# Paratingent Derivative Applied to the Measure of the Sensitivity in Multiobjective Differential Programming

Authors: F. García; M. A. Melguizo Padial
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We analyse the sensitivity of differential programs of the formMinf(x) subject to g(x)=b,x∈D where f and g are 𝒞1 maps whose respective images lie in ordered Banach spaces. Following previous works on multiobjective programming, the notion of T-optimal solution is used. The behaviour of some nonsingleton sets of T-optimal solutions according to changes of the parameter b in the problem is analysed. The main result of the work states that the sensitivity of the program is measured by a Lagrange multiplier plus a projection of its derivative. This sensitivity is measured by means of the paratingent derivative.


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## Body

## 1. Introduction

The subject of this work is sensitivity analysis in vector programming. The model problem considered throughout the paper is of the form(1)Minf(x)subjecttog(x)=bx∈D.
Here, and throughout this work, f:D⊂X→Y and g:D⊂X→Z denote two 𝒞1 maps, X a Banach space and Y and Z two ordered Banach spaces. The sensitivity of the problem is analysed by studying the quantitative behaviour of a nonsingleton set of optimal points when the parameter b∈Z varies.As the mapsf and g lie in Banach spaces, the obtained results provide a general framework in which a wide range of problems can be studied; see [1].To perform our analysis we use the so-calledT-optimal solution; that is, solutions of the program characterized to become a minimum when the objective function is composed with a positive topological homomorphism T. For a fixed T, we measure the perturbation experienced by the whole set of the T-optimal solutions (not necessarily a singleton) when the right-hand side vector b varies. The study is carried out by means of set-valued derivatives.Tangent cones are the cornerstone of the notion of derivative of a set-valued map. There are different notions of tangency, and each of them provides a different cone. Experience shows that there are, mainly, four useful kinds of cones: Bouligand's contingent, adjacent, Clarke's tangent (or circatangent), and Bouligand's paratingent (see [2, 3]). All the four correspond to different regularity requirements and they carry in themselves a wide and particular information about the local behaviour of sets. Once a concept of tangent cone is chosen, we can associate with it a notion of the derivative of a set-valued map. Derivatives obtained with the former cones play an important role in several branches of mathematics, for example, nonsmooth analysis, control theory, viability theory, and so forth.The notions of derivative of set-valued maps have been used in many recent papers in the context of stability and sensitivity analysis; see, for example, [4–17]. In this paper we follow this research line, analysing the sensetivity of the differential program (1) by means of the paratingent derivative. In previous papers, the study of sensitivity of this problem has been carried out by using adjacent, contingent, and circatangent derivatives (see [18, 19]). Therefore, this study completes the sensitivity analysis of problem (1) with the four main derivatives mentioned above.Our view is that, on the one hand, the main theorem of this work (Theorem A) measures the specific kind of sensibility provided by the paratingent cone. On the other hand, this result measures the sensitivity of the problem in cases in which previous results do not. Next, we explain the former claim. Clarke cone has the nice property to be always a closed convex cone. Then, Clarke derivative is a closed convex process, that is, the set-valued analogous of a continuous linear operator. The price of this property, however, is quite high since this tangent cone may often be too small or even reduced to the singleton{0}; see [2, Chapters 2 and 4]. In that case, the corresponding derivative does not provide any information about the sensitivity of the problem. Paratingent cone is a natural generalization of Clarke cone. Although its definition is less restrictive, it holds the property of stability of the tangency with respect to perturbations around the point where the cone is taken, in the same way that Clarke cone does. Paratingent cone is bigger than Clarke cone, then paratingent derivative can measure stability when Clarke derivative fails. Contingent and adjacent cones are intermediate cones between Clarke and paratingent ones, but they lack the above-mentioned tangency's stability property. Finally, as the paratingent cone is the biggest one of the four considered cones, it can provide information about the sensitivity when the others fail; see Example 2 of Section 2. Let us note that paratingent derivative is also useful in works on optimality conditions [20], differential inclusions [21–23], dynamical systems and ergodic theory [24], differentiable maps [25], and differentiation theory [26].Before presenting the main results of the work, it is necessary to introduce some terminology and notation. Let us fix an order complete Banach latticeW and a positive linear and continuous surjective map T:Y→W such that its kernel has a topological supplement. It is said that a feasible xb∈D is a localT-optimal solution of (1) when there exists a neighbourhood Uxb⊂D of xb such that Tf(xb)≤Tf(x) for every feasible x∈Uxb. It is clear that every local T-optimal solution of (1) is a local optimal solution of the program, that is, f(xb)-f(x)∉Y+∖{0} for every feasible x∈Uxb. If T is a topological isomorphism satisfying some weak requirements, then the set of the T-optimal solutions is dense in the efficient line (see [27]), and thus, it constitutes a very suitable set to describe the full efficient line. An element x∈D is said to be regularwith respect to the problem (1) if the Fréchet differential, g′(x,·), of g at x is surjective. ℬ(Z,Y) will denote the Banach space of the continuous linear maps from Z on Y with the usual norm and by π, the natural projection from Y onto the kernel of T, KerT. Let xb∈D be a regular local T-optimal solution of (1) is represented, a map G∈ℬ(Z,Y) is said to be a T-Lagrange multiplier of (1) associated with xb if Tf′(xb,·)=TGg′(xb,·) and πf(xb)=πG(b). In [6], it is proved that for every regular local T-optimal solution xb∈D of (1), there exists a T-Lagrange multiplier Gxb∈ℬ(Y,Z) associated with it. Now we can define the following key set-valued maps of the work.Definition 1.
In the context of the precedent paragraph, it is defined that(i)
T-perturbation map of (1), Υ:V⊂Z⇝Y, by
(2)Υ(b)≔{f(xb):xbisaregularlocalT-optimalsolutionof(1)};(ii)
T-dual perturbation map of (1), Ψ:V⊂Z⇝ℬ(Z,Y), by
(3)Ψ(b)≔{Gxb:itisaT-Lagrangemultiplierof(1)associatedwithxb}.Now we can state the main result of the work. In this, it is represented byPLΥΥ (resp., PLΨΨ), the paratingent derivative of Υ (resp., Ψ) relative to LΥ (resp., LΨ).Theorem A.
Let one fixesLΥ⊂Graph(Υ),  (b*,f(xb*))∈LΥ, and  LΨ={(b,Gxb)∈Graph(Ψ):(b,f(xb))∈LΥ and Gxb is associated with xb}. If Ψ is  is lower semicontinuous, paraderivable relative to LΨ at (b*,Gxb*)∈LΨ and TΨ is continuously Fréchet differentiable at b*, then Υ is  lower semicontinuous, paraderivable relative to LΥ at (b*,f(xb*)), and
(4)PLΥΥ(b*,f(xb*))(z)=Gxb*(z)+πPLΨΨ(b*,Gxb*)(z)(b*),∀z∈Z.IfLΥ=Graph(Υ), the derivatives of the statement of the former result are denoted by PΥ and PΨ. In this case, we obtain a more particular but simpler version.Corollary B.
Let  one supposes thatΨ is lower semicontinuous, paraderivable relative to  Graph(Ψ) at (b*,Gxb*)∈Graph(Ψ), TΨ is continuously Fréchet differentiable at b*, then Υ is lower semicontinuous, paraderivable relative to  Graph(Υ) at (b*,f(xb*)), and
(5)PΥ(b*,f(xb*))(z)=Gxb*(z)+πPΨ(b*,Gxb*)(z)(b*),∀z∈Z.The proof of Theorem A is based on the fact that the set-valued mapΨˇ, defined by Ψˇ(b)≔Ψ(b)(b), is paraderivable when T-dual perturbation map Ψ is and TΨ is Fréchet differentiable. This is stated in Theorem 12 of Section 4. Besides, on that result, the paratingent derivative of  Ψˇ is expressed in terms of the paratingent derivative of Ψ, formula (58). This formula also holds even when Ψ is not paraderivable. The former fact and the proof of Theorem A allows us to claim that (4) also holds even when Ψ is not paraderivable. As a consequence, (4) provides a way to measure sensitivity of problem (1) when the other derivatives fail.The paper is organized as follows. In Section2,  the necessary mathematical background is reviewed; mainly, basic definitions and characterizations of cones and derivatives are provided in a precise way. Section 3 is a technical part devoted to establish some results which will be useful in Section 5. In particular, condition 𝒯 is introduced and after that we prove Theorem 4 which allows us, via property 𝒯, to transfer the condition of paraderivability from an arbitrary set-valued map Σ:V⇝ℬ(Z,Y) to that one  Σˇ:V⇝Y defined by Σˇ(b)=Σ(b)(b). After that we obtain some more technical results. For example, in Lemma 6, it is proved when the paraderivability can be transferred from Σˇ to πΣˇ, and Proposition 9, which is the paratingent version of the useful Proposition 5.1.2 of [2], provides the paratingent derivative of the sum of a set-valued map and a single-valued map. The objective of Section 4 is to prove the main result of the work about the sensitivity of the problem (1). After some technical considerations, Theorem 12 is stated and proved. In its statement it can be seen that it is possible to transfer the paraderivability property from the set-valued Ψ to Ψˇ. After that, Theorem A is proved. At the end of Section 4 an example that illustrates the main theorem can be seen.


## 2. Cones and Derivatives

Before introducing the notions of cone and derivative, we will recall (and will go into details) some definitions which were scarcely given in the former section. Let us recall thatX represents a Banach space, Y, Z, and W ordered Banach spaces such that W is an order complete Banach lattice (i.e., every nonempty bounded from below subset has an infimum in W). Let Y+,Z+, and W+ denote the respective positive convex cones of Y,Z, and W, and suppose that Y+ and Z+ are closed. The dual space of a Banach space Z will be denoted by Z*. The map T:Y→W is fixed throughout the paper and it is a positive (T(Y+∖{0})⊂W+∖{0}) linear and continuous surjective map such that Ker T has a topological supplement denoted by YT. The symbol T^ represents the restriction of T to YT. It follows from the open mapping theorem (Theorem 2.11 in [28]) that the inverse operator T^-1 is continuous.Now let us introduce the notions and characterizations of cones and derivatives that will be used throughout this work (see [2, 29] for further details). Let S be a normed space, A⊂S a nonempty set, A¯ its clausure in the norm topology, and x∈A¯. the natural distance map from the point s∈S to the set A will be denoted by d(A,s) and the first infinite ordinal by ω. The Bouligand contingent coneTA(x) to A at x is defined by
(6)TA(x)≔{v∈S:liminfh→0+d(A,x+hv)h=0}.
Therefore, v∈TA(x) if and only if, there exist two sequences: {hn}n<ω⊂ℝ converging to 0+ (hn→0+ for short) and {vn}n<ω⊂S converging to v (S∋vn→v for short), such that x+hnvn∈A for every n<ω. Let L⊂S a nonempty set and x∈A¯∩L¯. The Bouligand paratingent conePAL(x) to A relative to L at x is defined by
(7)PAL(x)≔{v∈S:liminfh→0+,L∋x^→xd(A,x^+hv)h=0}.
When L=A, we set PA(x)≔PAA(x). Therefore, v∈PAL(x) if and only if, there exist three sequences: hn→0+, L∋xn→x, and S∋vn→v such that xn+hnvn∈A for every n<ω. Then PAL(x)⊃TA(x). Besides, when L={x} we have TA(x)=PAL(x). Hence, behaviour of paratingent cone respect to perturbations around the point x is stabler than that of contingent cone.In Section1, we noted that paratingent cone is the biggest cone among all of the cones cited there. The following example shows a situation in which contingent cone vanishes, whereas paratingent cone does not. On that, we consider the Banach space of all bounded real sequences, ℓ∞={(xn)∈ℝω:supn|xn|<ω}, endowed with the supremum norm, ∥x∥∞=supn|xn| for every x=(xn)∈ℓ∞. In addition, we will denote by 0 the real number and by 0ℓ∞ the zero element of ℓ∞.Example 2.
Define the set-valued mapΣ:{0}∪{2-i:i<ω}⊂ℝ⇝ℓ∞ by Σ(0)={0ℓ∞} and, for every i<ω,
(8)Σ(2-i)≔{(-2-i-1,…,-2-i-1︸ithcoordinate,0ℓ∞),(0,…,0,2-i︸i+1-thcoordinate,2-i,…)}.
Then
(9)TGraph(Σ)(0,0ℓ∞)={(0,0ℓ∞)},
but
(10)(-12,(12,12,…))∈PGraph(Σ)(0,0ℓ∞).Finally, we now introduce the type of derivatives that we will handle. LetS1 and S2 be two normed spaces, F:A⊂S1⇝S2 a set-valued map, and x0∈Dom(F)≔{x∈A:F(x)≠∅}. It is said that F is lower semicontinuous at x0 if for every y∈F(x0) and any sequence Dom(F)∋xn→x*, there exists a sequence Y∋yn→y such that yn∈F(xn) for every n<ω. Let us fix (x,y)∈Graph(F)≔{(x,y)∈S1×S2:y∈F(x)}, the contingent derivative of F at (x,y)∈Graph(F) is the set-valued map DF(x,y):S1⇝S2 defined by Graph(DF(x,y))≔TGraph(F)(x,y). Now let us fix also a subset L⊂Graph(F), the paratingent derivative of F relative to L at (x,y)∈L is the set-valued map PLF(x,y):S1⇝S2 defined by Graph(PLF(x,y))≔PGraph(F)L(x,y). If L=Graph(F), then PLF(x,y) is written as PF(x,y). Finally, F is said to be paraderivable relative to L⊂Graph(F) at (x,y)∈L if DF(x,y)=PLF(x,y). As can be seen in [11], if F is single-valued and Fréchet differentiable at x, then F is derivable at (x,F(x)) and DF(x,F(x))(u)=F′(x,u) for every u∈S1. However, in our framework, a single-valued map F has to be continuously Fréchet differentiable at x in order to be paraderivable relative to L⊂Graph(F) at (x,F(x))∈L; in this case we have PLF(x,F(x))(u)=F′(x,u) for every u∈S1.


## 3. Regularity Condition and First Results on Paraderivability

In this section the regularity condition𝒯 will be established. It will allow us to transfer the condition of paraderivability from a set-valued map  Σ:V⇝ℬ(Z,Y) to the corresponding  Σˇ:V⇝Y defined by Σˇ(b)≔Σ(b)(b). It is known that if Σ is a single-valued and Fréchet differentiable map then Σˇ is also Fréchet differentiable (see Lemma 11 in [4]).From now on, the following notation will be used. Given a set-valued mapΣ:V⇝ℬ(Z,Y), the set-valued map Σˇ:V⇝Y will be defined by Σˇ(b)=Σ(b)(b) for every b∈V. Given a set L⊂Graph(Σ), the set Lˇ will be defined by Lˇ={(b,G(b)):(b,G)∈L}⊂Graph(Σˇ).Definition 3.
A set-valued mapΣ:V⇝ℬ(Z,Y) is said to have property 𝒯(L) at (b*,G*)∈L⊂Graph(Σ) if after fixing the following four sequences: (i)
hn→0+;(ii)
Z∋bn→b∈Z;(iii)
Lˇ∋(b^n,G^n(b^n))→(b*,G*(b*));(iv)
{Gn}n<ω⊂ℬ(Z,Y) such that each Gn∈Σ(b^n+hnbn) and there exists
(11)limn→ωGn(b^n+hnbn)-G^n(b^n)hn∈Y;there exist three sequences(1)
Z∋b¯n→b∈Z;(2)
ℬ(Z,Y)∋G~n→G* such that each G~n∈Σ(b^n) and G~n(b^n)=G^n(b^n);(3)
{G¯n}n<ω⊂ℬ(Z,Y) verifying that each G¯n∈Σ(b^n+hnb¯n), in such a way that
(12)limn→ωG¯n(b^n+hnb¯n)-G~n(b^n)hn=limn→ωGn(b^n+hnbn)-G^n(b^n)hn∈Y,such that there exits
(13)limn→ωG¯n-G~nhn∈ℬ(Z,Y).The interpretation of the former definition can be as follows. It is not restrictive to suppose that the two sequences of linear and continuous maps,{Gn}n and {G^n}n, related by the pointwise limit (11) are, in fact, related by the uniform version of this limit. In the next result we will see how, by means of property 𝒯, it is possible to transfer the paraderivability from the set-valued map Σ to the set-valued map Σˇ defined above.Theorem 4.
LetΣ:V⇝ℬ(Z,Y) be a set-valued map with property 𝒯(L) at (b*,G*)∈L⊂Graph(Σ) and paraderivable relative to L at (b*,G*). Then Σˇ is also paraderivable relative to Lˇ at (b*,G*(b*))∈Lˇ and
(14)PLˇΣˇ(b*,G*(b*))(z)=PLΣ(b*,G*)(z)(b*)+G*(z),∀z∈Z.Proof.
The paraderivability ofΣˇ is a direct consequence of the equality (14). Thus, let us begin the proof by showing the equality. the inclusion ⊂ will be first proved. For this purpose we fix z∈Z and y∈PLˇΣˇ(b*,G*(b*))(z). Thus (z,y)∈PGraph(Σˇ)Lˇ(b*,G*(b*)) and there exist hn→0+, Z∋bn→z, Lˇ∋(b^n,G^n(b^n))→(b*,G*(b*)), and {Gn}n<ω⊂ℬ(Z,Y) such that each Gn∈Σ(b^n+hnbn) and
(15)limn→ωGn(b^n+hnbn)-G^n(b^n)hn=y.
Since Σ verifies property 𝒯(L) at (b*,G*), we can fix the three sequences of the second part of Definition 3. Let us recall that the following equality holds
(16)limn→ωG¯n(b^n+hnb¯n)-G~n(b^n)hn=limn→ωGn(bn+hnbn)-G^n(b^n)hn.
We define
(17)G≔limn→ωG¯n-G~nhn∈ℬ(Z,Y).
Hence, G∈PLΣ(b*,G*)(z) because L∋(b^n,G~n)→(b*,G*), hn→0+,
(18)(b¯n,G¯n-G~nhn)⟶(z,G),(b^n,G~n)+hn(b¯n,G¯n-G~nhn)=(b^n+hnb¯n,G¯n)∈Graph(Σ),∀n<ω.
From (15) and (16) we get
(19)y=limn→ωGn(b^n+hnbn)-G^n(b^n)hn=limn→ωG¯n(b^n+hnb¯n)-G~n(b^n)hn=limn→ω[G¯n-G~nhn(b^n)+G¯n(b¯n)]=G(b*)+G*(z).
Finally, since G∈PLΣ(b*,G*)(z), then y∈PLΣ(b*,G*)(z)(b*)+G*(z).
For the reverse inclusion,⊃, we fix z∈Z and y∈PLΣ(b*,G*)(z)(b*)+G*(z). Let us fix now G∈PLΣ(b*,G*)(z) such that y=G(b*)+G*(z). By definition, there exist L∋(b^n,G^n)→(b*,G*), hn→0+, Z×ℬ(Z,Y)∋(bn,Gn)→(z,G), such that each
(20)(b^n,G^n)+hn(bn,Gn)=(b^n+hnbn,G^n+hnGn)∈Graph(Σ).
Then Rn≔G^n+hnGn∈Σ(b^n+hnbn),  for all  n<ω. Moreover
(21)G=limn→ωRn-G^nhn,
and consequently,
(22)y=G(b*)+G*(z)=limn→ωRn-G^nhn(b^n)+Rn(bn)=limn→ωRn(b^n+hnbn)-G^n(b^n)hn.
Therefore, (z,y)∈PGraph(Σˇ)Lˇ(b*,G*(b*)), and consequently y belongs to the set PLˇΣˇ(b*,G*(b*))(z).
Let us begin now the last part of the proof. In this, the paraderivability ofΣˇ relative Lˇ at (b*,G*(b*))∈Lˇ will be shown; that is, it will be proved the inclusion PLˇΣˇ(b*,G*(b*))(z)⊂DΣˇ(b*,G*(b*))(z), for each z∈Z. For this goal let us fix (z,y)∈PGraph(Σˇ)Lˇ(b*,G*(b*)), the equality (14) and the paraderivability of Σ relative to L at (b*,G*)∈L yield that y∈DΣ(b*,G*)(z)(b*)+G*(z). Then there exists G∈DΣ(b*,G*)(z) such that y=G(b*)+G*(z). Hence, ∃Z×ℬ(Z,Y)∋(bn,Gn)→(z,G) and hn→0+ such that (b*,G*)+hn(bn,Gn)∈Graph(Σ), for every n<ω. Let us define now G^n≔G*+hnGn∈Σ(b*+hnbn), then there exists limn→+ω((G^n-G*)/hn)=G∈ℬ(Z,Y). Thus, if we define yn≔(G^n(b*+hnbn)-G*(b*))/hn, since Gn→G, we have that yn→y. Finally (b*,G*(b*))+hn(bn,yn)=(b*+hnbn,G^n(b*+hnbn))∈Graph(Σˇ), for all n<ω; therefore, the proof is over.The following example shows that in the former result neither assumptions can be dropped.Example 5.
(i) Let one definesΣ1:[0,+∞)→ℬ(ℝ,ℝ) by Σ1(x)≔x, then Σ1 has property 𝒯(Graph(Σ)) at (0,0ℬ(ℝ,ℝ)), but neither Σ1 nor Σˇ1 is paraderivable at (0,0ℬ(ℝ,ℝ)) and (0,0), respectively.
(ii) Let one definesΣ2:ℝ-∪{0}∪{2-i:i<ω}⇝ℬ(ℝ,ℓ∞) by Σ2(t)(x)≔0ℓ∞ if t∈ℝ-∪{0} and x∈ℝ, and for every i<ω,
(23)Σ2(2-i)(x)≔{(-x2,…,-x2︸ithcoordinate,0ℓ∞),(0,…,0,x︸i+1-thcoordinate,x,…)(-x2,…,-x2︸ithcoordinate,0ℓ∞)},
for every x∈ℝ. Then Σ2 is paraderivable at (0,0ℬ(ℝ,ℓ∞)), but it does not have property 𝒯(Graph(Σ)) at (0,0ℬ(ℝ,ℓ∞)) and Σˇ2 is not paraderivable at (0,0ℓ∞).In the statement of the following result, the previously fixed notation is used and, in addition, we will consider the setπLˇ={(b,πG(b)):(b,G)∈L} which is a subset of Graph(πΣˇ).Lemma 6.
Let one assumes thatTΣ is a single-valued and continuously Fréchet differentiable at b*∈V map. If Σˇ is paraderivable relative to Lˇ at (b*,G(b*))∈Lˇ, then πΣˇ is paraderivable relative to πLˇ at (b*,πG(b*)) and
(24)PπLˇ(πΣˇ)(b*,πG*(b*))(z)=πPLˇΣˇ(b*,G*(b*))(z),(b*)(b*)(b*)(b*)(b*)(b*)(b*)(b*)(b*)(b*)∀z∈Z.Proof.
In this first stage of the proof we are going to check thatπΣˇ is paraderivable relative to πLˇ at (b*,πG(b*)). To do this, we have to show the inclusion PGraph(πΣˇ)πLˇ(b*,πG*(b*))⊂TGraph(πΣˇ)(b*,πG*(b*)). For this purpose, let us fix (z,y)∈PGraph(πΣˇ)πLˇ(b*,πG*(b*))⊂Z×KerT, then there exists hn→0+, πLˇ∋(b^n,πG^n(bn))→(b*,πG*(b*)), and Z×KerT∋(zn,yn)→(z,y) such that
(25)(b^n,πG^n(bn))+hn(zn,yn)∈Graph(πΣˇ),∀n<ω.
Therefore, for any n<ω, there exists Gn∈Σ(b^n+hnzn) such that
(26)yn=πGn(b^n+hnzn)-πG^n(b^n)hn∈KerT,∀n<ω.
Besides, for all n<ω, the following equality holds:
(27)Gn(b^n+hnzn)-G^n(b^n)hn=πGn(b^n+hnzn)-πG^n(b^n)hn+T^-1[TGn(b^n+hnzn)-TG^n(b^n)hn].
Now, Lemma 11 of [4] yields that TΣˇ is continuously Fréchet differentiable at b*, which provides that
(28)limn→ωGn(b^n+hnzn)-G^n(b^n)hn=y+T^-1(TΣˇ)′(b*,z)∈KerT⊕YT=Y.
Also, since for all n<ω, we have G^n(b^n)=πG^n(b^n)+T^-1(TΣˇ(bn)), it is followed that limn→ωG^n(b^n)=G*(b*), and then
(29)(z,y+T^-1(TΣˇ)′(b*,z))∈PGraph(Σˇ)Lˇ(b*,G*(b*)).
Paraderivability of Σˇ yields
(30)(z,y+T^-1(TΣˇ)′(b*,z))∈TGraph(Σˇ)(b*,G*(b*)).
Hence, there exist h¯n→0+ and Z×Y∋(z¯n,y¯n)→(z,y+T^-1(TΣˇ)′(b*,z)) such that
(31)(b*,G*(b*))+hn(z¯n,y¯n)∈Graph(Σˇ),∀n<ω.
Therefore,
(32)(b*,πG*(b*))+hn(z¯n,πy¯n)∈Graph(πΣˇ),∀n<ω.
Now, y∈KerT implies limn→ωπy¯n=πy+πT^-1(TΣˇ)′(b*,z)=πy=y, and then (z,y)∈TGraph(πΣˇ)(b*,πG*(b*)).
At this point, we have shown the paraderivability ofπΣˇ and the inclusion
(33)PπLˇ(πΣˇ)(b*,πG*(b*))(z)⊂πPLˇΣˇ(b*,G*(b*))(z),(b*,G*(b*))(b*,G*(b*))(b*,G*(b*)(b*bb))∀z∈Z.
The proof of the lemma will be over by showing the reverse of the former inclusion. Let us fix now y∈πPLˇΣˇ(b*,G*(b*))(z)⊂KerT and  y¯∈PLˇΣˇ(b*,G*(b*))(z)⊂Y such that y=πy¯. Let us consider hn→0+, Lˇ∋(bn,Gn(bn))→(b*,G*(b*)) and Z×Y∋(zn,y¯n)→(z,y¯) such that
(34)(bn,Gn(bn))+hn(zn,y¯n)∈Graph(Σˇ),∀n<ω.
Hence,
(35)(bn,πGn(bn))+hn(zn,πy¯n)∈Graph(πΣˇ),∀n<ω,
and limn→ωπy¯n=πy¯=y. Then y∈PπLˇ(πΣˇ)(b*,πG*(b*))(z) which finishes the proof.Now we begin the second part of this section with a consequence of the medium value theorem. It shows the advantage of working with𝒞1-class maps.Lemma 7.
Letf:V⊂Z→Y be a continuously Fréchet differentiable map, V∋bn→b*∈V, Z∋zn→z*≠0, and hn→0+. Then
(36)limn→ωf(bn+hnzn)-f(bn)hn=f′(b*,z*).Proof.
Firstly let us define the auxiliary mapg(v)≔f(v)-f′(b*,v), for every v∈V. Then it is also continuously Fréchet differentiable and g′(b*,z)=0, for every z∈Z. Hence, given ϵ>0, there exists a neighbourhood U of b* such that
(37)∥g′(u,·)∥≤ϵ∥z*∥,∀u∈U.
Now, applying the medium value theorem, we can write
(38)0≤limn→ω∥g(bn+hnzn)-g(bn)hn∥≤limn→ωϵ∥z*∥hn∥zn∥hn=ϵ.
Since ϵ>0 was previously arbitrarily fixed, then
(39)limn→ωg(bn+hnzn)-g(bn)hn=0.
In fact, the former limit allows us to compute the limit of the statement because
(40)limn→ω[f(bn+hnzn)-f(bn)hn-f′(b*,z*)]=limn→ωf(bn+hnzn)-f(bn)hn-limn→ωf′(b*,zn)=limn→ωf(bn+hnzn)-f(bn)hn-limn→ωf′(b*,bn+hnzn)-f′(b*,bn)hn=limn→ωg(bn+hnzn)-g(bn)hn=0,
and the proof is over.In the following result we see how a condition in form of inclusion for a set-valued mapΣ:V⊂Z⇝ℬ(Z,Y) allows us to turn some pointwise limits, of linear and continuous maps, into uniform ones in a stronger way than property 𝒯 does. It will be useful in the proof of the main theorem of the work, in Section 4.Proposition 8.
Letα:V⊂Z→ℬ(Z,Y) and β:V⊂Z→Z* be two continuously Fréchet differentiable maps such that β(b)(b)=1 for every b∈V. Let one fixes also a set-valued map Σ:V⇝ℬ(Z,Y) such that for every b∈V  Σ(b)⊂{α(b)+yβ(b):y∈Y}, a point (b*,G*)∈L⊂Graph(Σ), and four sequences: Z∋bn→z, hn→0+, Lˇ∋(b^n,G^n(b^n))→(b*,G*(b*)), and {Gn}n<ω⊂ℬ(Z,Y) such that each Gn∈Σ(b^n+hnbn). Now, if there exists
(41)limn→ωGn(b^n+hnbn)-G^n(b^n)hn∈Y,
then there exists
(42)limn→ωGn-G^nhn∈ℬ(Z,Y)aswell.Proof.
In the first place, the inclusion ofΣ(b) in the statement yields that there are y*∈Y and two sequences {yn}n<ω and {y^n}n<ω of elements of Y such that G*=α(b*)+y*β(b*), Gn=α(b^n+hnbn)+ynβ(b^n+hnbn), and G^n=α(b^n)+y^nβ(b^n) for every n<ω.
Now, on the one hand,(43)Gn(b^n+hnbn)-G^n(b^n)hn=α(b^n+hnbn)(b^n+hnbn)-α(b^n)(b^n)hn+yn-y^nhn,∀n<ω.
On the other hand, sinceα is continuously Fréchet differentiable, the former lemma and Lemma 10 of [6] yield
(44)limn→ωyn-y^nhn=y-α′(b*,z)(b*)-α(b*)(z),
where z=limn→ωbn∈Z and y=limn→ω((Gn(b^n+hnbn)-G^n(b^n))/hn)∈Y.
To conclude, for everyn<ω, we decompose
(45)Gn-G^nhn=α(b^n+hnbn)+ynβ(b^n+hnbn)-α(b^n)-y^nβ(b^n)hn=α(b^n+hnbn)-α(b^n)hn+ynβ(b^n+hnbn)-β(b^n)hn+yn-y^nhnβ(b^n).
The former decomposition allows us to arrive at
(46)limn→ωGn-G^nhn=α′(b*,z)+y*β′(b*,z)+[y-α′(b*,z)(b*)-α(b*)(z)]β(b*),
and thus, the proof is over.Let us compute now the formula for the paratingent derivative of the sum of a set-valued map and a single-valued map. For the formula of the sum with other derivatives, we refer the reader to [2].Proposition 9.
Letf:V⊂Z→Y be a continuously Fréchet differentiable map, F:V⊂Z⇝Y a set-valued map, L⊂Graph(F), (b*,y*)∈L, and L′={(b,f(b)+y)∈V×Y:(b,y)∈L}. Then
(47)PL′(f+F)(b*,f(b*)+y*)(z)=f′(b*,z)+PLF(b*,y*)(z),∀z∈Z.Proof.
Let us begin by checking the inclusion⊂. For this purpose we fix arbitrary elements z∈Z and y∈PL′(f+F)(b*,f(b*)+y*)(z). Now, by the usual characterization of the paratingent cone we have three sequences: hn→0+, Z×Y∋(zn,yn)→(z,y), and L′∋(bn,f(bn)+y*)→(b*,f(b*)+y*). Moreover, the following condition holds true for every n<ω:
(48)f(bn)+yn*+hnyn∈f(bn+hnzn)+F(bn+hnzn).
Then, for every n<ω, there exists z^n∈F(bn+hnzn) such that
(49)yn=f(bn+hnzn)-f(bn)hn+z^n-yn*hn∈Y,
which yields
(50)limn→ωz^n-yn*hn=y-f′(b*,z).
Finally, the definition of paratingent cone assures that the last limit belongs to PLF(b*,y*)(z), and hence, y-f′(b*,z) also does.
In order to prove the reverse inclusion, we fix arbitraryz∈Z and y∈f′(b*,z)+PLF(b*,y*)(z). We use again the characterization of paratingent cones which provides us the usual three sequences: hn→0+, Z×Y∋(zn,yn)→(z,y-f′(b*,z)), and L∋(bn,yn*)→(b*,y*) in such away that
(51)yn*+hnyn∈F(bn+hnzn),∀n<ω.
Now, for every n<ω, there exists z^n∈F(bn+hnzn) such that yn=(z^n-yn*)/hn, and the sequence defined by the general term
(52)wn=f(bn+hnzn)+z^n-f(bn)-yn*hn∈Y
converges to y. To conclude, since
(53)(bn,f(bn)+yn*)+hn(zn,wn)∈Graph(f+F),∀n<ω,
certainly
(54)y∈PL′(f+F)(b*,f(b*)+y*)(z).


## 4. Sensitivity Analysis

Theorem A will be proved in this section. However, before this, it will be necessary to state and prove Theorem12. In this theorem, it is shown how the Fréchet differentiability of the single-valued map TΨ allows us to transfer the paraderivability from the T-dual perturbation map Ψ to the set-valued map Ψˇ. In its proof, the obtained results in Section 3 are applied, taking Σ as Ψ. In addition, we will need to establish also some previous technical results which will constitute the first part of this section.Throughout this section we will assume, firstly, that the parameterb belongs to an open convex set V⊂Z such that 0∉V. This condition is not a restriction because the problem (1) with b=0 is equivalent to some problem (1) with b≠0. If we keep this assumption in mind, the following claim can be proved. There exists a continuously Fréchet differentiable map β:V→Z* such that β(v)(v)=1 for each v∈V. This is a consequence of the Hanh-Banach theorem, Theorem 3.4 of [28].In the second place, we will assume that for everyb∈V there exists a regular local T-optimal solution xb∈D of (1) in such a way that the map λ:V→X given by λ(b)=xb is Fréchet differentiable. The existence of this kind of map has been studied by several authors; the linear case can be seen in [30]. This assumption jointly with Proposition 8 of [6] implies that TΨ is a single-valued map, that is, TG=TG′∈ℬ(Z,W) for every two elements G, G′∈Ψ(b) and b∈V.Definition 10.
Letb∈V, xb∈D a regular local T-optimal solution of (1), and Gxb∈ℬ(Z,Y) a T-Lagrange multiplier of (1) associated with xb. The (T,β)-modification of Gxb is the map defined by
(55)Gxb(T,β)(z)≔T^-1TGxb(z)+πGxb(b)β(b)(z),∀z∈Z.It is easy to check thatGxb(T,β) is a T-Lagrange multiplier of (1) associated with xb. Moreover, (T,β)-modifications of two different T-Lagrange multipliers of (1) associated to the same regular local T-optimal solution coincide. Moreover, we have the following result.Proposition 11 (see Proposition 10 of [19]).
Letb∈V, xb∈D a regular local T-optimal solution of (1), Gxb∈ℬ(Z,Y) a T-Lagrange multiplier of (1) associated with xb, and 𝒰xb the set of the T-Lagrange multipliers of (1) associated with xb. Then
(56)𝒰xb={Gxb(T,β)}+ℋb,
where ℋb={H∈ℬ(Z,KerT):H(b)=0}.Next we will prove the following.Theorem 12.
Let one supposes that theT-dual perturbation map Ψ is paraderivable relative to L at (b*,G*)∈L⊂Graph(Ψ) and TΨ is Fréchet differentiable at b*∈V. Then Ψˇ is paraderivable relative to
(57)Lˇ≔{(b,G(b))∈Z×Y:(b,G)∈L}
at (b*,G*(b*)) and
(58)PLˇΨˇ(b*,G*(b*))(z)=PLΨ(b*,G*)(z)(b*)+G*(z),∀z∈Z.Proof.
The statement will be proved by applying Theorem4. Hence, it is enough to prove that Ψ verifies property 𝒯(L) at (b*,G*)∈L. For this, let us fix the four sequences which appear in the first part of the definition of Definition 3, that is, hn→0+, Z∋bn→b∈Z, Lˇ∋(b^n,G^n(b^n))→(b*,G*(b*)), and {Gn}n<ω⊂ℬ(Z,Y) such that each Gn∈Σ(b^n+hnbn) and the below limit exists
(59)limn→ωGn(b^n+hnbn)-G^n(b^n)hn∈Y.
Now we can consider, on the one hand, the continuously Fréchet differentiable mapβ:V→Z* defined at the beginning of this section and such that β(b)(b)=1 for every b∈V. On the other hand, we consider the auxiliary maps Gn(T,β) and G^n(T,β) for every n<ω. By Proposition 11 we have
(60)Gn(b^n+hnbn)=Gn(T,β)(b^n+hnbn),G^n(b^n)=G^n(T,β)(b^n),
for every n<ω. Moreover, the below limit exists
(61)limn→ωGn(T,β)(b^n+hnbn)-G^n(T,β)(b^n)hn∈Y.
Then Proposition 8 applied to the set-valued map
(62)Ψ(T,β):V⇝ℬ(Z,Y),b⇝Ψ(T,β)(b)={G(T,β)∈ℬ(Z,Y):G∈Ψ(b)}
implies that the following also exists:
(63)limn→ωGn(T,β)-G^n(T,β)hn∈ℬ(Z,Y).
Now, we have all necessary ingredients at hand in order to check thatΨ enjoys property 𝒯(L) at (b*,G*)∈L. For this purpose, we will define three sequences which verify the conditions stated in the second part of Definition 3. In fact, for each n<ω, we consider b¯n≔bn, and define the maps G~n and G¯n by
(64)G~n(z)≔G^n(T,β)(z)+(G*-G*(T,β))(z)+(G*(T,β)-G*)(b^n)β(b^n)(z),G¯n(z)≔Gn(T,β)(z)+(G*-G*(T,β))(z)+(G*(T,β)-G*)(b^n+hnbn)β(b^n+hnbn)(z),
for every z∈Z. Clearly {G¯n}n<ω,{G~n}n<ω⊂ℬ(Z,Y). Again Proposition 11 yields G¯n∈Ψ(b^n+hnbn) and G~n∈Ψ(b^n) for every n<ω. Furthermore, since each G¯n(b^n+hnbn)=Gn(T,β)(b^n+hnbn) and G~n(b^n)=G^n(T,β)(b^n), from (60)  we get G¯n(b^n+hnbn)=Gn(b^n+hnbn) and G~n(b^n)=G^n(b^n). This provides the equality
(65)limn→ωG¯n(b^n+hnb¯n)-G~n(b^n)hn=limn→ωGn(b^n+hnbn)-G^n(b^n)hn∈Y.
The last step in the proof is to check that limn→ω(G¯n-G~n)/hn exists. Indeed, for every n<ω,
(66)G¯n-G~nhn=Gn(T,β)-G^n(T,β)hn+(G*-G*(T,β))(b^n)β(b^n+hnbn)-β(b^n)hn+(G*-G*(T,β))(bn)β(b^n+hnbn).
Now, taking into account thatβ is continuously differentiable at b*, from Lemma 7, we obtain that
(67)limn→ωβ(b^n+hnbn)-β(b^n)hn=β′(b*,b).
Therefore,
(68)limn→ωG¯n-G~nhn=limn→ωGn(T,β)-G^n(T,β)hn+(G^*-G^*(T,β))(b*)β′(b*,b)+(G*-G*(T,β))(b)β(b*),
and the proof is over.Now we can prove Theorem A.Proof of Theorem A.
In the first part of the proof we will check thatπΥ is paraderivable relative to πLˇΨ at (b*,πf(xb*)). After that we will finish the proof by means of the equality
(69)Υ(b)=T^-1TΥ(b)+πΥ(b),∀b∈V.
Since πΥ(b)=πΨˇ(b) for every b∈V, it is enough to check that the set-valued map πΨˇ is paraderivable relative to πLˇΨ at (b*,πGxb*(b*)). Applying Theorem 12 we get that Ψˇ is paraderivable relative to
(70)LˇΨ={(b,G(b))∈Z×X:(b,G)∈LΨ},
at (b*,Gxb*(b*)) and
(71)PLˇΨΨˇ(b*,Gxb*(b*))(z)=PLΨΨ(b*,Gxb*)(z)(b*)+Gxb*(z),∀z∈Z.
By Lemma 6  πΨˇ is paraderivable relative to πLˇΨ at (b*,πGxb*(b*)) and
(72)πPLˇΨΨˇ(b*,Gxb*(b*))(z)=PπLˇΨπΨˇ(b*,πGxb*(b*))(z),∀z∈Z.
Now, the equality πΥ=πΨˇ and (71) yield
(73)PπLˇΨπΥ(b*,πf(xb*))(z)=πGxb*(z)+πPLΨΨ(b*,Gxb*)(z)(b*),∀z∈Z.
In order to finish the proof we consider the equality (69). On the one hand, by Proposition 8 of [6], the function TΥ is continuously Fréchet differentiable at b* and [TΥ]′(b*,z)=TGxb*(z), for every z∈Z. Thus, [T^-1TΥ]′(b*,z)=T^-1TGxb*(z) for every z∈Z. On the other hand, if we apply Proposition 9 to equality (69), we obtain that the set-valued map Υ is paraderivable relative to LΥ at (b*,f(xb*)) and
(74)PLΥΥ(b*,f(xb*))(z)=T^-1TGxb*(z)+πGxb*(z)+πPLΨΨ(b*,Gxb*)(z)(b*)=Gxb*(z)+πPLΨΨ(b*,Gxb*)(z)(b*),∀z∈Z.This section is finished by illustrating Theorem A with the aid of an example.Example 13.
Let one considersX=Y=ℝ3,Z=W=ℝ, the interval  V=(-11/10,9/10)⊂ℝ,D={(x,y,z)∈ℝ3:2coshy2>z}, and the problem
(75)Min(x-2y+z,x+2y+z,-2x+2z);-log(2coshy2-z)=b,(x,y,z)∈D,
for every b∈V.
Let us takeT=(1,1,2). Solving the problem we obtain the T-perturbation map
(76)Υ(b)={(2eb-1eb-λ,2eb-1eb-λ,22eb-1eb+2λ)∈ℝ3:λ∈ℝ(2eb-1eb-λ,2eb-1eb-λ,22eb-1eb+2λ)},
for every b∈V.
Let us study the sensitivity of the problem (75) at b=-1, the point x-1=(0,0,2-e), and so f(x1)=(2-e,2-e,2(2-e)), taking L=Graph(Υ). We first analyze directly the sensitivity by calculating
(77)PΥ(-1,f(x-1))(u)={(eu+λ,eu+λ,2eu-2λ):λ∈ℝ},∀u∈ℝ.
Let us now apply Theorem A to verify (77). Since KerT is the linear space generated by the vectors (-2,0,1) and (-1,1,0), taking YT=(KerT)⊥ we obtain Gx-1(u)=(eu,eu,2eu) and the T-dual perturbation map
(78)Ψ(b)={(e-b+λb,e-b+λb,2e-b-2λb):λ∈ℝ}.
Thus, we have
(79)PΨ(-1,Gx-1)(u)(b)={(-e-bu-λb2,-e-bu-λb2,-2e-bu+2λb2):λ∈ℝ},
for every u∈ℝ, and therefore
(80)πPΨ(-1,Gx-1)(u)(-1)={(-λ,-λ,2λ):λ∈ℝ}.
Finally we obtain that
(81)Gx-1(u)+πPΨ(-1,Gx-1)(u)(-1)=(eu,eu,2eu)+{(-λ,-λ,2λ):λ∈ℝ}={(eu-λ,eu-λ,2eu+2λ):λ∈ℝ}=PΥ(-1,f(x-1))(u)
Consequently, equality (4) holds as Theorem A states.


## 5. Concluding Remarks

The objective of this article is to analyze the sensitivity of a multiobjective differential program with equality constraints. Given a family of parameterized programs, theT-perturbation set-valued map is defined as the correspondence that assigns to each right-hand side parameter value the set of minimal points of its associated program, on which a positive linear continuous map, T, takes a minimum value. The behavior of the T-perturbation map is analyzed quantitatively by making use of the paratingent derivative. The main result of the paper is stated in Theorem A, where we assert that the sensitivity of the program is measured by a Lagrange multiplier plus the projection of its derivative onto the kernel of T. The use of the paratingent derivative in the analysis transmits to the obtained result its distinctive stability with respect to disturbances around the point where the analysis is performed. In previous papers, some authors have carried out this kind of analysis by means of other derivatives. By doing so, this research completes the study of sensitivity of T-optimal solutions of this program by means of the four main set-valued derivatives, that is, contingent, adjacent, circatangent, and paratingent derivatives.



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