 , u0 , λ0 , λ1 ), G1 (x1 , u1, λ1 , λ2 ), . . . , GN (xN , σ)) . Next, we start describing the feasibility phase of an IR iteration. In this phase, the feasibility of the current point (xπ , uπ ) ∈ Ω is improved, i.e., one finds a point (yπ , vπ ) ∈ Ω which satisfies h(yπ , vπ ) ≤ θh(xπ , uπ ) ,

θ ∈ [0, 1[ .

The new point (yπ , vπ ) is required to be not too far from (xπ , uπ ) within the set Ω, in some sense. The condition of “being not too far” is expressed by the following inequality. yπ − xπ + vπ − uπ ≤ K1 h(xπ , uπ ), where K1 is a positive constant. An IR iteration continues with the optimality phase for finding a point (zπ , wπ ) ∈ Ω in the plane tangent to the constraints described by ∇h(yπ , vπ ) (zπ − yπ , wπ − vπ ) = 0 ,

(30)

such that optimality of the point (yπ , vπ ) is improved. The tangent plane given in (30) can be expressed equivalently by z0 − y0 = 0 , Δt [Ai (zi − yi ) + Bi (wi − vi )] − (zi+1 − yi+1 ) + (zi − yi ) = 0 , ∂ξ(xN ) (zN − yN ) = 0 , ∂x

(31) (32) (33)

where we have used the short-hand notation Ai = ∂f /∂x(yi , vi ) and Bi = ∂f /∂u(yi , vi ). As in the case of the feasibility phase, the “more optimal” point (zπ , wπ ) is also required to be “not so far”, in some sense, from the point (yπ , vπ ) within the set Ω. A condition to meet this requirement will be given later. This condition will involve not only the “more optimal” point (zπ , wπ ) but also the Lagrange multipliers, which are defined below.


12

In order to find (zπ , wπ ), one minimizes the Lagrangian L(zπ , wπ , σ, λπ ) constrained to the tangent plane (31)-(33) as well as to Ω. Now for all (zπ , wπ ) ∈ Ω, we define the Lagrangian function for the optimality phase by π , wπ , σ ˜ , μπ ) := L(zπ , wπ , σ, λπ ) L(z + (μT0 − λT0 ) (z0 − y0 ) +

N −1

(μTi+1 − λTi+1 ) {Δt [Ai (zi − yi) + Bi (wi − vi )]

i=0

σT − σT ) −(zi+1 − yi+1 ) + (zi − yi)} + ( = Δt

N −1

f0 (zi , wi) +

λT0 (z0

0

−x )+

i=0

N −1

∂ξ(xN ) (zN − yN ) ∂x

λTi+1 hi+1 (zi , zi+1 , wi ) + σ T ξ(zN )

i=0

+ (μT0 − λT0 )(z0 − y0 ) + φ(zN ) +

N −1

(μTi+1 − λTi+1 ){Δt [Ai (zi − yi ) + Bi (wi − vi )]

i=0

− (zi+1 − yi+1 ) + (zi − yi )} + ( σT − σT )

∂ξ(xN ) (zN − yN ) , ∂x

∈ IRn . The ( σ −σ) and (μi −λi ) are the Lagrange mulwhere μπ = (μ0 , ..., μN ) ∈ IRn(N +1) and σ tipliers corresponding to the linear constraints (31)-(33). The necessary condition of optimality for this subproblem can be expressed as ˜ G(zπ , wπ , σ , μπ , λπ , vπ , yπ , σ) = P (zπ , wπ ) − ∇L(zπ , wπ , σ , μπ ) − (zπ , wπ ) = 0 . (34) 1 (zi , wi, σ In order to expand Equation (34), we define the vectors G , μi , λi, vi , yi , σ) ∈ IRn and i m 2 (zi , wi, σ G , μi , λi, vi , yi , σ) ∈ IR as follows. i μ0 + μ1 ∂L ∂f0 T ∂f T T + λ1 (z0 , w0 ) + (z0 , w0 ) + (μ1 − λ1 ) A0 , = z0 − = z0 − Δt ∂zi Δt ∂z ∂z ∂L ∂f0 ∂f 1 T T T (zi , wi) + (zi , wi ) + (μi+1 − λi+1 ) Ai , Gi = zi − = zi − Δt μi + λi+1 ∂zi ∂z ∂z ∂f ∂ L ∂f 0 2 T T T = wi − G = wi − Δt λi+1 (zi , wi) + (zi , wi ) + (μi+1 − λi+1 ) Bi , i ∂wi ∂w ∂w ˜ ∂φ(zN ) ∂L 1 T ∂ξ(zN ) + , GN = zN − = zN − σ ˜ ∂zN ∂x ∂x 1 G 0

where μi = (μi+1 − μi )/Δt. We also define the Pi2 (zi , wi , σ , μi, λi , vi , yi, σ) as follows. ⎧ 1 − zij , G ⎪ ⎪ ⎨ ij Pij1 = aj − zij , ⎪ ⎪ ⎩ aj − zij ,

vector functions Pi1 (zi , wi, σ , μi , λi, vi , yi , σ) and 1 ≤ aj , if aj ≤ G ij 1 > aj , if G ij 1 < aj , if G ij


13

⎧ 2 − wik , if bj ≤ G 2 ≤ bk , G ⎪ ik ⎪ ⎨ ik 2ij > bk , Pik2 = bk − wik , if G ⎪ ⎪ ⎩ 2 > bk , bk − wik , if G ij

and

for j = 1, . . . , n and k = 1, . . . , m. Now for i = 0, . . . , N − 1, i (zi , wi , σ G , μi, λi , vi , yi , σ) = (Pi1(zi , wi, σ , μi , λi, vi , yi, σ), Pi2(zi , wi, σ , μi , λi, vi , yi , σ)) , N (zN , σ G ) = (PN1 (zN , σ ), 0) , and so 0 (z0 , w0 , σ π , wπ , σ , μπ , λπ , vπ , yπ , σ) = (G , μ0 , λ0 , v0 , y0 , σ) , G(z 1 (z1 , w1 , σ N (zN , σ G , μ1 , λ1 , v1 , y1 , σ), . . . , G )) . Birgin and Mart´ınez [10] proved local convergence of the IR method to a local solution, if a set of conditions that they devised were satisfied. We adopt their conditions to Problem (PIR). We say that an IR iteration, starting from (xπ , uπ , σ, λπ ) which is sufficiently close to the local solution (x∗π , u∗π , σ ∗ , λ∗π ), can be completed if one can compute (yπ , vπ ), (zπ , wπ ) ∈ Ω, ∈ IRn such that it satisfies the following conditions. μπ ∈ IRn(N +1) and σ h(yπ , vπ ) ≤ θ h(xπ , uπ ) ,

(35)

yπ − xπ + vπ − uπ ≤ K1 h(xπ , uπ ) ,

(36)

∇h(yπ , vπ )(zπ − yπ , wπ − vπ ) ≤ K2 G(yπ , vπ , σ, λπ )2 ,

(37)

˜ π , wπ , μπ , yπ , vπ , σ, σ , λπ ) ≤ η G(yπ , vπ , σ, λπ ) , G(z

(38)

zπ − yπ + wπ − vπ + μπ − λπ + σ − σ ≤ K3 G(yπ , vπ , σ, λπ ) ,

(39)

where θ, η ∈ [0, 1[, K1 , K3 > 0, K2 ≥ 0 and · is any norm in the relevant finite dimensional space. Note that Conditions (37) and (38) are the inexact versions of Conditions (30) and (34), respectively. The “more optimal” point (zπ , wπ , σ ¯ , μπ ) will remain close (in the sense given in (39)) to the “more feasible” point (yπ , vπ , σ, λπ ) by applying (39). Kaya and Mart´ınez [1] proved that, for the unconstrained optimal control problems with free terminal state, the more feasible point (yπ , vπ , σ, λπ ) will remain in the neighbourhood of the current point (xπ , uπ , σ, λπ ). This property holds for Problem (PIR) in this paper if one poses a bound on vπ − uπ . We express this property by the following lemma. Lemma 4.2 Suppose that Assumptions (A1) and (A2) hold. If Condition (35) is satisfied and there exists a positive constant Kc such that vπ − uπ ≤ Kc h(xπ , uπ ) ,

(40)

then Condition (36) holds for all (xi , ui ) and (yi , vi ) sufficiently close to the continuous-time solution (x∗ (ti ), u∗(ti )), i = 0, . . . , N − 1. Proof. The functions f (x(t), u(t)) and ξ are Lipschitz continuous, which means that there exists K > 0 such that for all (xπ , uπ ) and (yπ , vπ ) f (yπ , vπ ) − f (xπ , uπ ) ≤ K (yπ − xπ + vπ − uπ ) ,


14

and ξ(yN ) − ξ(xN ) ≤ K yN − xN . Condition (35) implies that h(yπ , vπ ) ≤ θh(xπ , uπ )) ≤ h(xπ , uπ ) . For i = 1, . . . , N, rewrite the equality constraints of Problem (PIR) as follows. xi = xi−1 + Δt f (xi−1 , ui−1) + hi (xi−1 , xi , ui−1) , and yi = yi−1 + Δt f (yi−1, vi−1 ) + hi (yi−1 , yi, vi−1 ) . So yi − xi = yi−1 − xi−1 + Δt (f (yi−1, vi−1 ) − f (xi−1 , ui−1)) + ri , where ri = hi (yi−1 , yi , vi−1 ) − hi (xi−1 , xi , ui−1 ). Therefore, yi − xi ≤ yi−1 − xi−1 + Δt K (yi−1 − xi−1 + vi−1 − ui−1) + ri , or yi − xi ≤ (1 + Δt K) yi−1 − xi−1 + Δt K vi−1 − ui−1 + ri .

(41)

Now define r0 = h0 (x0 ) − h0 (y0 ). So, r0 = h0 (x0 ) − h0 (y0 ) = y0 − x0 .

(42)

Also define rN +1 = hN +1 (yN ) − hN +1 (xN ), to get rN +1 = hN +1 (yN ) − hN +1 (xN ) = ξ(yN ) − ξ(xN ) . By using (42), the inequality (41) for i = 1 yields y1 − x1 ≤ (1 + Δt K) r0 + Δt K v0 − u0 + r1 .

(43)

Similarly, by using (43), the inequality (41) for i = 2 gives y2 − x2 ≤ (1 + Δt K)2 r0 + (1 + Δt K) r1 + r2 + Δt K [(1 + Δt) v0 − u0 + v1 − u1 ] . Proceeding inductively, one gets yi − xi ≤ (1 + Δt K)i r0 + (1 + Δt K)i−1 r1 + . . . + (1 + Δt K) ri−1 + ri + Δt K (1 + Δt K)i−1 v0 − u0 + . . . + (1 + Δt K) vi−2 − ui−2 + vi−1 − ui−1 ] . Adding the term vi − ui to both sides, and using the fact that (1 + Δt K)N ≥ (1 + Δt K)k for all k = 1, . . . , N, yi − xi + vi − ui ≤ (1 + Δt K)N (r0 + r1 + . . . + rN ) + (1 + Δt K)N (v0 − u0 + v1 − u1 + . . . + vN −1 − uN −1 ) ≤ N (1 + Δt K)N sup rj + N (1 + Δt K)N sup vj − uj 1≤j≤N +1

1≤j≤N −1

≤ N (1 + Δt K)N rj L∞ + N (1 + Δt K)N vj − uj L∞


15

This is valid for any i. So, noting that rπ ∞ = h(yπ , vπ ) − h(xπ , uπ )L∞ ≤ 2 h(xπ , uπ )L∞ , and by using (40), we get yπ − xπ L∞ + vπ − uπ L∞ ≤ (2 + Kc ) N (1 + Δt K)N h(xπ , uπ )L∞ , which completes the proof.

(44) 2

Remark 4.1 Inequality (44) implies that the coefficient K1 in Inequality (36) depends on the Lipschitz constant, K, the number of subdivisions, N, and the constant Kc as defined in (40). Note that, as Δt → 0, (1 + Δt K)N → e(tf −t0 )K , and that (1 + Δt K)N is increasing in N. Therefore, if K1 ≥ (2 + Kc ) N e(tf −t0 )K , then (36) is satisfied, for any N. In summary, Lemma 4.2 states that, in the feasibility phase, if the “more feasible” point vπ is not “too far” from uπ (in the sense described by (40)), then, automatically, the “more feasible” point yπ will not be “too far” from xπ . Now we present the result which guarantees local convergence of a solution of the IR method to a local solution of Problem (PIR), as in [1]. Theorem 4.1 Suppose that Assumptions (A1)-(A6) hold and Conditions (35) and (37)-(40) are satisfied. Consider all (xπ , uπ ) ∈ Ω such that xπ −x∗ ≤ δ and uπ −u∗ ≤ ε for small and positive constants δ and ε. Then for all sufficiently small Δt > 0, the sequence of IR iterates (j) (j) (j) {xπ , uπ , σ (j) , λπ } is locally convergent to the local solution {x∗π , u∗π , σ ∗ , λ∗π } of Problem (PIR). Furthermore, if θ = η = 0, convergence of the iterates is r-quadratic. Proof. By the hypotheses of the theorem and Lemma 4.2, (35)-(39) are satisfied, and so the hypotheses of Theorem 3 in [10] hold, yielding the first conclusion of the theorem. The second conclusion is provided by Theorem 5 in [10]. 2 Corollary 4.1 Suppose Assumptions (A1)-(A6) hold and Conditions (35) and (37)-(40) are satisfied. As Δt → 0, a local solution (x∗π , u∗π , σ ∗ , λ∗π ) of Problem (PIR) found by the IR method will converge to a local solution (x∗ , u∗ , σ ∗ , λ∗ ) of the original problem (P), where λ∗ is the adjoint variable for the direct adjoining approach. One obtains the adjoint variable for the ∗ ∗ indirect adjoining approach as ψ ∗ = λ∗ + ν 1 − ν 2 . Proof. The conclusion follows from Theorem 4.1, Corollary 3.1, and (19).

5

2

IR Algorithm

In [10], Birgin and Mart´ınez implement the IR method in an algorithm, where globally convergent sofware is employed for solving the subproblems in the feasibility and optimality phases of the IR method. Furthermore, they test their algorithm against the popular solver Lancelot. They show that the IR algorithm exhibits a more robust behaviour than that of Lancelot. We adopt their approach and implement the IR method for optimal control problems in an algorithm similar to that in [10]. The IR Algorithm we describe further below uses π , wπ , vπ , μπ , λ, σ, σ G(z ) = P((xπ , uπ ) − ∇L(xπ , uπ , λπ , σ)) − (xπ , uπ ) ,


16

where P is the Euclidian projection operator onto the box n m 3 max {1, (yπ , uπ )∞ } , Ω ∩ {(zπ , wπ ) ∈ (IRN , IRN ) | zπ − yπ ∞ + wπ − uπ ∞ ≤ K

in any IR iteration. IR Algorithm Step 0 (Initialization) Set the inexact feasibility parameter 0 < θ < 1, inexact optimality parameter 0 < η < 1, the other algorithmic parameters K1 , K2 , K3 , and set k = 0. Set a small stopping criterion parameter > 0, as well as the accuracy parameters f > 0 and (0) (0) (0) o > 0. Take the initial guesses xπ , uπ , σ (0) and λπ . Step 1 (Feasibility phase) Step 1.0 Solve approximately (with accuracy f ) the minimization problem ⎧ h(yπ , vπ )2 min ⎪ ⎪ ⎨ (yπ ,vπ )∈Ω (k) (k) (k) (Pfeas) subject to yπ − xπ ∞ ≤ K1 h(xπ , uπ ) , ⎪ ⎪ ⎩ (k) 1 h(xπ(k) , uπ(k) ) , vπ − uπ ∞ ≤ K (k)

(k)

to obtain (yπ , vπ ) . (k)

(k)

Step 1.1 If the approximate solution (yπ , vπ ) of Problem (Pfeas) satisfies h(yπ(k) , vπ(k) ) ≤ max {, θh(xπ(k) , uπ(k))} ,

(45)

then go to Step 2; otherwise set f = f /10 and go to Step 1.0. If not able to find such an approximate solution which satisfies (45), stop and declare: ”Failure at the feasibility phase.” (k)

(k)

Step 2 (Stopping criteria) If the approximate solution (yπ , vπ ) satisfies h(yπ(k), vπ(k) )∞ ≤ and G(yπ(k) , vπ(k) , λπ(k) , σ (k) )∞ ≤ , then stop the execution of the algorithm, and declare that the quadruplet (k) (k) (k) (yπ , vπ , λπ , σ (k) ) is an approximate solution of the problem. Step 3 (Optimality Phase) Step 3.0 Solve approximately (with accuracy o ) the problem

(Popt)

⎧ (k) min L(zπ , wπ , λπ , σ (k) ) ⎪ ⎪ ⎨ (zπ ,wπ )∈Ω (k) (k) (k) (k) subject to ∇h(yπ , vπ ) (zπ − yπ , wπ − vπ ) = 0 , ⎪ ⎪ ⎩ (k) (k) 3 max {1, (yπ(k), vπ(k) )∞ }, zπ − yπ + wπ − vπ ≤ K (k)

σ − σ (k) ) ∈ IRn are the Lagrange multipliers where (μπ − λπ ) ∈ IR(N +1)n and (˜ associated with the equality constraints of Problem (Popt).

17


Step 3.1 If the approximate solution (zπ , wπ , σ ˜ , μπ ) of Problem (Popt) satisfies the inequalities ∇h(yπ(k) , vπ(k) ) (zπ − yπ(k), wπ − vπ(k) ) ≤ max {, K2 G(yπ(k), vπ(k) , λπ(k) , σ)2∞ }, ˜ π , wπ , v (k) , μπ , λ(k) , σ, σ G(z ˜ ) ≤ max {, η G(yπ(k) , vπ(k) , λπ(k) , σ)∞ } , π (k+1)

(k+1)

(k+1)

then set xπ = zπ , uπ = wπ , σ = σ and λπ set f = f /10 and go to Step 3.0.

(46)

= μπ and go to Step 1; otherwise

If not able to find such an approximate solution which satisfies the conditions (46), stop and declare: ”Failure at the optimality phase.”

6

Numerical Experiments

In this section, we solve two computationally challenging optimal control problems from the literature by using the IR Algorithm we have described in the preceding section. For coding the algorithm we use AMPL [19], which is a modelling language for general finite-dimensional optimization problems. For solving the subproblems (Pfeas) and (Popt) in the IR algorithm, we employ (interfaced with AMPL) the optimization software Ipopt [20], version 3.2.4s. We also solve the same optimal control problems by using Ipopt on its own (again interfaced with AMPL), in order to make comparisons between the computational performances of this conventional approach, i.e. Ipopt on its own, and the IR algorithm employing Ipopt in its feasibility and optimality phases.

6.1

Container Crane

Container cranes are used to handle cargo in ship or rail road terminals. Optimal control of container cranes have been widely studied; see, for example, [23-26]. In order to improve the efficiency of crane operations, Sakawa and Shindo [23] developed a dynamical control model and other authors [24-26] used this model as a test problem for numerical optimal control algorithms. Augustin and Maurer [24] presented a numerical solution to this problem which satisfied optimality conditions with a high accuracy. The problem studied in the aforementioned references is briefly described as follows. The container crane is equipped with a trolley drive motor and a hoist motor, which independently generate torque to actuate the crane; see Figure 1. The aim of the control process is to keep the swing angle as small as possible since a large swing of the container load during the transfer may become dangerous. The critical part of the displacement is the diagonal displacement from point B to point C as shown in Figure 2, where the vertical displacement is connected with the horizontal displacement. The model is described by means of six state and two control variables: x1 : horizontal displacement, x2 : vertical displacement, x3 : swing angle,

x4 : horizontal velocity, x5 : vertical velocity, x6 : swing (angular) velocity,

u1 : control via trolley drive motor,

u2 : control via hoist motor.


u2

18

hoist b2 motor

trolley drive motor u1

b1 m

M

Figure 1: A schematic diagram of the container crane. The dynamical model, boundary conditions and control and state constraints are considered in the time interval [t0 , tf ] with fixed final time tf > 0 : x˙ 1 (t) = x4 (t) , x˙ 2 (t) = x5 (t) , x˙ 3 (t) = x6 (t) , x˙ 4 (t) = u1 (t) + 17.2656 x3(t) , x˙ 5 (t) = u2 (t) , x˙ 6 (t) = −

1 (u1 (t) + 27.0756 x3(t) + 2 x5 (t) x6 (t)) , x2 (t)

x(0) = (0, 22, 0, 0, −1, 0) , |x4 (t)| ≤ 2.5 ,

x(tf ) = (10, 14, 0, 2.5, 0, 0) ,

(47) (48)

|x5 (t)| ≤ 1 ,

|u1(t)| ≤ 2.83374 ,

−0.80865 ≤ u2 (t) ≤ 0.71265 .

(49)

The boundary conditions in (48) correspond to the points B and C in Figure 2. Note that the load arrives at point B with maximum velocity, which gives the initial condition x5 (0) = −1 at point B. Thus, the state constraint |x5 (t)| ≤ 1 becomes active at t0 = 0. The goal of the control process is to keep the swing angle and velocity small. The final time tf is chosen to be 9 sec for this example. Augustin and Maurer [24] consider the minimization of the functional, 1 9 2 F (x, u) = x3 (t) + x26 (t) + c (u21 (t) + u22 (t)) , 2 0

(50)

with c = 0.01, in order to keep the swing angle and velocity small. We choose to minimize the same functional as in (50). The numerical results, obtained by applying the IR algorithm, and Ipopt on its own (i.e., Ipopt alone), are listed in Table 1. In order to compare the robustness of the two approaches,


d1

l3

B l1

d3

d2

l2

0 1 0x5 = x5 max 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0x5 = 0 1

19

l4 D1 C1 0 0 0 1 0 1 1111111111 0000000000 x3 = 0 x3 = 0 x4 = x4 max x4 = x4 max x x5 = 0 5 = 0 0 1 0 x4 = 0 E 1 x5 = x5 max l5 00 F11 00 11 x5 = 0

A

11 00 111 000 1 0 00 11

Figure 2: The diagonal motion from B to C of the container crane, with l2 = 22 m, l3 = 14 m and d1 = 10 m. both the IR algorithm and Ipopt alone are run 100 times, and convergence from randomly generated initial guesses in each run to an optimal solution is examined. More precisely, the initial guesses for the unknowns xji and urj are generated uniformly in given intervals, such that and urj ∈ [−0.8, 0.7] , (51) xpi ∈ [0, 1] for p = 1, . . . , 6, r = 1, 2, i = 0, . . . , N, j = 0, . . . , N − 1. Each approach is repeated 100 times with these (random) initial guesses. We set the IR (algorithmic) parameters to be 1 = 103 , K 3 = 0.3, K2 = 106 and = 10−6 . Additionally, η = 0.9, θ = 0.9, K1 = 106 , K 3 in Table 1, which are calculated from we show the average values (over 100 runs) of K1 and K (1) (2) (k) (1) 3 = max {K 3 , K 3(2) , ..., K 3(k) }, where k is the maximum K1 = max {K1 , K1 , ..., K1 } and K (i) 3(i) , i = 1, ..., k, are obtained by using the formulas number of IR iterations and K1 and K (i)

K1

(i)

=

(i−1)

yπ − xπ

(i−1)

h(xπ (i)

(i)

(i)

(i−1)

∞ + vπ − uπ (i−1)

, uπ

(i)

∞

,

)∞ (i)

(i) = zπ − yπ ∞ + wπ − vπ ∞ , K 3 (i) yπ ∞ where the value of the variable at the current iteration of IR is shown by superscript (i). Because the IR algorithm is run 100 times, the success rate (or percentage) of the IR approach in Table 1 is simply the number of times IR has found a solution. The success rate for the approach using Ipopt alone is computed similarly. As can be seen from Table 1, the success rate of the IR algorithm becomes much higher than that of Ipopt alone as N increases. Thus the IR algorithm exhibits a better performance in terms of randomly generated initial guesses, or in terms of robustness, for this example. Table 2 is generated similarly to Table 1, with the


20

difference that a wider range for random initial guesses are used, such that x1i ∈ [−10, 10] , x4i ∈ [−2.5, 2.5] , u1j ∈ [−2.8, 2.8] ,

x2i ∈ [−15, 15] , x3i ∈ [−0.1, 0.1] , x5i ∈ [−1, 1] , x6i ∈ [−0.01, 0.1] , u2j ∈ [−0.8, 0.7] ,

(52)

for i = 0, . . . , N, j = 0, . . . , N − 1. These intervals are roughly consistent with the ranges of the minimum and maximum values of the control and state variables. We observe that with these wider initial guesses, the IR algorithm appears to be even more successful. In Tables 1 and 2 we also report the average elapsed CPU times as well as the average number of IR iterations, both over the number of successful runs out of 100 attempts. CPU time [sec] N IR Ipopt 100 2.1 1.2 500 5.7 18.0 1000 10.8 75.0 1500 16.3 121.4

IR param. Cost K1 K3 IR, Ipopt 3129 0.08 0.036951 4481 0.09 0.037375 5689 0.09 0.037445 31094 0.09 0.037469

# of IR iter’ns 4 4 4 4

Success IR 100 100 100 100

rate [%] Ipopt 100 100 91 50

Table 1: Numerical results for the IR algorithm and Ipopt for the container crane, with c = 0.01 and initial guesses generated uniformly as in (51). CPU time [sec] N IR Ipopt 100 7.5 2.4 500 281.6 64.8 1000 4789 -

IR param. Cost # of IR 3 IR, Ipopt iter’ns K1 K 559.8 0.08 0.036950 4.1 4451 0.08 0.037375 7.2 5729 0.09 0.037445 13.6

Success rate [%] IR Ipopt 100 100 63 53 20 0

Table 2: Numerical results for the IR algorithm and Ipopt for the container crane, with c = 0.01 and initial guesses generated uniformly as in (52). Figures 3, 4 and 5 depict the graphs for the optimal state, costate and control variables of the container crane example with 1000 discretization points. These graphs are in agreement with those in [24], except the graph of λ5 in Figure 4. In other words, ψi = λi , for i = 1, 2, 3, 4, 6, but ψ5 = λ5 . Here, λ5 is the multiplier of the fifth ODE in the direct adjoining approach. Never the less, one can still obtain the multiplier ψ5 of the fifth ODE in the indirect adjoining approach, by using Remark 3.1 and the transformation in (19). In Figure 6, we provide the resulting graphs of (i) the multiplier μ 5 of the state constraint −x5 − 1 ≤ 0 in the direct adjoining approach, (ii) the multiplier ν5 of the (indirect) state constraint −x˙ 5 ≤ 0 in the indirect adjoining approach and (iii) the multiplier ψ5 , found by using (19). Now, the graph of the multiplier ψ5 does agree with that presented in [24]. Moreover, the necessary condition, ν(t) ˙ ≥ 0, as well as the fact that ν5 (t) is constant whenever −x5 (t) − 1 ≤ 0 is not active, is verified by the graph of ν5 . It should be noted that the graph of the discretized μ 5 has been obtained by dividing the multiplier in the Ipopt-alone implementation, by Δt. Otherwise, the multiplier μ 5 cannot be obtained using the IR algorithm, because all box constraints are embedded into the set Ω, and so, no multipliers are employed for these constraints.


21

It is straightforward to check that Assumption (A4) is satisfied: The matrix in (10) is simply ∂f5 /∂u(x∗ , u∗ ) = [0 1], whose rank is 1, obtained when −x5 − 1 ≤ 0 is the only active state constraint. Note that the second derivative of the Hamiltonian function, Huu [t] = c I2 , where I2 is the 2 × 2 identity matrix, is positive definite, i.e., the strict Legendre condition holds. Therefore, Assumption (A6) can be verified. This has been done in [24] by producing a bounded solution to an associated Riccati equation in [0, tf ], which asserts sufficiency for optimality. Note also that, formally, Assumption (A1) is not fully satisfied, because one of the state constraints is active at t = 0, namely that −x5 (0) + 1 = 0. However, this does not seem to affect the overall results. By setting c = 0 in the cost functional (50), one gets a much harder problem to solve. In this case, the control constraints (49) become active; moreover, we observe from the solution graphs that both control variables become discontinuous. The optimal control variables computed are of bang–bang and singular type, whose graphs are depicted in Figure 9. For such problems, where control variables are discontinuous, convergence theory for discretization does not exist, except in some special cases; see, for example, [27]. Then, it is common practice to use methods tested for continuous controls for finding, at least an estimate of the structure of, discontinuous (bang–bang and singular) controls. That is what has been done in [24], for example. Once the structure of the discontinuous controls are “estimated”, it is again common practice to parameterize the switching times and find the times at which control has discontinuities (or jumps) by shooting-type techniques developed for this purpose, for example, those techniques presented in [16-18]. For the case when c = 0, i.e., when optimal controls are discontinuous, we apply the IR algorithm, as well as Ipopt on its own. Figures 7-9 present the graphs of optimal state, costate and control variables which have been obtained by taking 1000 discretization points. Note again that here we also have that ψi = λi , for i = 1, 2, 3, 4, 6, but ψ5 = λ5 , which we will address in the same way as we did in the case of c = 0.01. We have used the same values for the IR parameters as in the case of c = 0.01. The random initial guesses have been generated uniformly as in (51). Table 3 illustrates the performance. The IR approach is again seen to be more robust, i.e., the success rate is higher, compared to Ipopt alone. Note that, in this case, the average CPU times required for the IR solutions (of course, only counting the successful runs) are also less than those for Ipopt. CPU time [sec] N IR Ipopt 100 2.1 1.3 500 6.7 16.8 1000 9.1 66.6 1500 15.5 108.3

IR param. Cost # of IR K1 K3 IR, Ipopt iter’ns 239.3 0.12 0.005008 4 1999 0.13 0.005117 4 2173 0.14 0.005134 4 3470 0.14 0.005139 4

Success IR 100 100 100 98

rate [%] Ipopt 100 100 78 53

Table 3: Numerical results for the IR algorithm and Ipopt for container-crane, with c = 0 and initial guesses generated uniformly as in (52). In Figure 9, the control u1 (when singular) appears to have jump discontinuities at the switching times of the control u2. This can be explained by examining the switching function σ1 for u1 : Note that the minimization of the Hamiltonian function can simply be written as min H =

b1 ≤u1 ≤b1

min σ1 (t) u1 ,

b1 ≤u1 ≤b1

22


10

22 20

x1

5

18

x2

16 0 0

5

14 0

10

t

5 t

10

5 t

10

3

0

2

−0.02

x4

x3

1

−0.04 −0.06 0

5

0 0

10

t

0

0.04

x6

x5 −0.5

0.02 0

−1 0

5 t

10

−0.02 0

5

10

t

Figure 3: Optimal state variables for the container crane problem, with c = 0.01, N = 1000.

23


−3

2

3

x 10

2

1

λ2

λ1

1

0 −1 0

5

0 0

10

5

10

t t

0.2

λ3

−0.01 −0.02

0

λ4

−0.03 −0.2 0

5

10

−0.04 0

5

t

t

−3

5

10

−0.1

x 10

−0.2 0 λ5

λ6

−5 −10 0

−0.3 −0.4

5 t

10

−0.5 0

5

10

t

Figure 4: Optimal costate variables for the container crane problem, with c = 0.01, N = 1000.

24


1

1.5

0.5

1

u1

u2

0

0.5 0 0

5

−0.5 0

10

t

5

10

t

Figure 5: Optimal control variable for container-crane problem, with c = 0.01, N = 1000.

−3

2.5

x 10

0.01

2

μ 25

0.005

1.5

ψ5

1 0.5

0 −0.005

0 0

5

10

−0.01 0

5

10

t

t 0 −0.002

ν52

−0.004 −0.006 −0.008 −0.01 0

5

10

t Figure 6: Multipliers for the container crane problem, with c = 0.01, N = 1000.


25

22

10

20

x1

x2 18

5

16 0 0

5

10

14 0

t

5

10

t 3

0

2

−0.02

x4

x3 −0.04

1

−0.06 0

5

0 0

10

t

5

10

t 0.06

0

0.04

x6

x5 −0.5

0.02 0

−1 0

5

10

−0.02 0

t

5

10

t

Figure 7: Optimal state variables for container-crane problem, with c = 0, N = 1000. where σ1 (t) := ψ4 − ψ6 /x2 . One has a singular arc over a time interval [t1 , t2 ], if, and only if, σ1 (t) = 0 for a.e. t ∈ [t1 , t2 ]. Then over [t1 , t2 ], σ˙ 1 (t) ≡ 0. Similarly, σ¨1 (t) ≡ 0, in which both u1 and u2 appear. Solving for the singular u1 , one gets u1 = −ψ6 u2 + g(x, ψ) . This expression points that the singular u1 might have jumps at points where u2 has jumps. As in the case of c = 0.01, we provide the graphs of the multipliers μ 5 , ν5 and ψ5 , in Figure 10. Now, ψ5 for c = 0 also agrees with that presented in [24]. The graph of ν5 again verifies the necessary conditions, ν(t) ˙ ≥ 0, and the fact that ν5 is constant whenever −x5 (t) − 1 < 0.

6.2

Free-Flying Robot

The second example is the Free-Flying Robot (FFR) problem, studied earlier by Sakawa [28] and Vossen and Maurer [29]. In this problem, a robot is moved at a constant height from an


2

−4

2

x 10

1

λ1

0

λ2

0

−2

−1 0

5

10

−4 0

5

t

t

−3

0.1

0

0.05

−2

0

λ4 −4

−0.05

−6

λ3

10

−0.1 0

5

x 10

−8 0

10

t

5

10

t

−4

2

λ5

x 10

−0.02 −0.04

0

λ6 −0.06 −2 −0.08 −4 0

5

−0.1 0

10

5

t

10

t

Figure 8: Optimal costate variables for container-crane problem, with c = 0, N = 1000.

3 1

2

u1 1

u2

0

0 −1 0

5

10

−1 0

5

10

t Figure 9: Optimal control variable for container-crane problem , with c = 0, N = 1000.

26

27

Inexact Restoration for Euler Discretization of Constrained Optimal Control Problems by N. Banihashemi & C. Y. Kaya −4

1

−4

x 10

2 1

0.8

μ 25

x 10

0.6

ψ5

0

0.4

−1

0.2

−2

0 0

5

10

−3 0

5

10

t

t −4

0

x 10

−0.2

ν52

−0.4 −0.6 −0.8 −1 0

5

10

t Figure 10: Multipliers for the container crane problem, with c = 0, N = 1000. initial to a final equilibrium position. The robot can be controlled by the thrust of two jets. The planar motion of the FFR is depicted in Figure 11, where we use x1 and x2 for the co-ordinates of the FFR, x4 and x5 for the corresponding velocities, x3 for the direction of thrust, x6 for the angular velocity, and u1 and u2 for the thrusts of the two jets. This example is a constrained optimal control problem with box constraints on the control variable. Vossen and Maurer [29] formulate this problem as an Lp -minimization problem and they report numerical solutions with a high accuracy that satisfy relevant optimality conditions. As a test example, we consider the case of the regularized L2 -minimal control because u1 and u2 should be continuous in order for us to apply the IR algorithm. In [29], the FFR model is described with the box constraint ui ∈ [−1, 1], i = 1, 2. The optimal control solutions computed with these constraints for the L2 -minimization case all take values in the interior of the intervals; in other words, the control constraints never become active. Therefore, we tighten the control bounds to make the problem more challenging for both the IR algorithm and Ipopt on its own. So, the FFR problem we aim to study is described as follows. Minimize the cost tf (u21 (t) + u22 (t))dt , 0


28

x2 (x 4 , x 5 ) u1 + u2

u2 x3

P

u1

x1

Figure 11: The Free-Flying Robot subject to x˙ 1 (t) = x4 (t) , x˙ 2 (t) = x5 (t) , x˙ 3 (t) = x6 (t) , x˙ 4 (t) = (u1 (t) + u2 (t)) cos x3 (t) , x˙ 5 (t) = (u1 (t) + u2 (t)) sin x3 (t) , x˙ 6 (t) = α (u1 (t) − u2 (t)) , x(0) = (−10, −10, π/2, 0, 0, 0) , x(tf ) = (0, 0, 0, 0, 0, 0) , |u1(t)| ≤ 0.8 , |u2 (t)| ≤ 0.4, with tf = 12 and α = 0.2. To solve this example with both the IR algorithm and Ipopt alone, we start with random initial guesses generated uniformly in given intervals, such that xpi ∈ [−0.4, 0.4]

and

urj ∈ [−0.4, 0.4] ,

(53)

for p = 1, . . . , 6, r = 1, 2, i = 0, . . . , N, j = 0, . . . , N − 1. The performance of each approach, run 100 times, with randomly generated initial guesses, is presented in Table 4. As can be seen from the table, with increasing number of discretization points, Ipopt on its own becomes much less successful in finding an optimal solution compared to IR, although IR takes slightly longer time to find a solution when Ipopt alone can also find a solution. Of course, a much higher chance of finding a solution is more desirable than finding a solution (when it can) more quickly. Figures 12 and 13 depict the graphs for the optimal state, costate and control variables of the free-flying robot example with 1000 discretization points.


5

2

29

x3

0

x6

1 −5 x

−15 0

0

1

−10

x2

5

10

15

−1 0

t

5

10

15

t

3 2

λ1

−0.3505

x4 x5

λ2

−0.351

1 −0.3515

0 −1 0

5

10

−0.352 0

15

5

t

t

20

10

15

5

λ3 λ6

0

0 λ4

−20 0

5

t

10

−5 0

15

λ5

5

10

15

t

Figure 12: Optimal state and costate variables for the Free-Flying-Robot problem (N = 1000).

1

u1 u2

0

−1 0

5

10

15

t

Figure 13: Optimal control variables for the Free-Flying-Robot problem (N = 1000).


CPU time [sec] N IR Ipopt 100 3.1 0.63 500 5.7 1.6 1000 8.5 4.3 1500 12.2 6.6

IR param. Cost # of IR K3 IR, Ipopt iter’ns K1 341.2 0.05 6.160844 4.1 3190 0.05 6.154438 4.3 3570 0.05 6.154242 4 3789 0.05 6.154193 4

30

Success rate [%] IR Ipopt 100 92 100 36 100 7 100 4

Table 4: Numerical results for the IR algorithm and Ipopt for the Free-Flying Robot, with initial guesses generated uniformly as in (53).

7

Discussion and Conclusion

The IR method for Euler discretization of optimal control problems has been described and its convergence studied. An implementation of the IR method in an algorithm has been shown to be robust via numerical experiments. In particular, with randomly generated initial guesses, it has been observed that one is better-off using IR employing Ipopt for solving its subproblems rather than using Ipopt on its own. Future work should consider employing the same approach for the more general Runge-Kutta discretization as in [2]. The IR method has been further studied in recent years. An IR method making special use of the structure of bi-level optimization problems was studied in [30], and a recent application of an IR method where the structure of the specific problem is exploited has appeared in [31]. Another IR method was used as a generalization of the spectral projected gradient method in [32]. As a recent algorithmic development, a novel general scheme incorporating a simple line search for IR methods has appeared in [33]. Future work should also consider using the algorithmic development in [33] for discretization of optimal control problems.

Appendix A Proof of Proposition 2.1 It is not difficult to show that Vα (t)Vα (t)T ζ ≥ β ζ if, and only if, Vα (t)Vα (t)T is nonsingular. This, in turn, is equivalent to the condition that Vα (t) has full row rank. First, define the submatrices Vα1 (t) := Θ(t) Uα (t) 0 and Vα2 (t) := Υ(t)B(t) 0 Tα (t) . In what follows, we will not show, whenever appropriate, dependence of variables on t, for simplicity in appearance. Furthermore, all arguments below will be made for some fixed t. Denote the (2i − 1)st row of Vα1 (t) by r2i−1 , i = 1, . . . , m. Then, clearly, r2i−1 = ei + min{0, u∗i − bi + α} em+2i−1 and

r2i = −ei + min{0, −u∗i + bi + α} em+2i ,

where ei is the ith standard basis vector in IR3m+2n . Let p2j−1 , j = 1, . . . , n, denote the (2j −1)st row of Vα2 (t). Then p2j−1

m ∂fj ∗ ∗ = (x , u ) ek + min{0, x∗j − aj + α} e3m+2j−1 ∂uk k=1


and p2j = −

31

m ∂fj ∗ ∗ (x , u ) ek + min{0, −x∗j + aj + α} e3m+2j . ∂uk k=1

Suppose that i ∈ Ju := Ju \J u . Then bi < u∗i < bi and min{u∗i − bi , bi − u∗i } > 0. Next, choose α such that 0 < α < α i := min{u∗i − bi , bi − u∗i }. Since min{0, u∗i − bi + α} = 0, min{0, −u∗i + bi + α} = 0, and em+2i−1 = em+2i , the rows r2i−1 and r2i are linearly independent. Suppose that j ∈ Jx := Jx \J x . Via similar arguments, α can be chosen such that 0 < α < βj := min{x∗j −aj , aj −x∗j }, resulting in min{0, x∗i −ai +α} = 0, min{0, −x∗i +ai +α} = 0, and thus linearly independent p2j−1 and p2j . Suppose that i ∈ J u . Then either u∗i = bi or u∗i = bi (only one side of the constraint is active at

i := max{u∗i −bi , bi −u∗i }. a given t). So max{u∗i −bi , bi −u∗i } > 0. Choose α such that 0 < α < α 1 ∗ 2 ∗ Then one of ci := min{0, ui − bi + α} and ci := min{0, −ui + bi + α} is zero and the other nonzero. Let the index set of the rows with c1i = 0 or c2i = 0 be denoted by J u0 = {k1 , . . . , kq } and the index set of the rows with c1 = 0 or c2 = 0 by J u1 = {1 , . . . , q }. Then, for i ∈ J u and ∈ J u1 , we get γiu em+ , r = ±ei + where γix = 0, and the sign of the first term depends on which side of a constraint is active. For i ∈ J u and k ∈ J u0 , we have rk = ±ei . Suppose that i ∈ J x . Proceeding similarly, we choose α such that 0 < α < β i := max{x∗i − ai , ai − x∗i }. Let the index set of the rows corresponding to the active part of the two-sided constraints be denoted by J x0 = { k1 , . . . , ks } and the index set of the rows corresponding to the inactive part of the two-sided constraints be denoted by J x1 = { 1 , . . . , s }. Then, for j ∈ J x and ∈ J x1 , we get m ∂fj ∗ ∗ (x , u ) ek + γix e3m+ , p = ± ∂uk k=1

where γ ix = 0, and, for j ∈ J x and k ∈ J x0 , we have m ∂fj ∗ ∗ (x , u ) ek . p k = ± ∂uk k=1

The rows r2i−1 and r2i for all i ∈ Ju , p2j−1 and p2j for all j ∈ Jx , rk for all k ∈ J u1 , and p k for all k ∈ J x1 , are linearly independent, because each of these rows contains a nonzero element, which is either in the diagonal of the submatrix Uα or in the diagonal the submatrix Tα . Now let us look at the the remaining rows: r , for all ∈ J u0 , have a nonzero element only in the submatrix Θ, and p , for all ∈ J 0 , have nonzero elements only in the submatrix Υ B.

x


32

Therefore, Vα has full row rank if, and only if, the block ⎡ ⎤ r 1 ⎢ ⎥ .. ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ r q ⎢ ⎥ ⎢ ············ ⎥ ⎢ ⎥ ⎢ ⎥ p

1 ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎣ ⎦ p s has linearly independent rows, which is a condition equivalent to that in (10).

2

References 1. Kaya, C. Y., Mart´ınez, J. M.: Euler discretization for inexact restoration and optimal control. J. Optim. Theory Appl. 134, 191–206 (2007) 2. Kaya, C. Y.: Inexact Restoration for Runge-Kutta discretization of optimal control problems. SIAM J. Numer. Anal. 48(4), 1492–1517 (2010) 3. Hager, W. W.: Runge-Kutta methods in optimal control and the transformed adjoint system. Numer. Math. 87, 247–282 (2000) 4. Dontchev, A. L., Hager, W. W., Malanowski, K.: Error bound for Euler approximation of a state and control constrained optimal control problem. Numer. Funct. Anal. Optim. 21(6), 653–682 (2000) 5. Dontchev, A. L., Hager, W. W.: The Euler approximation in state constrained optimal control problems. Math. Comput. 70, 173–203 (2000) 6. Malanowski, K., B¨ uskens, C., Maurer, H.: Convergence of approximations to nonlinear optimal control problems. In: Fiacco, A.V. (ed.): Mathematical Programming with Data Perturbations V (Lecture Notes in Pure and Applied Mathematics), vol. 195, pp. 253– 284. (1997) 7. Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation: Applications, Volume II, Springer, Berlin, Heidelberg (2006) 8. Mart´ınez J. M., Pilotta, E. A.: Inexact Restoration algorithm for constrained optimization. J. Optim. Theory Appl. 104(1), 135–163 (2000) 9. Mart´ınez, J. M.: Inexact Restoration method with Lagrangian tangent decrease and new merit function for nonlinear. J. Optim. Theory Appl. 111, 39–58 (2001) 10. Birgin, E. G., Mart´ınez, J. M.: Local convergence of an Inexact-Restoration method and numerical experiments. J. Optim. Theory Appl. 127(2), 229–247 (2005) 11. Kaya, C. Y., Mart´ınez, J. M.: Euler discretization for inexact restoration and optimal control. Technical Report, 2006. See URL: http://people.unisa.edu.au/yalcin.kaya or URL: http://www.ime.unicamp.br/∼ martinez/


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Inexact Restoration for Euler Discretization of Box-constrained Optimal

Inexact Restoration for Euler Discretization of Box-constrained Optimal

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