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Optimality conditions for discreteoptimal control problemsBoban Marinković aa Department of Applied Mathematics, Faculty of Mining andGeology , Djušina 7, 11000, Belgrad, SerbiaPublished online: 28 Sep 2007.

To cite this article: Boban Marinković (2007) Optimality conditions for discrete optimal controlproblems, Optimization Methods and Software, 22:6, 959-969, DOI: 10.1080/10556780701485314

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Optimization Methods and SoftwareVol. 22, No. 6, December 2007, 959–969

Optimality conditions for discrete optimalcontrol problems

BOBAN MARINKOVIC*

Department of Applied Mathematics, Faculty of Mining and Geology,Djušina 7, 11000 Belgrad, Serbia

(Received 11 November 2005; in final form 12 January 2007)

Discrete optimal control problems with variable endpoints and with equality type constraints on thecontrol are considered. We derive first- and second-order necessary optimality conditions, which aremeaningful under the assumptions weaker than those previously used in the literature.

Keywords: Discrete optimal control; Optimality conditions; Mathematical programming; 2-regularity

1. Introduction

Consider the following discrete optimal control problem:

minimizeN−1∑i=0

fi(xi, ui); (1)

xi+1 = ϕi(xi, ui), i = 0, N − 1, (2)

gi(ui) = 0, i = 0, N − 1, (3)

K(x0, xN) = 0, (4)

where

fi(x, u) : Rn × Rr −→ R, ϕi(x, u) : Rn × Rr −→ Rn,

gi(u) : Rr −→ Rm, K(x0, xN) : Rn × Rn −→ Rk

are twice continuously differentiable functions.Here xi ∈ Rn is a state variable, ui ∈ Rr is a control parameter, N is a given number of

steps. Vector ξ = (x0, x1, . . . , xN) is called a trajectory, w = (u0, u1, . . . , uN−1) is called acontrol, x0 is a starting point and xN is an end point of corresponding trajectory.

*Corresponding author. Email: mboban@verat.net

Optimization Methods and SoftwareISSN 1055-6788 print/ISSN 1029-4937 online © 2007 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/10556780701485314

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960 B. Marinkovic

Let x0 be a starting point and let w be a control. Then the pair (x0, w) defines the corre-sponding trajectory ξ = (x0, x1, . . . , xN). If the conditions (2), (3) and (4) are satisfied thenwe say that the pair (x0, w) is feasible.

The discrete optimization problem is to minimize the function

J (x0, w) =N−1∑i=0

fi(xi, ui),

on the set of feasible pairs. A feasible pair (x0, w) is called a weak local minimum for theproblem (1)–(4) if for some ε > 0, the pair (x0, w) minimizes J (x0, w) over all feasible pairs(x0, w) satisfying

||xi − xi ||n < ε, i = 0, N, ||ui − ui ||r < ε, i = 0, N − 1,

where ||·||n and ||·||r are any norms in Rn and Rr .The aim of this paper is to obtain first- and second-order necessary optimality conditions

for the problem (1)–(4) which are meaningful under the assumptions substantially weakerthen those previously used in the optimal control theory. Our results are based on the generaltheory developed in refs. [1,2].

2. First-order optimality conditions

Let (x0, w) be a feasible pair and let ξ = (x0, x1, . . . , xN ) be a corresponding trajectory.Suppose that the pair (x0, w) is optimal.

Put

∂ϕi

∂x(xi , ui) = Ci,

∂ϕi

∂u(xi , ui) = Di,

∂gi

∂u(ui) = Ei

and

∂2ϕi

∂x2(xi , ui) = C2

i ,∂2ϕi

∂u2(xi , ui) = D2

i ,∂2ϕi

∂x∂u(xi , ui) = M2

i ,∂2gi

∂u2(ui) = E2

i .

Similarly, put

∂K

∂x0(x0, xN ) = K0,

∂K

∂xN

(x0, xN ) = KN

and

∂2K

∂x20

(x0, xN ) = K20 ,

∂2K

∂x2N

(x0, xN ) = K2N,

∂2K

∂x0∂xN

(x0, xN ) = K20N.

Note that for the given linear operator L : Rn → Rm, L(x) stands for the value of the linearoperator, i.e., L(x) = L · (x1, . . . , xn)

T , where L is the corresponding matrix.

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Discrete optimal control problems 961

For a fixed vector (h, v) ∈ Rn(N+1) × RrN let us define the linear operator

N(h, v) : Rn(N+1) × RrN −→ RnN × RmN × Rk

by

N(h, v)(h, v) = (x0, . . . , xN−1, y0, . . . , yN−1, z)T

where

xk = −C2k [hk, hk] − D2

k [vk, vk] − M2k [hk, vk] − M2

k [hk, vk], k = 0, N − 1,

yk = E2k [vk, vk], k = 0, N − 1,

and

z = K20 [h0, h0] + K2

N [hN , hN ] + K20N [h0, hN ] + K2

0N [h0, hN ].

Let M(h, v) : Rn × RrN → RnN × RmN × Rk be the linear operator defined by

M(h, v)(h0, v) = N(h, v)(h0, h1, . . . , hN, v) (5)

where h1, h2, . . . , hN are given by

hk =k−1∏s=0

Csh0 +k−2∑j=0

∏j<s≤k−1

CsDjvj + Dk−1vk−1, k = 1, N. (6)

Specifically,

M(h, v)(h0, v) = N(h, v)(h0, h1, . . . , hN , v)

where h1, h2, . . . , hN are given by

hk =k−1∏s=0

Csh0 +k−2∑j=0

∏j<s≤k−1

CsDj vj + Dk−1vk−1, k = 1, N.

Let Q = {(u, v, w) | u ∈ RnN, v ∈ RmN, w ∈ Rk} be the solution set of the followingsystem of the linear equations:

−C∗0 u0 + K∗

0 w = 0, (7)

ui − C∗i+1ui+1 = 0, i = 0, N − 2, (8)

uN−1 + K∗Nw = 0, (9)

−D∗i ui + E∗

i vi = 0, i = 0, N − 1. (10)

Denote by P orthogonal projector onto the linear subspace Q ⊂ RnN × RmN × Rk .

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962 B. Marinkovic

Let Bi , i = 0, N − 1, be the matrices defined by

B0 = K0 + KN

N−1∏s=0

Cs,

Bi = KNCiCi+1 · · · CN−1Di−1, i = 1, N − 1,

BN = KNDN−1.

Let B be the linear operator given by the following block matrix

B =

⎡⎢⎢⎢⎢⎣

B0 B1 · · · BN

0 E0 · · · 0...

...

0 0 · · · EN−1

⎤⎥⎥⎥⎥⎦,

where by 0 we denote the zero matrix of the corresponding size.Let H(ξ , w) be the cone of all (h, v) satisfying the following conditions:

1.

PM(h, v)(h0, v) = 0.

2.

B(h0, v) = 0.

3. h1, h2, . . . , hN are given by (6).

Let (h, v) ∈ H(ξ , w) be some fixed vector and let L be the block matrix

L =[

B

PM(h, v)

].

Put

l = dim(ker L).

Note that dim(ker L) is the dimension of the set of all vectors (h0, v)T such that

L(h0, v) = 0.

Denote by C(ξ , w) the set of all (h, v) ∈ H(ξ , w) such that

l = n + rN − mN − k. (11)

Define the generalized Pontryagin’s functions

Hi2(x, u, p1, p2, q1, q2, h, v) : Rn × Rr × Rn × Rn × Rm × Rm × Rn × Rr −→ R,

i = 0, N − 1,

and the generalized small Lagrangian

l2(x0, xN, λ1, λ2, h0, hN) : Rn × Rn × Rk × Rk × Rn × Rn −→ R

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Discrete optimal control problems 963

by

Hi2(x, u, p1, p2, q1, q2, λ0, h, v) = 〈p1, ϕi(x, u)〉 − 〈p2, gi(u)〉 − fi(x, u)

+⟨q1,

∂ϕi

∂x(x, u)h + ∂ϕi

∂u(x, u)v

⟩−

⟨q2,

∂gi

∂u(u)v

⟩

and

l2(x0, xN, λ1, λ2, h0, hN) = 〈λ1, K(x0, xN)〉 +⟨λ2,

∂K

∂(x0, xN)(x0, xN)(h0, hN)

⟩.

THEOREM 2.1 Let (x0, w) be the optimal solution for the problem (1)–(4). Then there existsLagrange multiplier λ = (λ1, λ2, p

0, p1, p2, q1, q2), λ1, λ2 ∈ Rk, p0 ∈ Rn, p1 ∈ RnN, p2 ∈RmN, q1 ∈ RnN and q2 ∈ RmN, such that for every (h, v) ∈ C(ξ , w) the following conditionsare satisfied:

(i)

p0 = ∂l2

∂x0(x0, xN , λ1, λ2, h0, hN), (12)

p1i = ∂H i

2

∂x(xi , ui , p

1i+1, p

2i+1, q

1i+1, q

2i+1, hi, vi), i = 0, N − 1, (13)

p1N = − ∂l2

∂xN

(x0, xN , λ1, λ2, h0, hN), (14)

∂H i2

∂u(xi , ui , p

1i+1, p

2i+1, q

1i+1, q

2i+1, hi, vi) = 0, i = 0, N − 1. (15)

(ii) (q1, q2, λ2) is the solution of the system of the equation (7)–(10).

Proof We shall formulate the problem (1)–(4) as a mathematical programming problem, andthen we shall apply results from ref. [1].

Define the functions

f (ξ, w) : Rn(N+1) × RrN −→ R, Fi(ξ, w) : Rn(N+1) × RrN −→ Rn, i = 1, N,

F (ξ, w) : Rn(N+1) × RrN −→ RnN, G(w) : RrN −→ RmN

by

f (ξ, w) =N−1∑i=0

fi(xi, ui),

Fi+1(ξ, w) = xi+1 − ϕi(xi, ui), i = 0, N − 1,

F (ξ, w) = (F1(ξ, w), . . . , FN(ξ, w)),

G(w) = (g1(u1), . . . , gN−1(uN−1)).

Also, consider the function

F(ξ, w) : Rn(N+1) × RrN −→ RnN × RmN × Rk,

F(ξ, w) = (F (ξ, w), G(w), K(x0, xN))T .

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964 B. Marinkovic

Consider the following mathematical programming problem:

minimize f (ξ, w); (16)

F(ξ, w) = 0. (17)

Obviously that the point (ξ , w) is the local minimum for the preceding problem.Let us introduce the generalized Lagrangian function

L2(ξ, w, λ, h, v) : Rn(N+1) × RrN × R × RnN+mN+k × RnN+mN+k × H (ξ , w) −→ R

by

L2(ξ, w, λ, h, v) = λ0f (ξ, w) + 〈p, F(ξ, w)〉 +⟨q,

∂F∂(ξ, w)

(ξ, w)(h, v)

⟩,

where λ = (λ0, p, q), λ0 ∈ R, p, q ∈ RnN+mN+k and H (ξ , w) is the set of all (h, v) ∈Rn(N+1) × RrN such that the following conditions are satisfied:

1)

∂F∂(ξ, w)

(ξ , w)(h, v) = 0,

2)

∂2F∂(ξ, w)2

(ξ , w)[(h, v), (h, v)] ∈ im∂F

∂(ξ, w)(ξ , w).

Note that the vector (h, v) is a parameter in the generalized Lagrange function.Put

A = ∂F∂(ξ, w)

(ξ , w).

It is easy to see that the operator A is given by the matrix

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−C0 I 0 . . . −D0 . . . 0...

...

0 . . . −CN−1 I 0 . . . −DN−1

0 . . . E0 . . . 0...

...

0 . . . EN−1

K0 0 . . . KN 0 . . . 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

where by I , resp. 0, we denote the identity matrix, resp. the zero matrix, of thecorresponding size.

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Discrete optimal control problems 965

From ref. [1], theorem 11.1, we have that ∃λ, λ0 ≥ 0, λ0 + |q| �= 0, such that for every(h, v) ∈ H (ξ , w),

∂L2

∂(ξ, w)(ξ , w, λ, h, v) = 0, p ∈ im A, q ∈ (im A)⊥. (18)

First, we shall prove that λ0 > 0, i.e., we shall prove that the operator F is 2-regular at thepoint (ξ , w) with respect to a direction (h, v) ∈ C(ξ , w). Let

G(h, v) : Rn(N+1) × RrN −→ RnN × RmN × Rk

be the linear operator defined by

G(h, v)(h, v) = A(h, v) + π∂2F

∂(ξ, w)2(ξ , w)[(h, v), (h, v)],

where (h, v) is a fixed vector and where π is the orthogonal projector onto (im A)⊥ in RnN ×RmN × Rk . As is known from [1,3,4], the operator F is 2-regular at the point (ξ , w) withrespect to a direction (h, v) if

im G(h, v) = RnN × RmN × Rk.

We shall prove that the preceding condition holds. From the fact that

(im A)⊥ = ker A∗,

and from the equations (7)–(10), we have that

(im A)⊥ = Q.

Now, we shall prove that

dim (ker G(h, v)) = l.

Indeed, ker G(h, v) is the set of all (h, v) satisfying

A(h, v) = 0, (19)

and∂2F

∂(ξ, w)2(ξ , w)[(h, v), (h, v)] ∈ im A. (20)

From (19) we obtain the equations

hk = Ck−1hk−1 + Dk−1vk−1, k = 1, N, (21)

Ekvk = 0, k = 0, N − 1, (22)

and

K0h0 + KNhN = 0. (23)

Solving the equations (21) we obtain (6). From (6), for k = N , we obtain

hN =N−1∏s=0

Csh0 +N−2∑j=0

∏j<s≤N−1

CsDjvj + DN−1vN−1. (24)

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966 B. Marinkovic

From (24), (23) and (22) we obtain that

dim (ker A) = dim (ker B).

According to the definition of the operator M(h, v) we obtain that the relation

∂2F∂(ξ, w)2

(ξ , w)[(h, v), (h, v)] ∈ im A

is equivalent to the fact that

PM(h, v)(h0, v) = 0.

We conclude that

dim (ker G(h, v)) = dim (ker L).

Now we shall calculate codim (ker G(h, v)). We have that

codim (ker G(h, v)) = n(N + 1) + rN − l = n(N + 1) + rN

− (n + rN − mN − k) = nN + mN + k.

From the known equality

dim (im G(h, v)) = codim (ker G(h, v)),

and from the fact that H (ξ , w) = H(ξ , w) we have that the operator F is 2-regular at thepoint (ξ , w) with respect to a direction (h, v). In ref. [1] was proved that in that case holdsthat λ0 > 0. It follows that, without loss of generality, we may take that λ0 = 1.

Put

∂L2

∂xi

= ∂L2

∂xi

(ξ , w, λ, h, v),∂L2

∂ui

= ∂L2

∂ui

(ξ , w, λ, h, v).

From the fact that

L2(ξ , w, λ, h, v) =N−1∑i=0

fi(xi , ui) +N−1∑i=0

〈p1i+1, xi+1 − ϕi(xi , ui)〉

+N−1∑i=0

〈p2i+1, gi(ui)〉 + 〈λ1, K(x0, xN )〉

+N−1∑i=0

⟨q1

i+1, hi+1 − Cihi − Divi

⟩

+N−1∑i=0

⟨q2

i+1, Eivi

⟩ + ⟨λ2,

∂K

∂(x0, xN)(x0, xN )(h0, hN)

⟩,

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Discrete optimal control problems 967

where p = (p1, p2, λ1) and q = (q1, q2, λ2), we have

∂L2

∂x0= ∂ l2

∂x0− ∂H 0

2

∂x= 0, (25)

∂L2

∂xi

= p1i − ∂H i

2

∂x= 0, i = 1, N − 1, (26)

∂L2

∂xN

= p1N + ∂l2

∂xN

= 0, (27)

∂L2

∂ui

= −∂H i2

∂u= 0, i = 0, N − 1, (28)

where (∂H i2/∂x), (∂H i

2/∂u), (∂l2/∂xN) and (∂l2/∂x0) are introduced analogously as above.Put

p0 = ∂H 02

∂x(x0, u0, p

11, p

21, q

11 , q2

1 , λ0, h, v). (29)

From (29) and (25) we have

p0 = ∂l2

∂x0(x0, xN , λ1, λ2, h0, hN).

Obviously that from (26)–(28) we obtain that (13)–(15) hold.Put λ = (λ1, λ2, p

0, p1, p2, q1, q2). We proved that for considered λ the assertions oftheorem 2.1 hold. �

3. Second-order optimality conditions

Suppose that the functions fi(x, u), ϕi(x, u), gi(u) and K(x0, xN) are three times continuouslydifferentiable.

For a given Lagrange multiplier λ define the bilinear form

�λ[(h, v), (h, v)] = ∂2 l2

∂x20

[h0, h0] + ∂2 l2

∂x2N

[hN, hN ] + 2∂2 l2

∂x0∂xN

[h0, hN ]

−N−1∑i=0

∂2H i2

∂x2[hi, hi] −

N−1∑i=0

∂2H i2

∂u2[vi, vi] − 2

N−1∑i=0

∂2H i2

∂x∂u[hi, vi],

where (∂2 l2/∂x20 ), (∂2 l2/∂x2

N), (∂2 l2/∂x0∂xN), (∂2H i2/∂x2), (∂2H i

2/∂u2) and (∂2H i2/∂x∂u)

are introduced analogously as in the previous section only with (q1i /3), (q2

i /3) and (λ2/3)

instead of q1i , q2

i and λ2.

THEOREM 3.1 Let (x0, w) be the optimal solution for the problem (1)–(4). Then there existsLagrange multiplier λ such that the assertions of theorem 1 hold, and for every (h, v) ∈C(ξ , w) holds

�λ[(h, v), (h, v)] ≥ 0. (30)

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968 B. Marinkovic

Proof Analogously as in the proof of theorem 2.1 we consider the mathematical programmingproblem (16)–(17). It is easy to see that for the generalized Lagrangian function hold

∂2L2

∂xi∂xj

≡ 0, ∀(i, j) : i �= j, (i, j) �= (0, N); ∂2L2

∂xi∂uj

≡ 0,∂2L2

∂ui∂uj

≡ 0, ∀i �= j.

From ref. [1], theorem 11.2 and from the preceding facts we obtain that the assertions oftheorem 3.1 hold. �

Remark 1 Optimality conditions in calculus of variations and optimal control problems,based on 2-regularity concept, can be found in papers byAvakov (see comments and referencesin ref. [1]). Obtained optimality conditions in that case are analogue to the conditions in thecorresponding discrete case.

Remark 2 Optimality conditions for discrete optimal control problems with inequality con-straints on control may also be obtained with 2-regularity condition, but that approach requireddifferent techniques and will be considered by the author elsewhere.

4. Concluding remarks

First, we shall compare the number of the variables and the number of the equations fromtheorem 2.1. We have that the number of the variables ξ , w, and λ from theorem 2.1 is equal to

n(N + 1) + rN + 2(nN + mN + k) = 3nN + 2mN + rN + 2k + n.

From (2)–(4), (12)–(15) and from the fact that (q1, q2, λ2) is the solution of the system ofthe equation (7)–(10), we obtain

nN + mN + k + n(N + 1) + rN + nN + mN + k = 3nN + 2mN + rN + 2k + n

equations. It follows that we have the complete system of the equations associated with ξ , w,

and λ.First-order necessary optimality conditions for the initial problem are well known and it can

be found in refs. [5,6]. Second order optimality conditions are obtained in, e.g., [7] (see alsothe references there) with the regularity assumption, i.e., with discrete variant of Robinson’scondition. Also, when Robinson’s condition is not satisfied then known first order optimalityconditions become noninformative.

Assume that for the problem (1)–(4) Robinson’s condition holds, i.e., we assume that

im∂F

∂(ξ, w)(ξ , w) = RnN × RmN × Rk

holds. Then we have that

l = dim (ker L) = dim (ker B).

It follows that

l = n(N + 1) + rN − codim (ker B) = n(N + 1) + rN − (nN + mN + k)

= n + rN − mN − k,

and we have that the condition (11) holds. Note that in the case when Robinson’s conditionholds then the generalized Pontryagin’s function becomes classical Pontryagin’s function and

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Discrete optimal control problems 969

the generalized small Lagrangian becomes classical small Lagrangian. Indeed, in that case wehave that (q1, q2, λ2) is the trivial solution of the system (7)–(10).

The most important fact is that our theorems contained nontrivial first- and second-order optimality conditions which are meaningful under the assumptions weaker then thosepreviously used in the literature.

Discrete optimal control problems are also considered in [8,9] where are obtained first-and second-order necessary optimality conditions and developed sensitivity analysis, withoutregularity assumptions.

Acknowledgement

The author would like to thank A.V. Arutyunov for the valuable and constructive comments.

References[1] Arutyunov, A.V., 2000, Optimality Conditions: Abnormal and Degenerate Problems (Dordrecht: Kluwer

Academic Publishers).[2] Avakov, E.R., 1985, Extremum conditions for smooth problems with equality-type constraints. USSR Computa-

tional Mathematics and Mathematical Physics, 25, 24–32 (in Russian).[3] Izmailov, A.F. and Solodov, M., 2002, The theory of 2-regularity for mappings with Lipschitzian derivatives and

its applications to optimality conditions. Mathematics of Operations Research, 27, 614–635.[4] Izmailov,A.F. and Tretyakov,A.A., 1994, Factor Analysis of Nonlinear Mappings (Moscow: Nauka) (in Russian).[5] Boltyanskii, V.G., 1978, Optimal Control of Discrete Systems (New York: Wiley).[6] Ioffe, A.D. and Tikhomirov, V.M., 1979, Theory of Extremal Problems (Amsterdam: North-Holland).[7] Hilscher, R. and Zeidan,V., 2002, Second order sufficiency criteria for a discrete optimal control problem. Journal

of Difference Equations and Applications, 8(6), 573–602.[8] Arutyunov, A.V. and Marinkovic, B., 2005, Necessary conditions for optimality in discrete optimal control

problems. Vestnik MGU. Series 15, 1, 43–48 (in Russian).[9] Marinkovic, B., 2006, Sensitivity analysis for discrete optimal control problems. Mathematical Methods of

Operations Research, 63(3), 513–524.

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