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

Abstract

We analyse the sensitivity of differential programs of the form Min𝑓(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.

1. Introduction

The subject of this work is sensitivity analysis in vector programming. The model problem considered throughout the paper is of the form (𝑥)(𝑥)Min𝑓subjectto𝑔=𝑏𝑥∈𝐷. (1) 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 maps 𝑓 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-called 𝑇-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 lattice 𝑊 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 𝑥_𝑏∈𝐷 is a local 𝑇-optimal solution of (1) when there exists a neighbourhood 𝑈_𝑥𝑏⊂𝐷 of 𝑥_𝑏 such that 𝑇𝑓(𝑥_𝑏)≤𝑇𝑓(𝑥) for every feasible 𝑥∈𝑈_𝑥𝑏. It is clear that every local 𝑇-optimal solution of (1) is a local optimal solution of the program, that is, 𝑓(𝑥_𝑏)−𝑓(𝑥)∉𝑌₊⧵{0} for every feasible 𝑥∈𝑈_𝑥𝑏. 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 regular with 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 𝑥_𝑏∈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 𝑥_𝑏 if 𝑇𝑓^′(𝑥_𝑏,⋅)=𝑇𝐺𝑔^′(𝑥_𝑏,⋅) and 𝜋𝑓(𝑥_𝑏)=𝜋𝐺(𝑏). In [6], it is proved that for every regular local 𝑇-optimal solution 𝑥_𝑏∈𝐷 of (1), there exists a 𝑇-Lagrange multiplier 𝐺_𝑥𝑏∈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 Υ(𝑏)∶=𝑓𝑥_𝑏∶𝑥_𝑏(1;isaregularlocal𝑇-optimalsolutionof) (2)(ii)𝑇-dual perturbation map of (1), Ψ∶𝑉⊂𝑍⇝ℬ(𝑍,𝑌), by Ψ(𝑏)∶=𝐺_𝑥𝑏(1)∶itisa𝑇-Lagrangemultiplierofassociatedwith𝑥_𝑏. (3)

Now we can state the main result of the work. In this, it is represented by 𝑃^𝐿ΥΥ (resp., 𝑃^𝐿ΨΨ), the paratingent derivative of Υ (resp., Ψ) relative to 𝐿_Υ (resp., 𝐿_Ψ).

Theorem A. Let one fixes 𝐿_Υ⊂𝐺𝑟𝑎𝑝ℎ(Υ),  (𝑏_∗,𝑓(𝑥_𝑏∗))∈𝐿_Υ, and 𝐿_Ψ={(𝑏,𝐺_𝑥𝑏)∈𝐺𝑟𝑎𝑝ℎ(Ψ)∶(𝑏,𝑓(𝑥_𝑏))∈𝐿_Υ and 𝐺_𝑥𝑏 is associated with 𝑥_𝑏}. If Ψ is is lower semicontinuous, paraderivable relative to 𝐿_Ψ at (𝑏_∗,𝐺_𝑥𝑏∗)∈𝐿_Ψ and 𝑇Ψ is continuously Fréchet differentiable at 𝑏_∗, then Υ is lower semicontinuous, paraderivable relative to 𝐿_Υ at (𝑏_∗,𝑓(𝑥_𝑏∗)), and 𝑃^𝐿ΥΥ𝑏_∗,𝑓𝑥_𝑏∗(𝑧)=𝐺_𝑥𝑏∗(𝑧)+𝜋𝑃^𝐿ΨΨ𝑏_∗,𝐺_𝑥𝑏∗(𝑧)𝑏_∗,∀𝑧∈𝑍. (4)

If 𝐿_Υ=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 that Ψ is lower semicontinuous, paraderivable relative to 𝐺𝑟𝑎𝑝ℎ(Ψ) at (𝑏_∗,𝐺_𝑥𝑏∗)∈𝐺𝑟𝑎𝑝ℎ(Ψ), 𝑇Ψ is continuously Fréchet differentiable at 𝑏_∗, then Υ is lower semicontinuous, paraderivable relative to 𝐺𝑟𝑎𝑝ℎ(Υ) at (𝑏_∗,𝑓(𝑥_𝑏∗)), and 𝑃Υ𝑏_∗,𝑓𝑥_𝑏∗(𝑧)=𝐺_𝑥𝑏∗(𝑧)+𝜋𝑃Ψ𝑏_∗,𝐺_𝑥𝑏∗(𝑧)𝑏_∗,∀𝑧∈𝑍. (5)

The proof of Theorem A is based on the fact that the set-valued map Ψ^̌‌, defined by Ψ^̌‌(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 Ψ^̌‌ 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 Section 2, 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 Σ^̌‌∶𝑉⇝𝑌 defined by Σ^̌‌(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 Σ^̌‌ to 𝜋Σ^̌‌, 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 Ψ^̌‌. 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 that 𝑋 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 𝑌_𝑇. The symbol 𝑇 represents the restriction of 𝑇 to 𝑌_𝑇. 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 cone 𝑇_𝐴(𝑥) to 𝐴 at 𝑥 is defined by 𝑇_𝐴(𝑥)∶=𝑣∈𝑆∶liminf_ℎ→0+𝑑(𝐴,𝑥+ℎ𝑣)ℎ=0. (6) Therefore, 𝑣∈𝑇_𝐴(𝑥) if and only if, there exist two sequences: {ℎ_𝑛}_𝑛<𝜔⊂ℝ converging to 0⁺ (ℎ_𝑛→0⁺ for short) and {𝑣_𝑛}_𝑛<𝜔⊂𝑆 converging to 𝑣 (𝑆∋𝑣_𝑛→𝑣 for short), such that 𝑥+ℎ_𝑛𝑣_𝑛∈𝐴 for every 𝑛<𝜔. Let 𝐿⊂𝑆 a nonempty set and 𝑥∈𝐴∩𝐿. The Bouligand paratingent cone 𝑃^𝐿_𝐴(𝑥) to 𝐴 relative to 𝐿 at 𝑥 is defined by 𝑃^𝐿_𝐴(𝑥)∶=𝑣∈𝑆∶liminf_{ℎ→0+,𝐿∋}𝑥_→𝑥𝑑(𝐴,𝑥+ℎ𝑣)ℎ=0. (7) When 𝐿=𝐴, we set 𝑃_𝐴(𝑥)∶=𝑃^𝐴_𝐴(𝑥). Therefore, 𝑣∈𝑃^𝐿_𝐴(𝑥) if and only if, there exist three sequences: ℎ_𝑛→0⁺, 𝐿∋𝑥_𝑛→𝑥, and 𝑆∋𝑣_𝑛→𝑣 such that 𝑥_𝑛+ℎ_𝑛𝑣_𝑛∈𝐴 for every 𝑛<𝜔. Then 𝑃^𝐿_𝐴(𝑥)⊃𝑇_𝐴(𝑥). Besides, when 𝐿={𝑥} we have 𝑇_𝐴(𝑥)=𝑃^𝐿_𝐴(𝑥). Hence, behaviour of paratingent cone respect to perturbations around the point 𝑥 is stabler than that of contingent cone.

In Section 1, 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_∞={(𝑥_𝑛)∈ℝ^𝜔∶sup_𝑛|𝑥_𝑛|<𝜔}, endowed with the supremum norm, ‖𝐱‖_∞=sup_𝑛|𝑥_𝑛| for every 𝐱=(𝑥_𝑛)∈ℓ_∞. In addition, we will denote by 0 the real number and by 𝟎_ℓ∞ the zero element of ℓ_∞.

Example 2. Define the set-valued map Σ∶{0}∪{2^−𝑖∶𝑖<𝜔}⊂ℝ⇝ℓ_∞ by Σ(0)={𝟎_ℓ∞} and, for every 𝑖<𝜔, Σ2^−𝑖∶=−2^−𝑖−1,…,−2^−𝑖−1_{𝑖thcoordinate},𝟎_ℓ∞,0,…,0,2^−𝑖_{𝑖+1-thcoordinate},2^−𝑖,…. (8) Then 𝑇_Graph(Σ)0,𝟎_ℓ∞=0,𝟎_ℓ∞, (9) but −12,12,12,…∈𝑃_Graph(Σ)0,𝟎_ℓ∞. (10)

Finally, we now introduce the type of derivatives that we will handle. Let 𝑆₁ 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)∋𝑥_𝑛→𝑥_∗, there exists a sequence 𝑌∋𝑦_𝑛→𝑦 such that 𝑦_𝑛∈𝐹(𝑥_𝑛) 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))∶=𝑇_Graph(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 𝑃^𝐿𝐹(𝑥,𝑦)∶𝑆₁⇝𝑆₂ defined by Graph(𝑃^𝐿𝐹(𝑥,𝑦))∶=𝑃^𝐿_Graph(𝐹)(𝑥,𝑦). If 𝐿=Graph(𝐹), then 𝑃^𝐿𝐹(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)=𝑃^𝐿𝐹(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 𝑃^𝐿𝐹(𝑥,𝐹(𝑥))(𝑢)=𝐹^′(𝑥,𝑢) for every 𝑢∈𝑆₁.

3. Regularity Condition and First Results on Paraderivability

In this section the regularity condition 𝒯 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 Σ^̌‌∶𝑉⇝𝑌 defined by Σ^̌‌(𝑏)∶=Σ(𝑏)(𝑏). It is known that if Σ is a single-valued and Fréchet differentiable map then Σ^̌‌ is also Fréchet differentiable (see Lemma 11 in [4]).

From now on, the following notation will be used. Given a set-valued map Σ∶𝑉⇝ℬ(𝑍,𝑌), the set-valued map Σ^̌‌∶𝑉⇝𝑌 will be defined by Σ^̌‌(𝑏)=Σ(𝑏)(𝑏) for every 𝑏∈𝑉. Given a set 𝐿⊂Graph(Σ), the set 𝐿^̌‌ will be defined by 𝐿^̌‌={(𝑏,𝐺(𝑏))∶(𝑏,𝐺)∈𝐿}⊂Graph(Σ^̌‌).

Definition 3. A set-valued map Σ∶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)  ℎ_𝑛→0⁺; (ii)  𝑍∋𝑏_𝑛→𝑏∈𝑍; (iii)  𝐿^̌‌∋(𝑏̂_𝑛,𝐺_𝑛(𝑏̂_𝑛))→(𝑏_∗,𝐺_∗(𝑏_∗)); (iv)  {𝐺_𝑛}_𝑛<𝜔⊂ℬ(𝑍,𝑌) such that each 𝐺_𝑛∈Σ(𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛) and there exists lim_𝑛→𝜔𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝐺_𝑛𝑏̂_𝑛ℎ_𝑛∈𝑌; (11)

there exist three sequences (1)𝑍∋𝑏_𝑛→𝑏∈𝑍; (2)ℬ(𝑍,𝑌)∋𝐺^∼_𝑛→𝐺_∗ such that each 𝐺^∼_𝑛∈Σ(𝑏̂_𝑛) and 𝐺^∼_𝑛(𝑏̂_𝑛)=𝐺_𝑛(𝑏̂_𝑛); (3){𝐺_𝑛}_𝑛<𝜔⊂ℬ(𝑍,𝑌) verifying that each 𝐺_𝑛∈Σ(𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛), in such a way that lim_𝑛→𝜔𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝐺^∼_𝑛𝑏̂_𝑛ℎ_𝑛=lim_𝑛→𝜔𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝐺_𝑛𝑏̂_𝑛ℎ_𝑛∈𝑌, (12)such that there exits lim_𝑛→𝜔𝐺_𝑛−𝐺^∼_𝑛ℎ_𝑛∈ℬ(𝑍,𝑌). (13)

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, {𝐺_𝑛}_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 Σ^̌‌ defined above.

Theorem 4. Let Σ∶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 Σ^̌‌ is also paraderivable relative to 𝐿^̌‌ at (𝑏_∗,𝐺_∗(𝑏_∗))∈𝐿^̌‌ and 𝑃Σ^̌‌𝐿̌𝑏_∗,𝐺_∗𝑏_∗(𝑧)=𝑃^𝐿Σ𝑏_∗,𝐺_∗(𝑧)𝑏_∗+𝐺_∗(𝑧),∀𝑧∈𝑍. (14)

Proof. The paraderivability of Σ^̌‌ 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∈𝑃Σ^̌‌𝐿̌(𝑏_∗,𝐺_∗(𝑏_∗))(𝑧). Thus (𝑧,𝑦)∈𝑃^̌‌𝐿_{Graph(̌‌Σ)}(𝑏_∗,𝐺_∗(𝑏_∗)) and there exist ℎ_𝑛→0⁺, 𝑍∋𝑏_𝑛→𝑧, 𝐿^̌‌∋(𝑏̂_𝑛,𝐺_𝑛(𝑏̂_𝑛))→(𝑏_∗,𝐺_∗(𝑏_∗)), and {𝐺_𝑛}_𝑛<𝜔⊂ℬ(𝑍,𝑌) such that each 𝐺_𝑛∈Σ(𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛) and lim_𝑛→𝜔𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝐺_𝑛𝑏̂_𝑛ℎ_𝑛=𝑦. (15) 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 lim_𝑛→𝜔𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝐺^∼_𝑛𝑏̂_𝑛ℎ_𝑛=lim_𝑛→𝜔𝐺_𝑛𝑏_𝑛+ℎ_𝑛𝑏_𝑛−𝐺_𝑛𝑏̂_𝑛ℎ_𝑛. (16) We define 𝐺∶=lim_𝑛→𝜔𝐺_𝑛−𝐺^∼_𝑛ℎ_𝑛∈ℬ(𝑍,𝑌). (17) Hence, 𝐺∈𝑃^𝐿Σ(𝑏_∗,𝐺_∗)(𝑧) because 𝐿∋(𝑏̂_𝑛,𝐺^∼_𝑛)→(𝑏_∗,𝐺_∗), ℎ_𝑛→0⁺, 𝑏_𝑛,𝐺_𝑛−𝐺^∼_𝑛ℎ_𝑛⟶(),𝑏̂𝑧,𝐺_𝑛,𝐺^∼_𝑛+ℎ_𝑛𝑏_𝑛,𝐺_𝑛−𝐺^∼_𝑛ℎ_𝑛=𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛,𝐺_𝑛∈Graph(Σ),∀𝑛<𝜔. (18) From (15) and (16) we get 𝑦=lim_𝑛→𝜔𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝐺_𝑛𝑏̂_𝑛ℎ_𝑛=lim_𝑛→𝜔𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝐺^∼_𝑛𝑏̂_𝑛ℎ_𝑛=lim_𝑛→𝜔𝐺_𝑛−𝐺^∼_𝑛ℎ_𝑛𝑏̂_𝑛+𝐺_𝑛𝑏_𝑛=𝐺𝑏_∗+𝐺_∗(𝑧). (19) Finally, since 𝐺∈𝑃^𝐿Σ(𝑏_∗,𝐺_∗)(𝑧), then 𝑦∈𝑃^𝐿Σ(𝑏_∗,𝐺_∗)(𝑧)(𝑏_∗)+𝐺_∗(𝑧).
For the reverse inclusion, ⊃, we fix 𝑧∈𝑍 and 𝑦∈𝑃^𝐿Σ(𝑏_∗,𝐺_∗)(𝑧)(𝑏_∗)+𝐺_∗(𝑧). Let us fix now 𝐺∈𝑃^𝐿Σ(𝑏_∗,𝐺_∗)(𝑧) such that 𝑦=𝐺(𝑏_∗)+𝐺_∗(𝑧). By definition, there exist 𝐿∋(𝑏̂_𝑛,𝐺_𝑛)→(𝑏_∗,𝐺_∗), ℎ_𝑛→0⁺, 𝑍×ℬ(𝑍,𝑌)∋(𝑏_𝑛,𝐺_𝑛)→(𝑧,𝐺), such that each 𝑏̂_𝑛,𝐺_𝑛+ℎ_𝑛𝑏_𝑛,𝐺_𝑛=𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛,𝐺_𝑛+ℎ_𝑛𝐺_𝑛∈Graph(Σ). (20) Then 𝑅_𝑛∶=𝐺_𝑛+ℎ_𝑛𝐺_𝑛∈Σ(𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛),  for all  𝑛<𝜔. Moreover 𝐺=lim_𝑛→𝜔𝑅_𝑛−𝐺_𝑛ℎ_𝑛, (21) and consequently, 𝑦=𝐺𝑏_∗+𝐺_∗(𝑧)=lim_𝑛→𝜔𝑅_𝑛−𝐺_𝑛ℎ_𝑛𝑏̂_𝑛+𝑅_𝑛𝑏_𝑛=lim_𝑛→𝜔𝑅_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝐺_𝑛𝑏̂_𝑛ℎ_𝑛. (22) Therefore, (𝑧,𝑦)∈𝑃^̌‌𝐿_{Graph(̌‌Σ)}(𝑏_∗,𝐺_∗(𝑏_∗)), and consequently 𝑦 belongs to the set 𝑃Σ^̌‌𝐿̌(𝑏_∗,𝐺_∗(𝑏_∗))(𝑧).
Let us begin now the last part of the proof. In this, the paraderivability of Σ^̌‌ relative 𝐿^̌‌ at (𝑏_∗,𝐺_∗(𝑏_∗))∈𝐿^̌‌ will be shown; that is, it will be proved the inclusion 𝑃Σ^̌‌𝐿̌(𝑏_∗,𝐺_∗(𝑏_∗))(𝑧)⊂𝐷Σ^̌‌(𝑏_∗,𝐺_∗(𝑏_∗))(𝑧), for each 𝑧∈𝑍. For this goal let us fix (𝑧,𝑦)∈𝑃^̌‌𝐿_{Graph(̌‌Σ)}(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)∋(𝑏_𝑛,𝐺_𝑛)→(𝑧,𝐺) and ℎ_𝑛→0⁺ such that (𝑏_∗,𝐺_∗)+ℎ_𝑛(𝑏_𝑛,𝐺_𝑛)∈Graph(Σ), for every 𝑛<𝜔. Let us define now 𝐺_𝑛∶=𝐺_∗+ℎ_𝑛𝐺_𝑛∈Σ(𝑏_∗+ℎ_𝑛𝑏_𝑛), then there exists lim_𝑛→+𝜔((𝐺_𝑛−𝐺_∗)/ℎ_𝑛)=𝐺∈ℬ(𝑍,𝑌). Thus, if we define 𝑦_𝑛∶=(𝐺_𝑛(𝑏_∗+ℎ_𝑛𝑏_𝑛)−𝐺_∗(𝑏_∗))/ℎ_𝑛, since 𝐺_𝑛→𝐺, we have that 𝑦_𝑛→𝑦. Finally (𝑏_∗,𝐺_∗(𝑏_∗))+ℎ_𝑛(𝑏_𝑛,𝑦_𝑛)=(𝑏_∗+ℎ_𝑛𝑏_𝑛,𝐺_𝑛(𝑏_∗+ℎ_𝑛𝑏_𝑛))∈Graph(Σ^̌‌), 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 defines Σ₁∶[0,+∞)→ℬ(ℝ,ℝ) by Σ₁(𝑥)∶=𝑥, then Σ₁ has property 𝒯(Graph(Σ)) at (0,𝟎_ℬ(ℝ,ℝ)), but neither Σ₁ nor Σ^̌‌₁ is paraderivable at (0,𝟎_ℬ(ℝ,ℝ)) and (0,0), respectively.
(ii) Let one defines Σ₂∶ℝ₋∪{0}∪{2^−𝑖∶𝑖<𝜔}⇝ℬ(ℝ,ℓ_∞) by Σ₂(𝑡)(𝑥)∶=𝟎_ℓ∞ if 𝑡∈ℝ₋∪{0} and 𝑥∈ℝ, and for every 𝑖<𝜔, Σ₂2^−𝑖(𝑥)−𝑥∶=2𝑥,…,−_𝑖thcoor2_dinate,𝟎_ℓ∞,0,…,0,𝑥^}_{𝑖+1-thcoordinate},𝑥,…, (23) for every 𝑥∈ℝ. Then Σ₂ is paraderivable at (0,𝟎_{ℬ(ℝ,ℓ∞)}), but it does not have property 𝒯(Graph(Σ)) at (0,𝟎_{ℬ(ℝ,ℓ∞)}) and Σ^̌‌₂ is not paraderivable at (0,𝟎_ℓ∞).

In the statement of the following result, the previously fixed notation is used and, in addition, we will consider the set 𝜋𝐿^̌‌={(𝑏,𝜋𝐺(𝑏))∶(𝑏,𝐺)∈𝐿} which is a subset of Graph(𝜋Σ^̌‌).

Lemma 6. Let one assumes that 𝑇Σ is a single-valued and continuously Fréchet differentiable at 𝑏_∗∈𝑉 map. If Σ^̌‌ is paraderivable relative to 𝐿^̌‌ at (𝑏_∗,𝐺(𝑏_∗))∈𝐿^̌‌, then 𝜋Σ^̌‌ is paraderivable relative to 𝜋𝐿^̌‌ at (𝑏_∗,𝜋𝐺(𝑏_∗)) and 𝑃^{𝜋̌‌𝐿}𝜋Σ^̌‌𝑏_∗,𝜋𝐺_∗𝑏_∗(𝑧)=𝜋𝑃Σ^̌‌𝐿̌𝑏_∗,𝐺_∗𝑏_∗(𝑧),∀𝑧∈𝑍. (24)

Proof. In this first stage of the proof we are going to check that 𝜋Σ^̌‌ is paraderivable relative to 𝜋𝐿^̌‌ at (𝑏_∗,𝜋𝐺(𝑏_∗)). To do this, we have to show the inclusion 𝑃^{𝜋̌‌𝐿}_{Graph(𝜋̌‌Σ)}(𝑏_∗,𝜋𝐺_∗(𝑏_∗))⊂𝑇_{Graph(𝜋̌‌Σ)}(𝑏_∗,𝜋𝐺_∗(𝑏_∗)). For this purpose, let us fix (𝑧,𝑦)∈𝑃^{𝜋̌‌𝐿}_{Graph(𝜋̌‌Σ)}(𝑏_∗,𝜋𝐺_∗(𝑏_∗))⊂𝑍×Ker𝑇, then there exists ℎ_𝑛→0⁺, 𝜋𝐿^̌‌∋(𝑏̂_𝑛,𝜋𝐺_𝑛(𝑏_𝑛))→(𝑏_∗,𝜋𝐺_∗(𝑏_∗)), and 𝑍×Ker𝑇∋(𝑧_𝑛,𝑦_𝑛)→(𝑧,𝑦) such that 𝑏̂_𝑛,𝜋𝐺_𝑛𝑏_𝑛+ℎ_𝑛𝑧_𝑛,𝑦_𝑛∈Graph𝜋Σ^̌‌,∀𝑛<𝜔. (25) Therefore, for any 𝑛<𝜔, there exists 𝐺_𝑛∈Σ(𝑏̂_𝑛+ℎ_𝑛𝑧_𝑛) such that 𝑦_𝑛=𝜋𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑧_𝑛−𝜋𝐺_𝑛𝑏̂_𝑛ℎ_𝑛∈Ker𝑇,∀𝑛<𝜔. (26) Besides, for all 𝑛<𝜔, the following equality holds: 𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑧_𝑛−𝐺_𝑛𝑏̂_𝑛ℎ_𝑛=𝜋𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑧_𝑛−𝜋𝐺_𝑛𝑏̂_𝑛ℎ_𝑛+𝑇⁻¹𝑇𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑧_𝑛−𝑇𝐺_𝑛𝑏̂_𝑛ℎ_𝑛. (27) Now, Lemma 11 of [4] yields that 𝑇Σ^̌‌ is continuously Fréchet differentiable at 𝑏_∗, which provides that lim_𝑛→𝜔𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑧_𝑛−𝐺_𝑛𝑏̂_𝑛ℎ_𝑛=𝑦+𝑇⁻¹𝑇Σ^̌‌′𝑏_∗,𝑧∈Ker𝑇⊕𝑌_𝑇=𝑌. (28) Also, since for all 𝑛<𝜔, we have 𝐺_𝑛(𝑏̂_𝑛)=𝜋𝐺_𝑛(𝑏̂_𝑛)+𝑇⁻¹(𝑇Σ^̌‌(𝑏_𝑛)), it is followed that lim_𝑛→𝜔𝐺_𝑛(𝑏̂_𝑛)=𝐺_∗(𝑏_∗), and then 𝑧,𝑦+𝑇⁻¹𝑇Σ^̌‌′𝑏_∗,𝑧∈𝑃^̌‌𝐿_Grapȟ‌Σ𝑏_∗,𝐺_∗𝑏_∗. (29) Paraderivability of Σ^̌‌ yields 𝑧,𝑦+𝑇⁻¹𝑇Σ^̌‌′𝑏_∗,𝑧∈𝑇_{Graph(̌‌Σ)}𝑏_∗,𝐺_∗𝑏_∗. (30) Hence, there exist ℎ_𝑛→0⁺ and 𝑍×𝑌∋(𝑧_𝑛,𝑦_𝑛)→(𝑧,𝑦+𝑇⁻¹(𝑇Σ^̌‌)^′(𝑏_∗,𝑧)) such that 𝑏_∗,𝐺_∗𝑏_∗+ℎ_𝑛𝑧_𝑛,𝑦_𝑛∈GraphΣ^̌‌,∀𝑛<𝜔. (31) Therefore, 𝑏_∗,𝜋𝐺_∗𝑏_∗+ℎ_𝑛𝑧_𝑛,𝜋𝑦_𝑛∈Graph𝜋Σ^̌‌,∀𝑛<𝜔. (32) Now, 𝑦∈Ker𝑇 implies lim_𝑛→𝜔𝜋𝑦_𝑛=𝜋𝑦+𝜋𝑇⁻¹(𝑇Σ^̌‌)^′(𝑏_∗,𝑧)=𝜋𝑦=𝑦, and then (𝑧,𝑦)∈𝑇_{Graph(𝜋̌‌Σ)}(𝑏_∗,𝜋𝐺_∗(𝑏_∗)).
At this point, we have shown the paraderivability of 𝜋Σ^̌‌ and the inclusion 𝑃^{𝜋̌‌𝐿}𝜋Σ^̌‌𝑏_∗,𝜋𝐺_∗𝑏_∗(𝑧)⊂𝜋𝑃Σ^̌‌𝐿̌𝑏_∗,𝐺_∗𝑏_∗(𝑧),∀𝑧∈𝑍. (33) The proof of the lemma will be over by showing the reverse of the former inclusion. Let us fix now 𝑦∈𝜋𝑃Σ^̌‌𝐿̌(𝑏_∗,𝐺_∗(𝑏_∗))(𝑧)⊂Ker𝑇 and 𝑦∈𝑃Σ^̌‌𝐿̌(𝑏_∗,𝐺_∗(𝑏_∗))(𝑧)⊂𝑌 such that 𝑦=𝜋𝑦. Let us consider ℎ_𝑛→0⁺, 𝐿^̌‌∋(𝑏_𝑛,𝐺_𝑛(𝑏_𝑛))→(𝑏_∗,𝐺_∗(𝑏_∗)) and 𝑍×𝑌∋(𝑧_𝑛,𝑦_𝑛)→(𝑧,𝑦) such that 𝑏_𝑛,𝐺_𝑛𝑏_𝑛+ℎ_𝑛𝑧_𝑛,𝑦_𝑛∈GraphΣ^̌‌,∀𝑛<𝜔. (34) Hence, 𝑏_𝑛,𝜋𝐺_𝑛𝑏_𝑛+ℎ_𝑛𝑧_𝑛,𝜋𝑦_𝑛∈Graph𝜋Σ^̌‌,∀𝑛<𝜔, (35) and lim_𝑛→𝜔𝜋𝑦_𝑛=𝜋𝑦=𝑦. Then 𝑦∈𝑃^{𝜋̌‌𝐿}(𝜋Σ^̌‌)(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 with 𝒞¹-class maps.

Lemma 7. Let 𝑓∶𝑉⊂𝑍→𝑌 be a continuously Fréchet differentiable map, 𝑉∋𝑏_𝑛→𝑏_∗∈𝑉, 𝑍∋𝑧_𝑛→𝑧_∗≠0, and ℎ_𝑛→0⁺. Then lim_𝑛→𝜔𝑓𝑏_𝑛+ℎ_𝑛𝑧_𝑛−𝑓𝑏_𝑛ℎ_𝑛=𝑓^′𝑏_∗,𝑧_∗. (36)

Proof. Firstly let us define the auxiliary map 𝑔(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 𝑔^′(≤𝜖𝑢,⋅)𝑧_∗,∀𝑢∈𝑈. (37) Now, applying the medium value theorem, we can write 0≤lim_𝑛→𝜔𝑔𝑏_𝑛+ℎ_𝑛𝑧_𝑛−𝑔𝑏_𝑛ℎ_𝑛≤lim_𝑛→𝜔𝜖𝑧_∗ℎ_𝑛𝑧_𝑛ℎ_𝑛=𝜖. (38) Since 𝜖>0 was previously arbitrarily fixed, then lim_𝑛→𝜔𝑔𝑏_𝑛+ℎ_𝑛𝑧_𝑛−𝑔𝑏_𝑛ℎ_𝑛=0. (39) In fact, the former limit allows us to compute the limit of the statement because lim_𝑛→𝜔𝑓𝑏_𝑛+ℎ_𝑛𝑧_𝑛−𝑓𝑏_𝑛ℎ_𝑛−𝑓^′𝑏_∗,𝑧_∗=lim_𝑛→𝜔𝑓𝑏_𝑛+ℎ_𝑛𝑧_𝑛−𝑓𝑏_𝑛ℎ_𝑛−lim_𝑛→𝜔𝑓^′𝑏_∗,𝑧_𝑛=lim_𝑛→𝜔𝑓𝑏_𝑛+ℎ_𝑛𝑧_𝑛−𝑓𝑏_𝑛ℎ_𝑛−lim_𝑛→𝜔𝑓^′𝑏_∗,𝑏_𝑛+ℎ_𝑛𝑧_𝑛−𝑓^′𝑏_∗,𝑏_𝑛ℎ_𝑛=lim_𝑛→𝜔𝑔𝑏_𝑛+ℎ_𝑛𝑧_𝑛−𝑔𝑏_𝑛ℎ_𝑛=0, (40) and the proof is over.

In the following result we see how a condition in form of inclusion for a set-valued map Σ∶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. Let 𝛼∶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∋𝑏_𝑛→𝑧, ℎ_𝑛→0⁺, 𝐿^̌‌∋(𝑏̂_𝑛,𝐺_𝑛(𝑏̂_𝑛))→(𝑏_∗,𝐺_∗(𝑏_∗)), and {𝐺_𝑛}_𝑛<𝜔⊂ℬ(𝑍,𝑌) such that each 𝐺_𝑛∈Σ(𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛). Now, if there exists lim_𝑛→𝜔𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝐺_𝑛𝑏̂_𝑛ℎ_𝑛∈𝑌, (41) then there exists lim_𝑛→𝜔𝐺_𝑛−𝐺_𝑛ℎ_𝑛∈ℬ(𝑍,𝑌)𝑎𝑠𝑤𝑒𝑙𝑙. (42)

Proof. In the first place, the inclusion of Σ(𝑏) in the statement yields that there are 𝑦_∗∈𝑌 and two sequences {𝑦_𝑛}_𝑛<𝜔 and {𝑦_𝑛}_𝑛<𝜔 of elements of 𝑌 such that 𝐺_∗=𝛼(𝑏_∗)+𝑦_∗𝛽(𝑏_∗), 𝐺_𝑛=𝛼(𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛)+𝑦_𝑛𝛽(𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛), and 𝐺_𝑛=𝛼(𝑏̂_𝑛)+𝑦_𝑛𝛽(𝑏̂_𝑛) for every 𝑛<𝜔.
Now, on the one hand, 𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝐺_𝑛𝑏̂_𝑛ℎ_𝑛=𝛼𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝛼𝑏̂_𝑛𝑏̂_𝑛ℎ_𝑛+𝑦_𝑛−𝑦_𝑛ℎ_𝑛,∀𝑛<𝜔. (43)
On the other hand, since 𝛼 is continuously Fréchet differentiable, the former lemma and Lemma 10 of [6] yield lim_𝑛→𝜔𝑦_𝑛−𝑦_𝑛ℎ_𝑛=𝑦−𝛼^′𝑏_∗,𝑧𝑏_∗−𝛼𝑏_∗(𝑧), (44) where 𝑧=lim_𝑛→𝜔𝑏_𝑛∈𝑍 and 𝑦=lim_𝑛→𝜔((𝐺_𝑛(𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛)−𝐺_𝑛(𝑏̂_𝑛))/ℎ_𝑛)∈𝑌.
To conclude, for every 𝑛<𝜔, we decompose 𝐺_𝑛−𝐺_𝑛ℎ_𝑛=𝛼𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛+𝑦_𝑛𝛽𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝛼𝑏̂_𝑛−𝑦_𝑛𝛽𝑏̂_𝑛ℎ_𝑛=𝛼𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝛼𝑏̂_𝑛ℎ_𝑛+𝑦_𝑛𝛽𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝛽𝑏̂_𝑛ℎ_𝑛+𝑦_𝑛−𝑦_𝑛ℎ_𝑛𝛽𝑏̂_𝑛. (45) The former decomposition allows us to arrive at lim_𝑛→𝜔𝐺_𝑛−𝐺_𝑛ℎ_𝑛=𝛼^′𝑏_∗,𝑧+𝑦_∗𝛽^′𝑏_∗+,𝑧𝑦−𝛼^′𝑏_∗,𝑧𝑏_∗−𝛼𝑏_∗(𝑧)𝛽𝑏_∗, (46) 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. Let 𝑓∶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 𝑃^𝐿′𝑓+𝐹𝑏_∗,𝑓𝑏_∗+𝑦_∗(𝑧)=𝑓^′𝑏_∗,𝑧+𝑃^𝐿𝐹𝑏_∗,𝑦_∗(𝑧),∀𝑧∈𝑍. (47)

Proof. Let us begin by checking the inclusion ⊂. For this purpose we fix arbitrary elements 𝑧∈𝑍 and 𝑦∈𝑃^𝐿′(𝑓+𝐹)(𝑏_∗,𝑓(𝑏_∗)+𝑦_∗)(𝑧). Now, by the usual characterization of the paratingent cone we have three sequences: ℎ_𝑛→0⁺, 𝑍×𝑌∋(𝑧_𝑛,𝑦_𝑛)→(𝑧,𝑦), and 𝐿^′∋(𝑏_𝑛,𝑓(𝑏_𝑛)+𝑦^∗)→(𝑏_∗,𝑓(𝑏_∗)+𝑦_∗). Moreover, the following condition holds true for every 𝑛<𝜔: 𝑓𝑏_𝑛+𝑦^∗_𝑛+ℎ_𝑛𝑦_𝑛∈𝑓𝑏_𝑛+ℎ_𝑛𝑧_𝑛+𝐹𝑏_𝑛+ℎ_𝑛𝑧_𝑛. (48) Then, for every 𝑛<𝜔, there exists 𝑧_𝑛∈𝐹(𝑏_𝑛+ℎ_𝑛𝑧_𝑛) such that 𝑦_𝑛=𝑓𝑏_𝑛+ℎ_𝑛𝑧_𝑛−𝑓𝑏_𝑛ℎ_𝑛+𝑧_𝑛−𝑦^∗_𝑛ℎ_𝑛∈𝑌, (49) which yields lim_𝑛→𝜔𝑧_𝑛−𝑦^∗_𝑛ℎ_𝑛=𝑦−𝑓^′𝑏_∗,𝑧. (50) Finally, the definition of paratingent cone assures that the last limit belongs to 𝑃^𝐿𝐹(𝑏_∗,𝑦_∗)(𝑧), and hence, 𝑦−𝑓^′(𝑏_∗,𝑧) also does.
In order to prove the reverse inclusion, we fix arbitrary 𝑧∈𝑍 and 𝑦∈𝑓^′(𝑏_∗,𝑧)+𝑃^𝐿𝐹(𝑏_∗,𝑦_∗)(𝑧). We use again the characterization of paratingent cones which provides us the usual three sequences: ℎ_𝑛→0⁺, 𝑍×𝑌∋(𝑧_𝑛,𝑦_𝑛)→(𝑧,𝑦−𝑓^′(𝑏_∗,𝑧)), and 𝐿∋(𝑏_𝑛,𝑦^∗_𝑛)→(𝑏_∗,𝑦_∗) in such away that 𝑦^∗_𝑛+ℎ_𝑛𝑦_𝑛∈𝐹𝑏_𝑛+ℎ_𝑛𝑧_𝑛,∀𝑛<𝜔. (51) Now, for every 𝑛<𝜔, there exists 𝑧_𝑛∈𝐹(𝑏_𝑛+ℎ_𝑛𝑧_𝑛) such that 𝑦_𝑛=(𝑧_𝑛−𝑦^∗_𝑛)/ℎ_𝑛, and the sequence defined by the general term 𝑤_𝑛=𝑓𝑏_𝑛+ℎ_𝑛𝑧_𝑛+𝑧_𝑛−𝑓𝑏_𝑛−𝑦^∗_𝑛ℎ_𝑛∈𝑌 (52) converges to 𝑦. To conclude, since 𝑏_𝑛,𝑓𝑏_𝑛+𝑦^∗_𝑛+ℎ_𝑛𝑧_𝑛,𝑤_𝑛∈Graph𝑓+𝐹,∀𝑛<𝜔, (53) certainly 𝑦∈𝑃^𝐿′𝑓+𝐹𝑏_∗,𝑓𝑏_∗+𝑦_∗(𝑧). (54)

4. Sensitivity Analysis

Theorem A will be proved in this section. However, before this, it will be necessary to state and prove Theorem 12. 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 Ψ^̌‌. 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 parameter 𝑏 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 every 𝑏∈V there exists a regular local 𝑇-optimal solution 𝑥_𝑏∈𝐷 of (1) in such a way that the map 𝜆∶𝑉→𝑋 given by 𝜆(𝑏)=𝑥_𝑏 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. Let 𝑏∈𝑉, 𝑥_𝑏∈𝐷 a regular local 𝑇-optimal solution of (1), and 𝐺_𝑥𝑏∈ℬ(𝑍,𝑌) a 𝑇-Lagrange multiplier of (1) associated with 𝑥_𝑏. The (𝑇,𝛽)-modification of 𝐺_𝑥𝑏 is the map defined by 𝐺_𝑥^(𝑇,𝛽)_𝑏(𝑧)∶=𝑇⁻¹𝑇𝐺_𝑥𝑏(𝑧)+𝜋𝐺_𝑥𝑏(𝑏)𝛽(𝑏)(𝑧),∀𝑧∈𝑍. (55)

It is easy to check that 𝐺_𝑥^(𝑇,𝛽)_𝑏 is a 𝑇-Lagrange multiplier of (1) associated with 𝑥_𝑏. 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]). Let 𝑏∈𝑉, 𝑥_𝑏∈𝐷 a regular local 𝑇-optimal solution of (1), 𝐺_𝑥𝑏∈ℬ(𝑍,𝑌) a 𝑇-Lagrange multiplier of (1) associated with 𝑥_𝑏, and 𝒰_𝑥𝑏 the set of the 𝑇-Lagrange multipliers of (1) associated with 𝑥_𝑏. Then 𝒰_𝑥𝑏=𝐺_𝑥^(𝑇,𝛽)_𝑏+ℋ_𝑏, (56) where ℋ_𝑏={𝐻∈ℬ(𝑍,Ker𝑇)∶𝐻(𝑏)=0}.

Next we will prove the following.

Theorem 12. Let one supposes that the 𝑇-dual perturbation map Ψ is paraderivable relative to 𝐿 at (𝑏_∗,𝐺_∗)∈𝐿⊂𝐺𝑟𝑎𝑝ℎ(Ψ) and 𝑇Ψ is Fréchet differentiable at 𝑏_∗∈𝑉. Then Ψ^̌‌ is paraderivable relative to 𝐿^̌‌∶={(𝑏,𝐺(𝑏))∈𝑍×𝑌∶(𝑏,𝐺)∈𝐿} (57) at (𝑏_∗,𝐺_∗(𝑏_∗)) and 𝑃Ψ^̌‌𝐿̌𝑏_∗,𝐺_∗𝑏_∗(𝑧)=𝑃^𝐿Ψ𝑏_∗,𝐺_∗(𝑧)𝑏_∗+𝐺_∗(𝑧),∀𝑧∈𝑍. (58)

Proof. The statement will be proved by applying Theorem 4. 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, ℎ_𝑛→0⁺, 𝑍∋𝑏_𝑛→𝑏∈𝑍, 𝐿^̌‌∋(𝑏̂_𝑛,𝐺_𝑛(𝑏̂_𝑛))→(𝑏_∗,𝐺_∗(𝑏_∗)), and {𝐺_𝑛}_𝑛<𝜔⊂ℬ(𝑍,𝑌) such that each 𝐺_𝑛∈Σ(𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛) and the below limit exists lim_𝑛→𝜔𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝐺_𝑛𝑏̂_𝑛ℎ_𝑛∈𝑌. (59)
Now we can consider, on the one hand, the continuously Fréchet differentiable map 𝛽∶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 𝐺_𝑛^(𝑇,𝛽) and 𝐺_𝑛^(𝑇,𝛽) for every 𝑛<𝜔. By Proposition 11 we have 𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛=𝐺_𝑛^(𝑇,𝛽)𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛,𝐺_𝑛𝑏̂_𝑛=𝐺_𝑛^(𝑇,𝛽)𝑏̂_𝑛, (60) for every 𝑛<𝜔. Moreover, the below limit exists lim_𝑛→𝜔𝐺_𝑛^(𝑇,𝛽)𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝐺_𝑛^(𝑇,𝛽)𝑏̂_𝑛ℎ_𝑛∈𝑌. (61) Then Proposition 8 applied to the set-valued map Ψ^(𝑇,𝛽)∶𝑉⇝ℬ(𝑍,𝑌),𝑏⇝Ψ^(𝑇,𝛽)(𝑏)=𝐺^(𝑇,𝛽)∈ℬ(𝑍,𝑌)∶𝐺∈Ψ(𝑏) (62) implies that the following also exists: lim_𝑛→𝜔𝐺_𝑛^(𝑇,𝛽)−𝐺_𝑛^(𝑇,𝛽)ℎ_𝑛∈ℬ(𝑍,𝑌). (63)
Now, we have all necessary ingredients at hand in order to check that Ψ 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∶=𝑏_𝑛, and define the maps 𝐺^∼_𝑛 and 𝐺_𝑛 by 𝐺^∼_𝑛(𝑧)∶=𝐺_𝑛^(𝑇,𝛽)(𝑧)+𝐺_∗−𝐺_∗^(𝑇,𝛽)(𝑧)+𝐺_∗^(𝑇,𝛽)−𝐺_∗𝑏̂_𝑛𝛽𝑏̂_𝑛(𝑧),𝐺_𝑛(𝑧)∶=𝐺_𝑛^(𝑇,𝛽)(𝑧)+𝐺_∗−𝐺_∗^(𝑇,𝛽)(𝑧)+𝐺_∗^(𝑇,𝛽)−𝐺_∗𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛𝛽𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛(𝑧), (64) for every 𝑧∈𝑍. Clearly {𝐺_𝑛}_𝑛<𝜔,{𝐺^∼_𝑛}_𝑛<𝜔⊂ℬ(𝑍,𝑌). Again Proposition 11 yields 𝐺_𝑛∈Ψ(𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛) and 𝐺^∼_𝑛∈Ψ(𝑏̂_𝑛) for every 𝑛<𝜔. Furthermore, since each 𝐺_𝑛(𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛)=𝐺_𝑛^(𝑇,𝛽)(𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛) and 𝐺^∼_𝑛(𝑏̂_𝑛)=𝐺_𝑛^(𝑇,𝛽)(𝑏̂_𝑛), from (60) we get 𝐺_𝑛(𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛)=𝐺_𝑛(𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛) and 𝐺^∼_𝑛(𝑏̂_𝑛)=𝐺_𝑛(𝑏̂_𝑛). This provides the equality lim_𝑛→𝜔𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝐺^∼_𝑛𝑏̂_𝑛ℎ_𝑛=lim_𝑛→𝜔𝐺_𝑛𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝐺_𝑛𝑏̂_𝑛ℎ_𝑛∈𝑌. (65) The last step in the proof is to check that lim_𝑛→𝜔(𝐺_𝑛−𝐺^∼_𝑛)/ℎ_𝑛 exists. Indeed, for every 𝑛<𝜔, 𝐺_𝑛−𝐺^∼_𝑛ℎ_𝑛=𝐺_𝑛^(𝑇,𝛽)−𝐺_𝑛^(𝑇,𝛽)ℎ_𝑛+𝐺_∗−𝐺_∗^(𝑇,𝛽)𝑏̂_𝑛𝛽𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝛽𝑏̂_𝑛ℎ_𝑛+𝐺_∗−𝐺_∗^(𝑇,𝛽)𝑏_𝑛𝛽𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛. (66)
Now, taking into account that 𝛽 is continuously differentiable at 𝑏_∗, from Lemma 7, we obtain that lim_𝑛→𝜔𝛽𝑏̂_𝑛+ℎ_𝑛𝑏_𝑛−𝛽𝑏̂_𝑛ℎ_𝑛=𝛽^′𝑏_∗,𝑏. (67) Therefore, lim_𝑛→𝜔𝐺_𝑛−𝐺^∼_𝑛ℎ_𝑛=lim_𝑛→𝜔𝐺_𝑛^(𝑇,𝛽)−𝐺_𝑛^(𝑇,𝛽)ℎ_𝑛+𝐺_∗−𝐺_∗^(𝑇,𝛽)𝑏_∗𝛽^′𝑏_∗+𝐺,𝑏_∗−𝐺_∗^(𝑇,𝛽)(𝑏)𝛽𝑏_∗, (68) 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 that 𝜋Υ is paraderivable relative to 𝜋𝐿^̌‌_Ψ at (𝑏_∗,𝜋𝑓(𝑥_𝑏∗)). After that we will finish the proof by means of the equality Υ(𝑏)=𝑇⁻¹𝑇Υ(𝑏)+𝜋Υ(𝑏),∀𝑏∈𝑉. (69) Since 𝜋Υ(𝑏)=𝜋Ψ^̌‌(𝑏) for every 𝑏∈𝑉, it is enough to check that the set-valued map 𝜋Ψ^̌‌ is paraderivable relative to 𝜋𝐿^̌‌_Ψ at (𝑏_∗,𝜋𝐺_𝑥𝑏∗(𝑏_∗)). Applying Theorem 12 we get that Ψ^̌‌ is paraderivable relative to 𝐿^̌‌_Ψ=(𝑏,𝐺(𝑏))∈𝑍×𝑋∶(𝑏,𝐺)∈𝐿_Ψ, (70) at (𝑏_∗,𝐺_𝑥𝑏∗(𝑏_∗)) and 𝑃Ψ^{̌‌𝐿Ψ̌‌}𝑏_∗,𝐺_𝑥𝑏∗𝑏_∗(𝑧)=𝑃^𝐿ΨΨ𝑏_∗,𝐺_𝑥𝑏∗(𝑧)𝑏_∗+𝐺_𝑥𝑏∗(𝑧),∀𝑧∈𝑍. (71) By Lemma 6  𝜋Ψ^̌‌ is paraderivable relative to 𝜋𝐿^̌‌_Ψ at (𝑏_∗,𝜋𝐺_𝑥𝑏∗(𝑏_∗)) and 𝜋𝑃Ψ^{̌‌𝐿Ψ̌‌}𝑏_∗,𝐺_𝑥𝑏∗𝑏_∗(𝑧)=𝑃^{𝜋̌‌𝐿Ψ}𝜋Ψ^̌‌𝑏_∗,𝜋𝐺_𝑥𝑏∗𝑏_∗(𝑧),∀𝑧∈𝑍. (72) Now, the equality 𝜋Υ=𝜋Ψ^̌‌ and (71) yield 𝑃^{𝜋̌‌𝐿Ψ}𝜋Υ𝑏_∗,𝜋𝑓𝑥_𝑏∗(𝑧)=𝜋𝐺_𝑥𝑏∗(𝑧)+𝜋𝑃^𝐿ΨΨ𝑏_∗,𝐺_𝑥𝑏∗(𝑧)𝑏_∗,∀𝑧∈𝑍. (73) 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)=𝑇𝐺_𝑥𝑏∗(𝑧), for every 𝑧∈𝑍. Thus, [𝑇⁻¹𝑇Υ]^′(𝑏_∗,𝑧)=𝑇⁻¹𝑇𝐺_𝑥𝑏∗(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 𝐿_Υ at (𝑏_∗,𝑓(𝑥_𝑏∗)) and 𝑃^𝐿ΥΥ𝑏_∗,𝑓𝑥_𝑏∗(𝑧)=𝑇⁻¹𝑇𝐺_𝑥𝑏∗(𝑧)+𝜋𝐺_𝑥𝑏∗(𝑧)+𝜋𝑃^𝐿ΨΨ𝑏_∗,𝐺_𝑥𝑏∗(𝑧)𝑏_∗=𝐺_𝑥𝑏∗(𝑧)+𝜋𝑃^𝐿ΨΨ𝑏_∗,𝐺_𝑥𝑏∗(𝑧)𝑏_∗,∀𝑧∈𝑍. (74)

This section is finished by illustrating Theorem A with the aid of an example.

Example 13. Let one considers 𝑋=𝑌=ℝ³,𝑍=𝑊=ℝ, the interval 𝑉=(−11/10,9/10)⊂ℝ,𝐷={(𝑥,𝑦,𝑧)∈ℝ³∶2cosh𝑦²>𝑧}, and the problem Min𝑥−√2𝑦+𝑧,𝑥+√2𝑦+𝑧,−√2𝑥+√2𝑧;−log2cosh𝑦²−𝑧=𝑏,𝑥,𝑦,𝑧∈𝐷, (75) for every 𝑏∈𝑉.
Let us take 𝑇=(1,1,√2). Solving the problem we obtain the 𝑇-perturbation map Υ(𝑏)=2𝑒^𝑏−1𝑒^𝑏−𝜆,2𝑒^𝑏−1𝑒^𝑏−𝜆,√22𝑒^𝑏−1𝑒^𝑏+√2𝜆∈ℝ³∶𝜆∈ℝ, (76) 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 𝑃Υ−1,𝑓𝑥₋₁(𝑢)=𝑒𝑢+𝜆,𝑒𝑢+𝜆,√2𝑒𝑢−√2𝜆∶𝜆∈ℝ,∀𝑢∈ℝ. (77) 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 𝑌_𝑇=(Ker𝑇)^⟂ we obtain 𝐺_𝑥−1(𝑢)=(𝑒𝑢,𝑒𝑢,√2𝑒𝑢) and the 𝑇-dual perturbation map Ψ(𝑏)=𝑒^−𝑏+𝜆𝑏,𝑒^−𝑏+𝜆𝑏,√2𝑒^−𝑏−√2𝜆𝑏∶𝜆∈ℝ. (78) Thus, we have 𝑃Ψ−1,𝐺_𝑥−1(𝑢𝑏)=)(−𝑒^−𝑏𝜆𝑢−𝑏²,−𝑒^−𝑏𝜆𝑢−𝑏²,−√2𝑒^−𝑏√𝑢+2𝜆𝑏²∶𝜆∈ℝ, (79) for every 𝑢∈ℝ, and therefore 𝜋𝑃Ψ−1,𝐺_𝑥−1(𝑢)(−1)=−𝜆,−𝜆,√2𝜆∶𝜆∈ℝ. (80) Finally we obtain that 𝐺_𝑥−1(𝑢)+𝜋𝑃Ψ−1,𝐺_𝑥−1(𝑢)=√)(−1𝑒𝑢,𝑒𝑢,2𝑒𝑢+−𝜆,−𝜆,√=√2𝜆∶𝜆∈ℝ𝑒𝑢−𝜆,𝑒𝑢−𝜆,2𝑒𝑢+√𝑥2𝜆∶𝜆∈ℝ=𝑃Υ−1,𝑓₋₁(𝑢) (81) 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, the 𝑇-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.

Acknowledgment

F. García was partially supported by the Universidad de Alicante Project GRE11-08.