Necessary and sufficient conditions for local Pareto optimality on time scales
aa r X i v : . [ m a t h . O C ] J a n NECESSARY AND SUFFICIENT CONDITIONS FOR LOCALPARETO OPTIMALITY ON TIME SCALES
AGNIESZKA B. MALINOWSKA AND DELFIM F. M. TORRES
Abstract.
We study a multiobjective variational problem on time scales.For this problem, necessary and sufficient conditions for weak local Paretooptimality are given. We also prove a necessary optimality condition for theisoperimetric problem with multiple constraints on time scales. Introduction
The calculus on time scales was initiated by Aulbach and Hilger (see e.g. [2]) inorder to create a theory that can unify discrete and continuous analysis. Since then,much active research has been observed all over the world (see e.g. [1, 3, 4, 7] andreferences therein). In this paper we consider multiobjective variational problemson time scales (Section 3.2). By developing a theory for multiobjective optimizationproblems on a time scale, one obtains more general results that can be applied todiscrete, continuous or hybrid domains. To the best of the authors’ knowledge, nostudy has been done in this field for time scales. The main results of the paperprovide methods for identifying weak locally Pareto optimal solutions; versions forcontinuous domain one can find e.g. in [6, 8, 9]. We show that necessary optimalityconditions for isoperimetric problems are also necessary for local Pareto optimalityfor a multiobjective variational problem on a time scale (Theorem 3.8), and thesufficient condition for local Pareto optimality can be reduced to the sufficientoptimal condition for a basic problem of the calculus of variations on a time scale(Theorem 3.7). We also prove a necessary optimality condition for the isoperimetricproblem with multiple constraints on time scales (Section 3.1).2.
Time scales calculus
In this section we introduce basic definitions and results that will be needed forthe rest of the paper. For a more general theory of calculus on time scales, we referthe reader to [5].A nonempty closed subset of R is called a time scale and it is denoted by T .The forward jump operator σ : T → T is defined by σ ( t ) = inf { s ∈ T : s > t } , for all t ∈ T , while the backward jump operator ρ : T → T is defined by ρ ( t ) = sup { s ∈ T : s < t } , for all t ∈ T , with inf ∅ = sup T (i.e. σ ( M ) = M if T has a maximum M ) and sup ∅ = inf T (i.e. ρ ( m ) = m if T has a minimum m ). Mathematics Subject Classification.
Key words and phrases.
Time scales, calculus of variations, isoperimetric problems with mul-tiple constraints, multiobjective variational problems, locally Pareto optimal solutions.Research partially supported by the
Centre for Research on Optimization and Control (CEOC)from the
Portuguese Foundation for Science and Technology (FCT), cofinanced by the EuropeanCommunity Fund FEDER/POCI 2010 and by KBN under Bia lystok Technical University GrantS/WI/1/07.
A point t ∈ T is called right-dense , right-scattered , left-dense and left-scattered if σ ( t ) = t , σ ( t ) > t , ρ ( t ) = t and ρ ( t ) < t , respectively.Throughout the paper we let T = [ a, b ] ∩ T with a < b and T a time scalecontaining a and b . Remark . The time scales T considered in this work have a maximum b and, bydefinition, σ ( b ) = b .The graininess function µ : T → [0 , ∞ ) is defined by µ ( t ) = σ ( t ) − t, for all t ∈ T . Following [5], we define T k = T \ ( ρ ( b ) , b ], T k = (cid:0) T k (cid:1) k .We say that a function f : T → R is delta differentiable at t ∈ T k if thereexists a number f ∆ ( t ) such that for all ε > U of t (i.e. U = ( t − δ, t + δ ) ∩ T for some δ >
0) such that | f ( σ ( t )) − f ( s ) − f ∆ ( t )( σ ( t ) − s ) | ≤ ε | σ ( t ) − s | , for all s ∈ U .
We call f ∆ ( t ) the delta derivative of f at t and say that f is delta differentiable on T k provided f ∆ ( t ) exists for all t ∈ T k .For delta differentiable functions f and g , the next formula holds:( f g ) ∆ ( t ) = f ∆ ( t ) g σ ( t ) + f ( t ) g ∆ ( t )= f ∆ ( t ) g ( t ) + f σ ( t ) g ∆ ( t ) , where we abbreviate here and throughout the text f ◦ σ by f σ .A function f : T → R is called rd-continuous if it is continuous at right-densepoints and if its left-sided limit exists at left-dense points. We denote the set of allrd-continuous functions by C rd and the set of all delta differentiable functions withrd-continuous derivative by C .It is known that rd-continuous functions possess an antiderivative , i.e. thereexists a function F with F ∆ = f , and in this case the delta integral is definedby R dc f ( t )∆ t = F ( c ) − F ( d ) for all c, d ∈ T . The delta integral has the followingproperty: Z σ ( t ) t f ( τ )∆ τ = µ ( t ) f ( t ) . We now present the integration by parts formulas for the delta integral:
Lemma 2.2. ( [5] ) If c, d ∈ T and f, g ∈ C rd , then Z dc f ( σ ( t )) g ∆ ( t )∆ t = [( f g )( t )] t = dt = c − Z dc f ∆ ( t ) g ( t )∆ t ; Z dc f ( t ) g ∆ ( t )∆ t = [( f g )( t )] t = dt = c − Z dc f ∆ ( t ) g ( σ ( t ))∆ t. We say that f : T → R n is a rd-continuous (a delta differentiable ) function if eachcomponent of f , f i : T → R , is a rd-continuous (a delta differentiable) function.By abuse of notation, we continue to write C rd for the set of all rd-continuousvector valued functions and C for the set of all delta differentiable vector valuedfunctions with rd-continuous derivative.The following Dubois-Reymond lemma for the calculus of variations on timescales will be useful for our purposes. Lemma 2.3. (Lemma of Dubois-Reymond [4] ) Let g ∈ C rd , g : [ a, b ] k → R n . Then, Z ba g ( t ) · η ∆ ( t )∆ t = 0 for all η ∈ C rd with η ( a ) = η ( b ) = 0 OCAL PARETO OPTIMALITY ON TIME SCALES 3 if and only if g ( t ) = c on [ a, b ] k for some c ∈ R n . Main Results
We begin by proving necessary optimality conditions for isoperimetric problemson time scales ( § § Isoperimetric problem on time scales.Definition 3.1.
For f : [ a, b ] → R n we define the norm k f k C rd = max t ∈ [ a,b ] k k f σ ( t ) k + max t ∈ [ a,b ] k k f △ ( t ) k , where k · k stands for any norm in R n .Let L : C rd → R be a functional defined on the function space C rd endowed withthe norm k · k C rd and let A ⊆ C rd . Definition 3.2.
A function ˆ f ∈ A is called a weak local minimum of L providedthere exists δ > L [ ˆ f ] ≤ L [ f ] for all f ∈ A with k f − ˆ f k C rd < δ .Now, let us consider a functional of the form(1) L [ y ] = Z ba L ( t, y σ ( t ) , y △ ( t )) △ t, where a, b ∈ T with a < b , L ( t, s, v ) : [ a, b ] k × R n × R n → R has partial continuousderivatives with respect to the second and third variables for all t ∈ [ a, b ] k , and L ( t, · , · ) and its partial derivatives are rd-continuous at t . The isoperimetric problem consists of finding a function y satisfying:(i) the boundary conditions(2) y ( a ) = α , y ( b ) = β , α, β ∈ R n ;and(ii) constraints of the form(3) G i [ y ] = Z ba G i ( t, y σ ( t ) , y △ ( t )) △ t = ξ i , i = 1 , . . . m, where ξ i , i = 1 , . . . m , are specified real constrains, G i ( t, s, v ) : [ a, b ] k × R n × R n → R , i = 1 , . . . m , have partial continuous derivatives with respect to the second andthird variables for all t ∈ [ a, b ] k , and G i ( t, · , · ) and their partial derivatives arerd-continuous at t ; that takes (1) to a minimum. Definition 3.3.
Let L be a functional defined on C rd . The first variation of L at y ∈ C rd in the direction η ∈ C rd , also called Gˆateaux derivative with respect to η at y , is defined as δ L [ y ; η ] = lim ε → L [ y + εη ] − L [ y ] ε = ∂∂ε L [ y + εη ] | ε =0 (provided it exists). If the limit exists for all η ∈ C rd , then L is said to be Gˆateauxdifferentiable at y .The existence of Gˆateaux derivative δ L [ y ; η ] presupposes that:(i) L [ y ] is defined;(ii) L [ y + εη ] is defined for all sufficiently small ε . A. B. MALINOWSKA AND D. F. M. TORRES
Theorem 3.4.
Let L , G , . . . , G m be functionals defined in a neighborhood of ˆ y andhaving continuous Gˆateaux derivative in this neighborhood. Suppose that ˆ y is a weaklocal minimum of (1) subject to the boundary conditions (2) and the isoperimetricconstrains (3) . Then, either:(i) ∀ v j ∈ C rd , j = 1 , . . . , m (4) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ G [ˆ y ; v ] δ G [ˆ y ; v ] · · · δ G [ˆ y ; v m ] δ G [ˆ y ; v ] δ G [ˆ y ; v ] · · · δ G [ˆ y ; v m ] ... ... ... ... δ G m [ˆ y ; v ] δ G m [ˆ y ; v ] · · · δ G m [ˆ y ; v m ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 or(ii) there exist constants λ i ∈ R , i = 1 , . . . , m for which (5) δ L [ˆ y ; η ] = m X i =1 λ i δ G i [ˆ y ; η ] ∀ η ∈ C rd . Proof.
This proof is patterned after the proof of Troutman [10, Theorem 5.16]. Letus consider, for fixed directions η, v , v , . . . , v m , the auxiliary functions: l ( p, q , . . . , q m ) = L [ˆ y + pη + q v + · · · + q m v m ] ,g ( p, q , . . . , q m ) = G [ˆ y + pη + q v + · · · + q m v m ] , ... g m ( p, q , . . . , q m ) = G m [ˆ y + pη + q v + · · · + q m v m ] , which are defined in some neighborhood of the origin in R m +1 , since L , G , . . . , G m themselves are defined in a neighborhood of ˆ y . Note that the partial derivative l p ( p, q , . . . , q m ) = ∂∂p l ( p, q , . . . , q m ) = ∂∂p L [ˆ y + pη + q v + · · · + q m v m ]= lim ε → L [ˆ y + ( p + ε ) η + q v + · · · + q m v m ] − L [ˆ y + pη + q v + · · · + q m v m ] ε = lim ε → L [ y + εη ] − L [ y ] ε , with y = ˆ y + pη + q v + · · · + q m v m . Therefore, l p ( p, q , . . . , q m ) = δ L [ˆ y ; η ]. Similarlywe have: l q i ( p, q , . . . , q m ) = δ L [ˆ y ; v i ] , i = 1 , . . . , m, ( g j ) p ( p, q , . . . , q m ) = δ G j [ˆ y ; η ] , j = 1 , . . . , m, ( g j ) q i ( p, q , . . . , q m ) = δ G j [ˆ y ; v i ] , i = 1 , . . . , m, j = 1 , . . . , m. Hence, the Jacobian determinant ∂ ( l,g ,...,g m ) ∂ ( p,q ,...,q m ) evaluated at ( p, q , . . . , q m ) = (0 , , . . . , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ L [ˆ y ; η ] δ L [ˆ y ; v ] · · · δ L [ˆ y ; v m ] δ G [ˆ y ; η ] δ G [ˆ y ; v ] · · · δ G [ˆ y ; v m ]... ... . . . ... δ G m [ˆ y ; η ] δ G m [ˆ y ; v ] · · · δ G m [ˆ y ; v m ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Note also that the vector valued function ( l, g , . . . , g m ) has continuous partialderivatives in a neighborhood of the origin, since L , G , . . . , G m have continuousGˆateaux derivative in the neighborhood of ˆ y . With this preparation we can proveour theorem. Assume condition (i) does not hold for one set of directions: v , v , . . . , v m and suppose there exists one direction η for which the determinant (6) is nonva-nishing. Therefore, the classical inverse function theorem applies, i.e. the appli-cation ( l, g , . . . , g m ) maps a neighborhood of the origin in R m +1 onto a region OCAL PARETO OPTIMALITY ON TIME SCALES 5 containing a full neighborhood of ( L [ˆ y ] , G [ˆ y ] , . . . , G m [ˆ y ]). That is, one can findpre-image points (´ p, ´ q , . . . , ´ q m ) and (` p, ` q , . . . , ` q m ) near the origin, for which thepoints ´ y = ˆ y + ´ pη + Σ mi =1 ´ q i v i and ` y = ˆ y + ` pη + Σ mi =1 ` q i v i satisfy the conditions: L [´ y ] < L [ˆ y ] < L [` y ] , G i [´ y ] = G i [ˆ y ] = G i [` y ] , i = 1 , . . . , m. This shows that ˆ y cannot be a local extremal for L subject to constraints (3),contradicting the hypothesis. Thus, for the specific set of directions: v , v , . . . , v m the determinant (6) must vanish for each η ∈ C rd . We expand it by minors of thefirst column(7) δ L [ˆ y ; η ] · cof δ L [ˆ y ; η ]+ δ G [ˆ y ; η ] · cof δ G [ˆ y ; η ]+ . . . + δ G m [ˆ y ; η ] · cof δ G m [ˆ y ; η ] = 0 , where we are using the notation cof to denote the cofactor. Dividing equation (7)by cof δ L [ˆ y ; η ], since it is precisely the nonvanishing determinant (cid:12)(cid:12)(cid:12)(cid:12) δ G i [ˆ y ; v j ] i, j = 1 , . . . , m (cid:12)(cid:12)(cid:12)(cid:12) ,and setting λ i = − cof δ G i [ˆ y ; η ] cof δ L [ˆ y ; η ]we obtain an equation equivalent to (5). (cid:3) Note that condition (ii) of Theorem 3.4 can be written in the form(8) δ L − m X i =1 λ i G i [ˆ y ; η ] ! = 0 ∀ η ∈ C rd , since the Gˆateaux derivative is a linear operation on the functionals (by the linear-ity of the ordinary derivative).Now, suppose that assumptions of Theorem 3.4 hold but condition (i) does nothold. Then, equation (8) is fulfilled for every η ∈ C rd . Let us consider function η such that η ( a ) = η ( b ) = 0 and denote by F the functional L − P mi =1 λ i G i . Then wehave 0 = δ F [ˆ y ; η ] = ∂∂ε F [ˆ y + εη ] | ε =0 = Z ba ( F s ( t, ˆ y σ ( t ) , ˆ y △ ( t )) η σ ( t ) + F v ( t, ˆ y σ ( t ) , ˆ y △ ( t )) η △ ( t )) △ t, where the function F : [ a, b ] k × R n × R n → R is defined by F ( t, s, v ) = L ( t, s, v ) − P mi =1 λ i G i ( t, s, v ) . Note that Z ba (cid:18)Z ta F s ( τ, ˆ y σ ( τ ) , ˆ y △ ( τ )) △ τ η ( t ) (cid:19) △ △ t = Z ta F s ( τ, ˆ y σ ( τ ) , ˆ y △ ( τ ) △ τ η ( t ) | t = bt = a = 0and Z ba (cid:18)Z ta F s ( τ, ˆ y σ ( τ ) , ˆ y △ ( τ )) △ τ η ( t ) (cid:19) △ △ t = Z ba ((cid:18)Z ta F s (cid:0) τ, ˆ y σ ( τ ) , ˆ y △ ( τ ) (cid:1) △ τ (cid:19) △ η σ ( t ) + Z ta F s ( τ, ˆ y σ ( τ ) , ˆ y △ ( τ )) △ τ η △ ( t ) ) △ t = Z ba (cid:26) F s ( t, ˆ y σ ( t ) , ˆ y △ ( t )) η σ ( t ) + Z ta F s ( τ, ˆ y σ ( τ ) , ˆ y △ ( τ )) △ τ η △ ( t ) (cid:27) △ t. Therefore,0 = Z ba (cid:26) F v ( t, ˆ y σ ( t ) , ˆ y △ ( t )) − Z ta F s ( τ, ˆ y σ ( τ ) , ˆ y △ ( τ )) △ τ (cid:27) η △ ( t ) △ t. A. B. MALINOWSKA AND D. F. M. TORRES
Since the function η is arbitrary, Lemma 2.3 implies that F v ( t, ˆ y σ ( t ) , ˆ y △ ( t )) − Z ta F s ( τ, ˆ y σ ( τ ) , ˆ y △ ( τ )) △ τ = c for some c ∈ R n and all t ∈ [ a, b ] k . Hence,(9) F △ v ( t, ˆ y σ ( t ) , ˆ y △ ( t )) = F s ( t, ˆ y σ ( t ) , ˆ y △ ( t ))for all t ∈ [ a, b ] k .We have just proved the following necessary optimality condition for the isoperi-metric problem with multiple constrains on time scales. Theorem 3.5.
Let us assumptions of Theorem 3.4 hold but condition (4) does nothold. If ˆ y ∈ C rd is a weak local minimum of the problem (1) - (3) , then it satisfiesthe Euler-Lagrange equation (9) for all t ∈ [ a, b ] k . Pareto optimality.
Let us consider a finite number d ≥ L i [ y ] = Z ba L i ( t, y σ ( t ) , y △ ( t )) △ t, i = 1 , . . . d, where a, b ∈ T with a < b , L i ( t, s, v ) : [ a, b ] k × R n × R n → R , i = 1 , . . . d , havepartial continuous derivatives with respect to the second and third variables forall t ∈ [ a, b ] k , and L i ( t, · , · ) and theirs partial derivatives, i = 1 , . . . d , are rd-continuous at t . We would like to find a function y ∈ C rd , satisfying the boundaryconditions (2), that renders the minimum value to each functional L i , i = 1 , . . . , d ,simultaneously. In general, there does not exist such a function, and one uses theconcept of Pareto optimality. Definition 3.6.
A function ˆ y ∈ C rd is called a weak locally Pareto optimal solution if there exists δ > y ∈ C rd with k y − ˆ y k C rd < δ and ∀ i ∈ { , . . . , d } : L i [ y ] L i [ˆ y ] ∧ ∃ j ∈ { , . . . , d } : L j [ y ] < L j [ˆ y ] . Theorem 3.7. If ˆ y is a weak local minimum of the functional P di =1 γ i L i [ y ] with γ i > for i = 1 , . . . , d and P di =1 γ i = 1 , then it is a weak locally Pareto optimalsolution of the multiobjective problem with functionals (10) .Proof. Let ˆ y be a weak local minimum of the functional P di =1 γ i L i [ y ] with γ i > i = 1 , . . . , d and P di =1 γ i = 1. Suppose on the contrary that ˆ y is not a weaklocally Pareto optimal. Then, for every δ > y with k y − ˆ y k C rd < δ such that ∀ i ∈ { , . . . , d } we have L i [ y ] L i [ˆ y ] and ∃ j ∈ { , . . . , d } such that L j [ y ] < L j [ˆ y ]. Since γ i > i = 1 , . . . , d , we obtain P di =1 γ i L i [ y ] < P di =1 γ i L i [ˆ y ].This contradicts our choice of ˆ y . (cid:3) Theorem 3.8. If ˆ y is a weak locally Pareto optimal solution of the multiobjectiveproblem with functionals (10) , then it minimizes each one of the scalar functionals L i [ y ] , i ∈ { , . . . , d } subject to the constraints L j [ y ] = L j [ˆ y ] , j = 1 , . . . , d and j = i . Proof.
Let ˆ y be a weak locally Pareto optimal solution of the problem on time scales(10) and suppose the contrary, i.e. that for some i ˆ y does not solve the problem L i [ y ] → min subject to L j [ y ] = L j [ˆ y ] , j = 1 , . . . , d ( j = i ) . Then, for every δ > y with k y − ˆ y k C rd < δ such that L i [ y ] < L i [ˆ y ] and L j [ y ] = L j [ˆ y ] , j =1 , . . . , d ( j = i ). This contradicts the weak local Pareto optimality of ˆ y . (cid:3) OCAL PARETO OPTIMALITY ON TIME SCALES 7
Example 3.9.
Let T = { , , } . We would like to find locally Pareto optimalsolutions for L [ y ] = Z y ( t + 1) △ t, L [ y ] = Z ( y ( t + 1) − △ t satisfying the boundary conditions y (0) = 0, y (2) = 0. Note that L [ y ] = X t =0 y ( t + 1) , L [ y ] = X t =0 ( y ( t + 1) − , and that the possible solutions are of the form y ( t ) = t = 0 a if t = 10 if t = 2 , where a ∈ R . On account of the above, we have L [ y ( t )] = a , and L [ y ( t )] =4 + ( a − . Using Theorem 3.7 we obtain that locally Pareto optimal solutionsfor functionals L , L are y ( t ) = t = 0 a if t = 1 , t = 2 a ∈ [0 , . References [1] C. D. Ahlbrandt, M. Bohner and J. Ridenhour, Hamiltonian systems on time scales, J. Math.Anal. Appl. (2000), no. 2, 561–578.[2] B. Aulbach and S. Hilger, Linear dynamic processes with inhomogeneous time scale, in
Non-linear dynamics and quantum dynamical systems (Gaussig, 1990) , 9–20, Akademie Verlag,Berlin.[3] Z. Bartosiewicz and D. F. M. Torres, Noether’s theorem on time scales, J. Math. Anal. Appl.(accepted) arXiv:0709.0400 [4] M. Bohner, Calculus of variations on time scales, Dynam. Systems Appl. (2004), no. 3-4,339–349.[5] M. Bohner and A. Peterson, Dynamic equations on time scales , Birkh¨auser Boston, Boston,MA, 2001.[6] Y. Censor, Pareto optimality in multiobjective problems, Appl. Math. Optim. (1977/78),no. 1, 41–59.[7] R. A. C. Ferreira and D. F. M. Torres, Higher-order calculus of variations on time scales. Proc.Workshop on Mathematical Control Theory and Finance, Lisbon, 10-14 April 2007, pp. 150–158. To appear in Springer—Business/Economics and Statistics (accepted). arXiv:0706.3141 .[8] D. T. Luc and S. Schaible, Efficiency and generalized concavity, J. Optim. Theory Appl. (1997), no. 1, 147–153.[9] K. Miettinen, Nonlinear multiobjective optimization , Kluwer Acad. Publ., Boston, MA, 1999.[10] J. L. Troutman,
Variational calculus and optimal control , Second edition, Springer, NewYork, 1996.
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