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We analyse the sensitivity of differential programs of the form

The subject of this work is sensitivity analysis in vector programming. The model problem considered throughout the paper is of the form

As the maps

To perform our analysis we use the so-called

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 [

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, [

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

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

In the context of the precedent paragraph, it is defined that

Now we can state the main result of the work. In this, it is represented by

Let one fixes

If

Let one supposes that

The proof of Theorem A is based on the fact that the set-valued map

The paper is organized as follows. In Section

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

Now let us introduce the notions and characterizations of cones and derivatives that will be used throughout this work (see [

In Section

Define the set-valued map

Finally, we now introduce the type of derivatives that we will handle. Let

In this section the regularity condition

From now on, the following notation will be used. Given a set-valued map

A set-valued map

there exist three sequences

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,

Let

The paraderivability of

For the reverse inclusion,

Let us begin now the last part of the proof. In this, the paraderivability of

The following example shows that in the former result neither assumptions can be dropped.

(i) Let one defines

(ii) Let one defines

In the statement of the following result, the previously fixed notation is used and, in addition, we will consider the set

Let one assumes that

In this first stage of the proof we are going to check that

At this point, we have shown the paraderivability of

Now we begin the second part of this section with a consequence of the medium value theorem. It shows the advantage of working with

Let

Firstly let us define the auxiliary map

In the following result we see how a condition in form of inclusion for a set-valued map

Let

In the first place, the inclusion of

Now, on the one hand,

On the other hand, since

To conclude, for every

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 [

Let

Let us begin by checking the inclusion

In order to prove the reverse inclusion, we fix arbitrary

Theorem A will be proved in this section. However, before this, it will be necessary to state and prove Theorem

Throughout this section we will assume, firstly, that the parameter

In the second place, we will assume that for every

Let

It is easy to check that

Let

Next we will prove the following.

Let one supposes that the

The statement will be proved by applying Theorem

Now we can consider, on the one hand, the continuously Fréchet differentiable map

Now, we have all necessary ingredients at hand in order to check that

Now, taking into account that

Now we can prove Theorem A.

In the first part of the proof we will check that

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

Let one considers

Let us take

Let us study the sensitivity of the problem (

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

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