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On higherorder adjacent derivative of perturbation map in parametric vector optimization
Journal of Inequalities and Applications volume 2016, Article number: 112 (2016)
Abstract
This paper deals with higherorder sensitivity analysis in terms of the higherorder adjacent derivative for nonsmooth vector optimization. The relations between the higherorder adjacent derivative of the minima/the proper minima/the weak minima of a multifunction and its profile map are given. Then the relationships between the higherorder adjacent derivative of the perturbation map/the proper perturbation map/the weak perturbation map, and the higherorder adjacent derivative of a feasible map in objective space are considered. Finally, the formulas for estimating the higherorder adjacent derivative of the perturbation map, the proper perturbation map, the weak perturbation map via the adjacent derivative of the constraint map, and the higherorder Fréchet derivative of the objective map are also obtained.
Introduction
Sensitivity analysis provides quantitative information as regards the solution map of a parameterized multiobjective optimization problem. A number of interesting results have been obtained in sensitivity analysis for multiobjective optimization problems. One of the first results was given by Tanino in [1, 2]. By using the firstorder contingent derivative, some results concerning the behavior of perturbation maps were obtained. The TPderivative was presented in [3] and used to weaken some assumptions in [1, 4]. References [5–8] investigated the perturbation map in nonsmooth convex problems. In [9–11], the Clarke derivatives were used for analyzing the sensitivity. The concept of the protodifferentiability of a multifunction, in which the contingent cone coincides with the adjacent cone at a point to its graph, was presented by Rockafellar in [12]. In [13, 14], the important results on the protodifferentiability of the efficient solution maps were obtained for generalized equations, a general model including optimization problems. Some developments were obtained in [15, 16]. A secondorder sensitivity analysis via the secondorder contingent derivatives were considered in [17, 18]. In [19], the secondorder protodifferentiability of a multifunction was proposed to discuss the secondorder sensitivity properties for generalized perturbation maps. The secondorder radialasymptotic derivative, introduced in [20], was used in qualification conditions to consider the secondorder protodifferentiability of the efficient solution map and the efficient frontier map of a parameterized vector optimization problem in [21]. Some results in higherorder sensitivity analysis using a higherorder adjacent derivative in [22] and a higherorder contingent derivative in [23] of perturbation maps in a parameterized vector optimization were given. Using higherorder variational sets, presented in [24], some results in higherorder sensitivity analysis were obtained in [25].
Unlike higherorder contingent derivatives based on encounter information, the mthorder variations of a map, based on the different rates of change of the point under consideration in the domain space and the range space of a map, were proposed to obtain the open mapping principle in [26] and consider Hölder metric regularity of setvalued maps in [27]. Another kind of mthorder derivatives, presented in [28], was used to establish the optimality condition for isolated local minima of nonsmooth functions and modified to characterize weak sharp minima in [29]. The mthorder derivatives in [28] were generalized to setvalued maps in [30–32] to establish higherorder optimality conditions. In [31], the higherorder sensitivity was consider by using the mthorder contingenttype derivatives. In [33], the lower Studniarski derivative of a perturbation map in vector optimization was considered.
To the best of our knowledge, there is no paper dealing with the sensitivity of the mthorder adjacent derivatives of perturbation maps of parameterized vector optimization problems. Moreover, the proper perturbation maps and the case that the objective function is higherorder Fréchet differentiable in constraint vector optimization have not been considered yet. Motivated by the above observations, in this paper, by making use of the mthorder adjacent derivatives of setvalued maps which were introduced in [31], we investigate quantitatively the perturbation map, the proper perturbation map, and the weak perturbation map of parameterized vector optimization problems. The paper is organized as follows. Section 2 contains preliminary facts we need in the paper. In Section 3, the relations between the mthorder adjacent derivatives of a setvalued map and those of its profile map are discussed. The obtained results are employed in Section 4 to investigate the relationships between the mthorder adjacent derivatives of the perturbation map/the proper perturbation map/the weak perturbation map and the mthorder adjacent derivative of the feasible map in the objective space. In Section 4, the formulas for estimating the mthorder adjacent derivatives of the perturbation map, the proper perturbation map, and the weak perturbation map via the adjacent derivative of constraint map and the mthorder Fréchet derivative of the objective map are also given.
Preliminaries
In this paper, if not otherwise stated, let X, Y, and Z be normed spaces, and \(C \subseteq Y\) be a pointed closed convex cone. \(\mathcal{U}(x_{0})\) is used for the set of neighborhoods of \(x_{0}\). \(\mathbb {R}\), \(\mathbb {R}_{+}\), and \(\mathbb {N}\) stand for the set of the real numbers, nonnegative real numbers, and natural numbers, respectively (shortly, resp.). For \(M \subseteq X\), intM, clM, bdM denote its interior, closure, and boundary, resp. A convex set \(B\subseteq Y\) is called a base of C iff \(0 \notin\operatorname{cl}B\) and \(C =\{tb \mid t \in\mathbb{R}_{+},b \in B\}\). Clearly C has a compact base B if and only if \(C\cap\operatorname{bd}B\) is compact. If Y is a finite dimensional space, then C has a compact base. For \(F : X\rightrightarrows Y\), the domain, graph, and epigraph of F are defined by, resp.,
The profile map of F is \(F+C\) (defined by \((F+C)(x):=F(x)+C\)). We recall some concepts of optimality/efficiency in vector optimization as follows, for \(a_{0}\in A\subseteq Y\).

(i)
\(a_{0}\) is called a local (Pareto) minimal/efficient point of A (with respect to C), and denoted by \(a_{0}\in\operatorname{Min}_{C}A\), iff there exists \(U\in\mathcal{U}(a_{0})\) such that
$$(A\cap Ua_{0})\cap\bigl(C \setminus\{0\}\bigr)= \emptyset. $$ 
(ii)
Supposing that \(\operatorname{int} C \ne\emptyset\), \(a_{0}\) is said to be a local weak minimal/efficient point of A, denoted by \(a\in\operatorname{WMin}_{C}A\), iff there exists \(U\in\mathcal{U}(a_{0})\) such that
$$(A\cap Ua_{0})\cap(\operatorname{int}C)= \emptyset. $$ 
(iii)
Assuming that C is pointed, \(a_{0}\) is termed a proper minimal/efficient point of A, denoted by \(a_{0} \in\operatorname{PrMin}_{C}A\), iff there exists a convex cone \(K\subsetneqq Y\) with \(C \setminus\{0\}\subseteq\operatorname{int}K\) and \(U\in\mathcal{U}(a_{0})\) such that
$$(A\cap Ua_{0})\cap(K)= \{0\}. $$
If \(U = Y\), the word ‘local’ is omitted, i.e., we have the corresponding global notions. For a subset \(A\subseteq Y\), A is said to have the domination property iff \(A\subseteq\operatorname{Min}_{C}A+C\) and A is said to have the proper domination property iff \(A\subseteq\operatorname{PrMin}_{C}A+C\). Similarly, when \(\operatorname{int}C\neq\emptyset\), A has the weak domination property iff \(A\subseteq\operatorname{WMin}_{C}A+\operatorname{int}C\cup\{ 0\}\).
Recall now the four kinds of higherorder derivatives which we are most concerned with in the sequel. Let \(F:X\rightrightarrows Y\), \(u\in X\), \(m\in\mathbb{N}\), and \((x_{0},y_{0})\in\operatorname{gr}F\).

(i)
([31]) The mthorder radialcontingent derivative of F at \((x_{0},y_{0})\) is defined by
$$\begin{aligned} D_{S}^{m}F(x_{0},y_{0}) (u):={}& \bigl\{ v\in Y\mid \exists t_{n}>0, \exists(u_{n},v_{n}) \to (u,v): t_{n}u_{n}\to0, \\ &{}y_{0}+t_{n}^{m}v_{n}\in F(x_{0}+t_{n}u_{n})\bigr\} . \end{aligned}$$ 
(ii)
([32]) The mthorder contingenttype derivative of F at \((x_{0},y_{0})\) is defined by
$$D^{m}F(x_{0},y_{0}) (u):=\bigl\{ v\in Y\mid \exists t_{n}\downarrow0, \exists(u_{n},v_{n}) \to(u,v), y_{0}+t_{n}^{m}v_{n}\in F(x_{0}+t_{n}u_{n})\bigr\} . $$ 
(iii)
([31]) The mthorder adjacent derivative of F at \((x_{0},y_{0})\) is defined by
$$D^{bm}F(x_{0},y_{0}) (u):=\bigl\{ v\in Y\mid \forall t_{n}\downarrow0 , \exists(u_{n},v_{n}) \to(u,v), y_{0}+t_{n}^{m}v_{n}\in F(x_{0}+t_{n}u_{n})\bigr\} . $$ 
(iv)
([32]) The mthorder lower Studniarski derivative of F at \((x_{0},y_{0})\) is defined by
$$D^{lm}F(x_{0},y_{0}) (u):=\bigl\{ v\in Y\mid \forall t_{n}\downarrow0 ,\forall u_{n}\to u, \exists v_{n}\to v, y_{0}+t_{n}^{m}v_{n} \in F(x_{0}+t_{n}u_{n})\bigr\} . $$
Remark 2.1
\(D^{lm}F(x_{0},y_{0})(u)\subseteq D^{bm}F(x_{0},y_{0})(u)\subseteq D^{m}F(x_{0},y_{0})(u)\), \(\forall u\in X\).
The reverse conclusions in Remark 2.1 may not hold. The following examples show the cases.
Example 2.1
Let \(\mathcal{I}=\{\frac{1}{n}:n\in\mathbb {N}\}\) and \(F:\mathbb{R}\rightrightarrows\mathbb{R}\) be defined by
Then, for \((x_{0},y_{0})=(0,0)\in\operatorname{gr}F\), we can check that
Taking \(t_{n}=\frac{2}{n}\), then \(t_{n}u_{n}=\frac{2u_{n}}{n}\nsubseteq\mathcal {I}\) for all \(u_{n}\to2\). Indeed, suppose to the contrary that there exists a subsequence \(\{\frac{1}{k} \}\subseteq\mathcal{I}\), \(k\ge n\), such that \(t_{n}u_{n}=\frac{1}{k}\). Then \(u_{n}=\frac{1}{2}.\frac {n}{k}\le\frac{1}{2}\), i.e., \(u_{n}\not\to2\), a contradiction. Hence, for the above \(t_{n}\), \(F(x_{0}+t_{n}u_{n})=\emptyset\). Consequently,
Hence,
Example 2.2
Let \(F:\mathbb{R}\rightrightarrows\mathbb{R}^{2}\) be defined by
Then, for \((x_{0},y_{0})=(0,(0,0))\),
and
Hence,
Definition 2.1
(see [31])
For \(u\in X\), \(F:X\rightrightarrows Y\) is called mthorder udirectionally contingent compact at \((x_{0},y_{0})\in\operatorname{gr}F\) iff, for any \(t_{n}\downarrow0\), \((u_{n},v_{n})\in X\times Y\) such that \(u_{n}\to u\), and \(y_{0}+t_{n}^{m}v_{n}\in F(x_{0}+t_{n}u_{n})\) for all n, there exists a convergent subsequence of \(\{v_{n}\}\).
Definition 2.2
Let \(F:X\rightrightarrows Y\) be a setvalued map, \(x_{0}\in X\), \(m\in\mathbb{N}\setminus\{0\}\), and \(\alpha>0\).

(i)
F is said locally Hölder continuous of order α at \((x_{0},y_{0})\in\operatorname{gr}F\) if there exist \(\lambda>0\), and \(U\in{\mathcal{U}}(x_{0})\) such that
$$F(x_{2})\subset F(x_{1})+\lambda\x_{1}x_{2} \^{\alpha}B_{Y}, \quad\forall x_{1},x_{2} \in U, $$where \(B_{Y}\) stands for the closed unit ball in Y.

(ii)
F is said locally pseudoHölder calm of order m (see [34]) at \((x_{0},y_{0})\in\operatorname{gr}F\) if there exist a real number \(\lambda>0\), \(\exists U\in{\mathcal{U}}(x_{0})\), and \(\exists V\in{\mathcal{U}}(y_{0})\) such that
$$F(x)\cap V\subset\{y_{0}\}+\lambda\xx_{0} \^{m}B_{Y},\quad \forall x\in U. $$
When \(m=1\), the word ‘Hölder’ is replaced by ‘Lipschitz’. If \(V=Y\), then ‘locally pseudoHölder calm’ is replaced by ‘locally Hölder calm’. In [13], F is called upper locally Lipschitz at \(x_{0}\in\operatorname{dom}F\) if there exist a real number \(\lambda>0\) and \(U\in{\mathcal{U}}(x_{0})\) such that \(F(x)\subset F(x_{0})+\lambda\xx_{0}\B_{Y}\), \(\forall x\in U\). It is easy to see that if F is upper locally Lipschitz at \(x_{0}\) and \(F(x_{0})=\{ y_{0}\}\) then F is locally Lipschitz calm at \((x_{0},y_{0})\).
Proposition 2.1
(see [31])
Let \(F:X\rightrightarrows Y\), \((x_{0},y_{0})\in\operatorname{gr}F\), and Y be finite dimensional space. If \(D_{S}^{m}F(x_{0},y_{0})(0)=\{0\}\), then F is mthorder udirectionally contingent compact at \((x_{0},y_{0})\) for all \(u\in X\).
Proposition 2.2
Let \(F:X\rightrightarrows Y\), \((x_{0},y_{0})\in\operatorname{gr}F\), and Y be finite dimensional space. If F is locally Hölder calm of order m at \((x_{0},y_{0}) \in\operatorname{gr}F\), then \(D^{m}_{S}F(x_{0},y_{0})(0)=\{0\}\).
Proof
Consider an arbitrary \(y\in D^{m}_{S}F(x_{0},y_{0})(0)\). Then there exist \(y_{n}\to y\), \(x_{n}\to0\), and \(t_{n}>0\) such that \(y_{0}+t_{n}^{m}y_{n}\in F(x_{0}+t_{n}x_{n})\) and \(t_{n}x_{n}\to0\). Since F is locally Hölder calm of order m at \((x_{0},y_{0})\), we derive that, for n large enough, there exists \(\lambda >0\) such that
Consequently,
It follows from the above equation that \(x_{n}\to0\), and \(y_{n}\to y\) that one has \(y=0\). □
Corollary 2.1
Let \(F:X\rightrightarrows Y\), \((x_{0},y_{0})\in\operatorname{gr}F\), and Y be finite dimensional space. If F is locally Hölder calm of order m at \((x_{0},y_{0})\in\operatorname{gr}F\), then F is mthorder udirectionally contingent compact at \((x_{0},y_{0})\) for all \(u\in X\).
Proof
It follows from Proposition 2.1 and Proposition 2.2 that the conclusion is obtained. □
Definition 2.3
(see [31])
Let \(f:X\to Y\) be a vectorvalued map. f is said to be mthorder Fréchet differentiable at \(x_{0}\in X\) iff there exists a linear continuous operator \(d^{m}F(x_{0}):X\times\cdots\times X\to Y\) (m times X), such that
where \(o(\xx_{0}\^{m})\) satisfies \(o(\xx_{0}\^{m})/\xx_{0}\^{m}\to0\) when \(x\to x_{0}\). \(d^{m}f(x_{0})\) is called the mthorder Fréchet derivative. f is said mthorder Fréchet differentiable on X if f is mthorder Fréchet differentiable at any \(x\in X\). If \(d^{m} f(\cdot)\) is continuous at \(x_{0}\) then f is said to be mthorder continuously Fréchet differentiable at \(x_{0}\).
Remark 2.2
(see [31])
For \(f:X\to Y\) and \(x_{0},u\in X\), if there exists \(d^{m}f(x_{0})\), then
Higherorder adjacent derivatives of setvalued maps
In this section, the relations between higherorder adjacent derivative of a setvalued map and those of its profile map are discussed. Such relations for various kinds of efficient points of these derivatives are also investigated.
Proposition 3.1
Let \((x_{0},y_{0})\in\operatorname{gr}F\). Then, for any \(u\in X\),
Proof
Let \(z=v+c\) for some \(v\in D^{bm}F(x_{0},y_{0})(u)\) and \(c\in C\). Then, for all \(t_{n}\downarrow0\), there exists \((u_{n},v_{n})\to(u,v)\) such that \(y_{0}+t_{n}^{m}v_{n}\in F(x_{0}+t_{n}u_{n})\) for all n. Setting \(\overline{v}_{n}:=v_{n}+c\), one has \(\overline{v}_{n}\to v+c\) and, for all n,
So, \(z=v+c\in D^{bm}(F+C)(x_{0},y_{0})(u)\). □
Note that the opposite inclusion of (1) may not hold. The following example illustrates the case.
Example 3.1
Let \(C=\mathbb{R}_{+}\), \(F: {\mathbb {R}}\rightrightarrows{\mathbb {R}}\) be defined by
Let \((x_{0},y_{0})=(0,0)\in\operatorname{gr}F\). Then
Hence, for all u,
Proposition 3.2
Suppose that either of the following conditions is satisfied:

(i)
for any \(u\in X\), F is mthorder udirectionally contingent compact at \((x_{0},y_{0})\);

(ii)
C has a compact base and \(D_{S}^{m}F(x_{0},y_{0})(0)\cap (C)=\{0\}\);

(iii)
C has a compact base and \(D^{bm}(F+C)(x_{0},y_{0})(u)\) has domination property.
Then, for all \(u\in X\),
Proof
It follows from Proposition 3.1 that we only need to show the reverse inclusion of (1).
(i) Let \(v\in D^{bm}(F+C)(x_{0},y_{0})(u)\). If \((u,v)=(0,0)\), we have \(0\in D^{bm}F(x_{0},y_{0})(0)+C\). For \((u,v)\ne(0,0)\), for all \(t_{n}\downarrow0\), there exists \((u_{n},v_{n})\to(u,v)\) such that \(y_{0}+t_{n}^{m}v_{n}\in(F+C)(x_{0}+t_{n}u_{n})\), ∀n. Hence, there exists \(c_{n}\in C\) such that \(y_{0}+t_{n}^{m}(v_{n}c_{n}/t_{n}^{m})\in F(x_{0}+t_{n}u_{n})\). By the udirectionally contingent compactness of F at \((x_{0},y_{0})\), we can assume that \(v_{n}c_{n}/t_{n}^{m}\to\bar{v}\in D^{bm}F(x_{0},y_{0})(u)\). Since \(c_{n}/t_{n}^{m}=v_{n}(v_{n}c_{n}/t_{n}^{m})\to v \bar{ v}\) and C is a closed convex cone, one gets \(v\bar{v} \in C\). Hence, \(v\in\bar {v}+C\subseteq D^{bm}F(x_{0},y_{0})(u)+C\).
(ii) Let \(u\in X\) and \(v\in D^{bm}(F+C)(x_{0},y_{0})(u)\) be arbitrary. As in (i), we need to consider only \((u,v)\neq(0,0)\). We see that, for all \(t_{n}\downarrow0\), there exist \((u_{n},v_{n})\to(u,v)\), and \(c_{n}\in C\) such that \(y_{0}+t_{n}^{m}v_{n}\in F(x_{0}+t_{n}u_{n})+c_{n}\) for all n. If there exists \(n_{0}\) such that \(c_{n}=0\) for all \(n>n_{0}\), then \(v\in D^{bm}F(x_{0},y_{0})(u)+0\subseteq D^{bm}F(x_{0},y_{0})(u)+C\). Now, assume that \(c_{n}\neq0\) and \(c_{n}/\c_{n}\\to\overline{c}\) for some \(\overline{c}\in C\) with norm one. There are only two cases for \(s_{n}:=\sqrt[m]{\c_{n}\}>0\).
Case 1: \(s_{n}/t_{n}\to+\infty\). Then \(s_{n}[(t_{n}/s_{n})u_{n}]=t_{n}u_{n}\to0\). Since
\((t_{n}/s_{n})^{m}y_{n}c_{n}/s_{n}^{m}\to\overline{c}\), and \((t_{n}/s_{n})u_{n}\to0\), one has \(\overline{c}\in D_{S}^{m}F(x_{0},y_{0})(0)\), an impossibility.
Case 2: \(\{s_{n}/t_{n}\}\) is bounded, and assume \(s_{n}/t_{n}\to\alpha \ge0\). Then, since
\(v_{n}(s_{n}/t_{n})^{m}(c_{n}/s_{n}^{m})\to v\alpha^{m}\overline{c}\), and \(u_{n}\to u\), one gets \(v\alpha^{m} \overline{c}\in D^{bm}F(x_{0},y_{0})(u)\), and hence \(v\in D^{bm}F(x_{0},y_{0})(u)+C\).
(iii) Since \(D^{bm}(F+C)(x_{0},y_{0})(u)\) has the domination property, for any \(u\in X\),
We claim that, for any \(u\in X\),
Indeed, let \(v\in\operatorname{Min}_{C}D^{bm}(F+C)(x_{0},y_{0})(u)\). Then, for all \(t_{n}\downarrow0\), there exist \((u_{n},v_{n})\to(u,v)\) and \(c_{n}\in C\) such that, for all n, \(y_{0}+t_{n}^{m}(v_{n}c_{n})\in F(x_{0}+t_{n}u_{n})\). Since C has a compact base, we may assume that \(c_{n}=\alpha_{n}b_{n}\) with \(\alpha_{n}>0\) and \(b_{n}\to b\neq0\). We now show that \(\alpha_{n}\to0\). Suppose to the contrary that \(\alpha_{n}\not\to0\). Then there exists \(\epsilon>0\) such that \(\alpha_{n}\ge\epsilon\) for all n. Setting \(\overline {c}_{n}=(\epsilon/\alpha_{n})c_{n}\). Then, for any n, \(c_{n}\overline {c}_{n}=(1\epsilon/\alpha_{n})c_{n}\in C\) and
Since \(v_{n}\overline{c}_{n}=v_{n}(\epsilon/\alpha_{n})c_{n}=v_{n}\epsilon b_{n}\to v\epsilon b\), we have \(v\epsilon b\in D^{bm}(F+C)(x_{0},y_{0})(u)\) and \(v(v\epsilon b)=\epsilon b\in C\setminus\{0\}\), which contradicts \(v\in\operatorname{Min}_{C}D^{bm}(F+C)(x_{0},y_{0})(u)\). Therefore, \(\alpha_{n}\to0\) and \(v_{n}c_{n}=v_{n}\alpha_{n}b_{n}\to v\), i.e., \(v\in D^{bm}F(x_{0},y_{0})(u)\). Thus, (3) holds. It follows from (2) and (3) that
The proof is complete. □
Proposition 3.3
Suppose that either of the following conditions holds:

(i)
for any \(u\in X\), F is mthorder udirectionally contingent compact at \((x_{0},y_{0})\);

(ii)
C has a compact base and \(D_{S}^{m}F(x_{0},y_{0})(0)\cap (C)=\{0\}\);

(iii)
C has a compact base and \(D^{bm}(F+C)(x_{0},y_{0})(u)\) has domination property.
Then, for all \(u\in X\),
Proof
Similarly to the proof of Proposition 4.3 in [31], we obtain the conclusion. □
Since the following propositions are proven similarly, the proofs are omitted.
Proposition 3.4
Suppose that either of the following conditions holds:

(i)
for any \(u\in X\), F is mthorder udirectionally contingent compact at \((x_{0},y_{0})\);

(ii)
C has a compact base and \(D_{S}^{m}F(x_{0},y_{0})(0)\cap (C)=\{0\}\);

(iii)
C has a compact base and \(D^{bm}(F+C)(x_{0},y_{0})(u)\) has proper domination property.
Then, for all \(u\in X\),
Proposition 3.5
Assume that \(\operatorname{int}C\neq \emptyset\) and K̃ is a closed convex cone with \(\widetilde {K}\subseteq\operatorname{int}C\cup\{0\}\). Suppose further that either of the following conditions holds:

(i)
for any \(u\in X\), F is mthorder udirectionally contingent compact at \((x_{0},y_{0})\);

(ii)
C has a compact base and \(D_{S}^{m}F(x_{0},y_{0})(0)\cap (\widetilde{K})=\{0\}\);

(iii)
C has a compact base and \(D^{bm}(F+\widetilde {K})(x_{0},y_{0})(u)\) has weak domination property.
Then, for all \(u\in X\),
The following example illustrates that we cannot replace K̃ by C in the conclusion of Proposition 3.5.
Example 3.2
Let \(X={\mathbb{R}}\), \(Y={\mathbb{R}}^{2}\), \((x_{0},y_{0})=(0,(0,0))\), \(C=\mathbb{R}^{2}_{+}\), and \(F:\mathbb {R}\rightrightarrows\mathbb{R}^{2}\) be defined by \(F(x)=\{(x^{2},x^{4})\}\). Then we can check that \(D^{2}_{S}F(x_{0},y_{0})(u)=D^{2b}F(x_{0},y_{0})(u)=\{(u^{2},0)\}\) for any \(u\in\mathbb{R}\), and
Hence, \(\operatorname{WMin}_{C} D^{2b}F(x_{0},y_{0})(u)=\{(u^{2},0)\}\) and
Since \(D^{2}_{S}F(x_{0},y_{0})(0)\cap(C) = \{0\}\) and C has a compact base \(B=\{(y_{1},y_{2})\in\mathbb{R}^{2}, y_{1}+y_{2}=1, y_{1}\ge0, y_{2}\ge0\}\), the assumption (ii) is fulfilled. We can check that
Higherorder adjacent derivatives of perturbations maps
In this section, we consider the following parameterized vector optimization problem:
where x is a ldimensional decision variable, u is a pdimensional parameter, \(f_{i}\) is a real valued objective function on \(\mathbb{R}^{l}\times\mathbb{R}^{p}\) for \(i=1,2,\ldots,q\), X is a setvalued map from \(\mathbb{R}^{p}\) to \(\mathbb{R}^{l}\), which defines a feasible decision set, and K is a nonempty pointed closed convex ordering cone in \(\mathbb{R}^{q}\). Let \(F(u)\) be the value at u of the feasible set map in the objective space, i.e.,
We define the perturbation/frontier map \(\mathcal{F}\), the weak perturbation/frontier map \(\mathcal{W}\), and the proper perturbation/frontier map \(\mathcal{P}\) of the considered problem as follows:
For \(u_{0}\in\mathbb{R}^{p}\) and a closed convex cone \(\widetilde {K}\subseteq\mathbb{R}^{q}\),

(i)
F is said to be Kdominated by \(\mathcal{F}\) near \(u_{0}\) iff \(F(u)\subseteq\mathcal{F}(u)+K\), for all u in some \(U\in\mathcal{U}(u_{0})\).

(ii)
F is said to be Kdominated by \(\mathcal{P}\) near \(u_{0}\) iff \(F(u)\subseteq\mathcal{P}(u)+K\), for all u in some \(U\in\mathcal{U}(u_{0})\).

(iii)
F is said to be K̃dominated by \(\mathcal{W}\) near \(u_{0}\) iff \(F(u)\subseteq\mathcal{W}(u)+\widetilde {K}\), for all u in some \(U\in\mathcal{U}(u_{0})\).
Remark 4.1
Since \(\mathcal{F}(u)\subseteq F(u)\), the Kdominatedness of F by \(\mathcal{F}\) near \(u_{0}\) implies that, for all \(u\in U\), \(\mathcal{F}(u)+K=F(u)+K\). Hence, if F is Kdominated by \(\mathcal{F}\) near \(u_{0}\), then for any \(u_{0}\in\mathbb{R}^{p}\), \(y_{0}\in\mathcal{F}(u_{0})\), and \(u\in U\),
Similar assertions are true for \(\mathcal{P}\) and \(\mathcal{W}\) as follows.

(i)
\(D^{bm}(\mathcal {P}+K)(u_{0},y_{0})(u)=D^{bm}(F+K)(u_{0},y_{0})(u)\) for any \(u_{0}\in\mathbb {R}^{p}\), \(y_{0}\in\mathcal{P}(u_{0})\), and \(u\in U\) if F is Kdominated by \(\mathcal{P}\) near \(u_{0}\).

(ii)
\(D^{bm}(\mathcal{W}+\widetilde {K})(u_{0},y_{0})(u)=D^{bm}(F+\widetilde{K})(u_{0},y_{0})(u)\) for any \(u_{0}\in \mathbb{R}^{p}\), \(y_{0}\in\mathcal{W}(u_{0})\), and \(u\in U\) if F is K̃dominated by \(\mathcal{W}\) near \(u_{0}\).
Higherorder adjacent derivatives of perturbation maps without constraints
Now, the relations between the higherorder adjacent derivative of feasible map and the higherorder adjacent derivative of perturbation/ weak perturbation maps are investigated in this subsection.
Proposition 4.1
Assume that F is mthorder udirectionally compact at \((u_{0},y_{0})\) for any \(u\in\mathbb{R}^{p}\).

(i)
If F is Kdominated by \(\mathcal{F}\) near \(u_{0}\), then, for u near \(u_{0}\),
$$\operatorname{Min}_{K} D^{bm}F(u_{0},y_{0}) (u)\subseteq D^{bm}\mathcal{F}(u_{0},y_{0}) (u). $$ 
(ii)
If F is Kdominated by \(\mathcal{P}\) near \(u_{0}\), then, for u near \(u_{0}\),
$$\operatorname{PrMin}_{K} D^{bm}F(u_{0},y_{0}) (u)\subseteq D^{bm}\mathcal{P}(u_{0},y_{0}) (u). $$ 
(iii)
If \(\operatorname{int}K\neq\emptyset\), there is a closed convex cone K̃ satisfying \(\widetilde{K} \subseteq\operatorname{int}K\cup\{0\}\), and F is K̃dominated by \(\mathcal{W}\) near \(u_{0}\), then, for u near \(u_{0}\),
$$\operatorname{WMin}_{K} D^{bm}F(u_{0},y_{0}) (u)\subseteq D^{bm}\mathcal{W}(u_{0},y_{0}) (u). $$
Proof
Since the proof is similar, we prove only assertion (iii). Observe that, being a pointed closed convex cone in \(\mathbb{R}^{q}\), K clearly has a compact base and hence so does K̃. Moreover, F is mthorder udirectionally compact at \((u_{0},y_{0})\) implies that \(\mathcal{W}\) is mthorder udirectionally compact at \((u_{0},y_{0})\). Therefore, one has
Here the first and third equalities are due to Propositions 3.5, and the second one follows from Remark 4.1. □
Now, we investigate the reverse conclusion in Proposition 4.1.
Proposition 4.2
Suppose that the following conditions are satisfied:

(i)
F is locally Hölder continuous of order m at \(u_{0}\);

(ii)
F is Kdominated by \(\mathcal{F}\) near \(u_{0}\);

(iii)
there is a neighborhood U of \(u_{0}\) such that for any \(u\in U\), \(\mathcal{F}(u)\) is a singlepoint set.
Then, for any \(u\in\mathbb{R}^{p}\),
Proof
Let \(u\in\mathbb{R}^{p}\) and \(v\in D^{bm}\mathcal {F}(u_{0},y_{0})(u)\). Then, for any sequence \(t_{n}\downarrow0\), there exists \((u_{n},v_{n})\to(u,v)\) such that
Suppose to the contrary that \(v\notin\operatorname{Min}_{K} D^{bm}F(u_{0},y_{0})(u)\). Then, there exists \(\overline{v}\in D^{bm}F(u_{0}, y_{0})(u)\) such that \(v\overline{v}\in K\setminus\{0\}\). Hence, for the preceding \(t_{n}\), there exists \((\overline{u}_{n},\overline {v}_{n})\to(u,\overline{v})\) such that
Since F is Kdominated by \(\mathcal{F}\) near \(u_{0}\), there exists \(U_{1}\in\mathcal{U}(u_{0})\) such that, for all \(u\in U_{1}\),
It follows from the locally Hölder continuity of order m of F that there exist \(U_{2}\in\mathcal{U}(u_{0})\) and \(L>0\) such that, for all \(u_{1},u_{2}\in U_{2}\) and
Naturally, since \(t_{n}\downarrow0\), there exists \(N>0\) such that
Therefore, from (5), (7), (8), and (6), there exists \(b_{n}\in B_{\mathbb{R}^{q}}\) such that, for all n large enough,
Thus, it follows from (4), (9), and assumption (iii) that
Since \(\overline{v}_{n}L\\overline{u}_{n}u_{n}\^{m}b_{n}v_{n}\to\overline {v}v\) and K is a pointed closed convex cone, one has \(\overline {v}v\in K\), which contradicts \(v\overline{v}\in K\setminus\{0\}\). This completes the proof. □
The following example shows that the assumption (iii) in Proposition 4.2 cannot be dropped.
Example 4.1
Let \(p=1\), \(q=2\), \(K=\{(y_{1},y_{2})\in\mathbb {R}^{2}\mid y_{1}= 0, y_{2}\ge0\}\), \((u_{0},y_{0})=(0,(0,0))\), and \(F:\mathbb {R}\rightrightarrows\mathbb{R}^{2}\) be defined by
Then
Hence, we can check that \(\mathcal{F}(u)\) is not a singlepoint set near \(u_{0}\), F is Kdominated by \(\mathcal{F}\) near \(u_{0}\), and F is locally Hölder continuous of order 2 at \(u_{0}\). By direct calculation, one has, for any \(u\in\mathbb{R}\),
and then
Therefore,
Remark 4.2
Similar properties to Proposition 4.2 for higherorder contingenttype derivatives of perturbation maps have not yet been investigated in [31]. With some suitable modifications, we can obtain similar properties for higherorder contingenttype derivatives of perturbation maps in [31].
Proposition 4.3
Suppose that the following conditions are satisfied:

(i)
F is locally Hölder continuous of order m at \(u_{0}\);

(ii)
F is Kdominated by \(\mathcal{P}\) near \(u_{0}\);

(iii)
there is a neighborhood U of \(u_{0}\) such that for any \(u\in U\), \(\mathcal{P}(u)\) is a singlepoint set.
Then, for any \(u\in\mathbb{R}^{p}\),
Proof
The proof is similar to that of Proposition 4.2. □
Proposition 4.4
Assume that \(\operatorname{int}K\neq \emptyset\). If F is locally Hölder continuous of order m at \(u_{0}\), then, for any \(u\in\mathbb{R}^{p}\),
Proof
Let \(u\in\mathbb{R}^{p}\) and \(v\in D^{bm}\mathcal {W}(u_{0},y_{0})(u)\). Then, for any sequence \(t_{n}\downarrow0\), there exists \((u_{n},v_{n})\to(u,v)\) such that
Suppose to the contrary that \(v\notin\operatorname{WMin}_{K} D^{bm}F(u_{0},y_{0})(u)\). Then there exists \(\overline{v}\in D^{bm}F(u_{0}, y_{0})(u)\) such that \(v\overline{v}\in\operatorname{int}K\). Hence, for the preceding \(t_{n}\), there exists \((\overline{u}_{n},\overline {v}_{n})\to(u,\overline{v})\) such that
It follows from the locally Hölder continuity of order m of F that there exist \(U\in\mathcal{U}(u_{0})\) and \(L>0\) such that, for all \(u_{1},u_{2}\in U\), one has
Naturally, since \(t_{n}\downarrow0\), there exists \(N>0\) such that
Therefore, from (12), (13), and (14), there exists \(b_{n}\in B_{\mathbb{R}^{q}}\) such that, for all n large enough,
It follows from \(v_{n}(\overline{v}_{n}L\\overline{u}_{n}u_{n}\^{m}b_{n})\to v\overline{v}\) and \(v\overline{v}\in\operatorname{int}K\) that we have \(v_{n}(\overline{v}_{n}L\\overline{u}_{n}u_{n}\^{m}b_{n})\in\operatorname{int}K\) for n large enough. Therefore, for n large enough,
which contradicts with (11). The conclusion is obtained. □
Higherorder adjacent derivatives of perturbation maps with constraints
In this section, the formulas for estimating higherorder adjacent derivative of perturbation map/ weak perturbation map via adjacent derivative of constraint map together with higherorder Fréchet derivative of the objective function are established.
Proposition 4.5
Let \(u_{0}\in\mathbb{R}^{p}\), \(x_{0}\in X(u_{0})\), \(y_{0}=f(x_{0},u_{0})\). Assume that f is mthorder Fréchet continuously differentiable at \((x_{0},u_{0})\). Then, for any \(u\in\mathbb{R}^{p}\),
Moreover, if X is (firstorder) udirectionally compact at \((u_{0},x_{0})\) for any \(u \in\mathbb{R}^{p}\), then the reverse inclusion of (16) is also valid.
Proof
Let y be as in the left hand side of (16). Then there exist \(u\in\mathbb{R}^{p}\) and \(x\in D^{b}X(x_{0},u_{0})(u)\) such that \(y=d^{m}f(x_{0},u_{0})((x,u),\ldots,(x,u))\). Since \(x\in D^{b}X(x_{0},u_{0})(u)\), for all \(t_{n}\downarrow0\), there exists \((u_{n},x_{n})\to(u,x)\) such that, for all n, \(x_{0}+t_{n}x_{n}\in X(u_{0}+t_{n}u_{n})\). Then
It follows from the mthorder Fréchet continuously differentiability of f and (17) that we have
Consequently,
It follows from (18) and
when \(n\to\infty\), one has \(y=d^{m}f(x_{0},u_{0})((x,u),\ldots,(x,u))\in D^{bm}F(u_{0},y_{0})(u) \). Hence, (16) has been established.
Now, let \(y\in D^{bm}F(u_{0},y_{0})(u)\). Then, for all \(t_{n}\downarrow0\), there exists \((u_{n},y_{n})\to(u,y)\) such that \(y_{0}+t_{n}^{m}y_{n}\in F(u_{0}+t_{n}u_{n})\), ∀n. Hence, there exists \(x_{n}\in X(u_{0}+t_{n}u_{n})\) such that
Setting \(\widetilde{x}_{n}:=\frac{x_{n}x_{0}}{t_{n}}\), we have
and
Since X is firstorder udirectionally compact at \((u_{0},x_{0})\), for preceding \(t_{n}\), \(u_{n}\), and \(\widetilde{x}_{n}\), by taking subsequence if necessary, one gets \(\widetilde{x}_{n}\to\widetilde{x}\in D^{b}X(x_{0},u_{0})(u)\). It follows from (21) and the mthorder Fréchet continuously differentiability of f at \((x_{0},u_{0})\) that one has
It implies that
Letting \(n\to\infty\), we have
The proof is complete. □
The following example illustrates that the assumption for the validity of the reverse inclusion of (16) in Proposition 4.5 cannot be omitted.
Example 4.2
Let \(p=q=l=1\), \(m=2\), \(f(x,u)=x^{4}\), and \(X:\mathbb{R}\rightrightarrows\mathbb{R}\) be defined by \(X(u)=\{x\in \mathbb{R}\mid 0\le x\le1\}\). Then \(F(u)=\{y\in\mathbb{R}\mid 0\le y\le1\}\).
Let \((x_{0},u_{0})=(0,0)\). Then \(y_{0}=f(x_{0},u_{0})=0\). We can check that X is not udirectionally compact at \((u_{0},x_{0})\) for any \(u\in \mathbb{R}\). Indeed, by taking \(t_{n}=\frac{1}{n}\), \(u_{n}\to u\), and \(x_{n}=\frac{1}{2}n\), we have \(x_{0}+t_{n}x_{n}=\frac{1}{2}\in F(u_{0}+t_{n}u_{n})\), and \(x_{n}\) has no convergent subsequence.
By direct calculation, one has, for any \(u\in\mathbb{R}\),
and then
Hence, for any \(u\in\mathbb{R}\),
Corollary 4.1
Let \(u_{0}\in\mathbb{R}^{p}\), \(x_{0}\in X(u_{0})\), \(y_{0}=f(x_{0},u_{0})\), and \(\widetilde{X}:\mathbb{R}^{p}\times\mathbb {R}^{q}\rightrightarrows\mathbb{R}^{l}\) be defined by \(\widetilde{X}(u,y):=\{x\in X(u):y=f(x,u)\}\). Assume that f is mthorder Fréchet continuously differentiable at \((x_{0},u_{0})\). Let one of the following conditions be fulfilled:

(i)
X is locally Lipschitz calm at \((u_{0},x_{0})\in\operatorname{gr}X\);

(ii)
\(D_{S}X(u_{0},x_{0})(0)=\{0\}\);

(iii)
X̃ is \((u,y)\)directionally compact at \(((u_{0},y_{0}),x_{0})\) for any \((u,y)\in\mathbb{R}^{p}\times\mathbb{R}^{q}\);

(iv)
X̃ is locally Lipschitz calm at \(((u_{0},y_{0}),x_{0})\in\operatorname{gr}\widetilde{X}\);

(v)
\(D_{S}\widetilde{X}((u_{0},y_{0}),x_{0})(0,0)=\{0\}\);

(vi)
X̃ is locally pseudoLipschitz at \(((u_{0},y_{0}),x_{0})\).
Then, for any \(u\in\mathbb{R}^{p}\),
Proof
From Proposition 2.1, Corollary 2.1, Proposition 4.5 and the analysis similar the proof of Corollary 4.1 and Proposition 4.2 in [35], we obtain the conclusion. □
Theorem 4.1
Let \(u_{0}\in\mathbb{R}^{p}\), \(x_{0}\in X(u_{0})\), \(y_{0}=f(x_{0},u_{0})\). Suppose that the following conditions are satisfied:

(i)
F is mthorder udirectionally compact at \((u_{0},y_{0})\) for any \(u\in\mathbb{R}^{p}\);

(ii)
F is Kdominated by \(\mathcal{F}\) near \(u_{0}\);

(iii)
F is locally Hölder continuous of order m at \(u_{0}\);

(iv)
there exists \(U\in\mathcal{U}(u_{0})\) such that for any \(u\in U\), \(\mathcal{F}(u)\) is a singlepoint set;

(v)
f is mthorder Fréchet continuously differentiable at \((x_{0},u_{0})\);

(vi)
X is firstorder udirectionally compact at \((u_{0},x_{0})\).
Then, for any \(u\in\mathbb{R}^{p}\),
Proof
It follows from Proposition 4.1(i), Proposition 4.2, and Proposition 4.5 that the proof is complete. □
The result in Theorem 4.1 is illustrated in the following example.
Example 4.3
Let \(p=q=l=1\), \(m=2\), \(K=\mathbb{R}_{+}\), \(f(x,u)=x^{2}\), and \(X:\mathbb{R}\rightrightarrows\mathbb{R}\) be defined by \(X(u)=\{x\in\mathbb{R}\mid u^{2}\le x\le2u^{2}\}\). Then
Let \((x_{0},u_{0})=(0,0)\). Then \(y_{0}=f(x_{0},u_{0})=0\). It is easy to see that the assumptions (ii) and (iv) in Theorem 4.1 are satisfied. Since \(D_{S}^{2}F(u_{0},y_{0})(0)=\{0\}\), from the Proposition 2.1, the assumption (i) Theorem 4.1 is fulfilled.
Moreover, we can check that F is locally Hölder continuous of order 2 at \(u_{0}\) and X is firstorder udirectionally compact at \((u_{0},x_{0})\). Hence, the assumptions (iii) and (vi) in Theorem 4.1 are fulfilled.
By direct calculation, one has, for any \(u\in\mathbb{R}\),
and then
Thus, all the assumptions in Theorem 4.1 are satisfied. For any \(u\in \mathbb{R}\) and \(x\in D^{b}X(u_{0}, x_{0})(u)\), i.e., \(x=0\), one has
Hence, for any \(u\in\mathbb{R}\),
Theorem 4.2
Let \(u_{0}\in\mathbb{R}^{p}\), \(x_{0}\in X(u_{0})\), \(y_{0}=f(x_{0},u_{0})\). Suppose that the following conditions are satisfied:

(i)
F is mthorder udirectionally compact at \((u_{0},y_{0})\) for any \(u\in\mathbb{R}^{p}\);

(ii)
F is Kdominated by \(\mathcal{P}\) near \(u_{0}\);

(iii)
F is locally Hölder continuous of order m at \(u_{0}\);

(iv)
there exists \(U\in\mathcal{U}(u_{0})\) such that for any \(u\in U\), \(\mathcal{P}(u)\) is a singlepoint set;

(v)
f is mthorder Fréchet continuously differentiable at \((x_{0},u_{0})\);

(vi)
X is firstorder udirectionally compact at \((u_{0},x_{0})\).
Then, for any \(u\in\mathbb{R}^{p}\),
Proof
It follows from Proposition 4.1(ii), Proposition 4.3, and Proposition 4.5 that the conclusion is obtained. □
Theorem 4.3
Let \(u_{0}\in\mathbb{R}^{p}\), \(x_{0}\in X(u_{0})\), \(y_{0}=f(x_{0},u_{0})\). Suppose that the following conditions are satisfied:

(i)
F is mthorder udirectionally compact at \((u_{0},y_{0})\) for any \(u\in\mathbb{R}^{p}\);

(ii)
\(\operatorname{int}K\neq\emptyset\), there is a closed convex cone K̃ satisfying \(\widetilde{K}\subseteq\operatorname{int}K\cup\{0\}\) and F is K̃dominated by \(\mathcal{W}\) near \(u_{0}\);

(iii)
F is locally Hölder continuous of order m at \(u_{0}\);

(iv)
f is mthorder Fréchet continuously differentiable at \((x_{0},u_{0})\);

(v)
X is udirectionally compact at \((u_{0},x_{0})\).
Then, for any \(u\in\mathbb{R}^{p}\),
Proof
It follows from Proposition 4.1(iii), Proposition 4.4, and Proposition 4.5 that the conclusion is given. □
Conclusions
Although there are some similar properties between the contingent derivatives and the adjacent derivatives, the adjacent derivatives have some advantages and drawbacks in some cases. When using adjacent derivatives instead of contingent derivatives in sensitivity analysis, the protodifferentiability assumption, such as in Theorem 3.3 in [4], Theorem 4.3 in [17], Theorem 5.1 in [18], can be omitted. The drawbacks of using of the adjacent derivatives is that the adjacent derivatives can be empty set in some cases such as in Example 2.1. Hence, in the case that the adjacent derivatives are not empty and avoiding the protodifferentiability assumption, the adjacent derivatives can be used. Based on the above observation, the mthorder adjacent derivatives was employed to consider higherorder sensitivity analysis for nonsmooth vector optimization in this paper. First of all, we considered the relationships between the mthorder adjacent derivatives of the perturbation map/the proper perturbation map/the weak perturbation map, and the mthorder adjacent derivative of feasible map in objective space. Then the above relations were used to establish the formulas for estimating the mthorder adjacent derivatives of the perturbation map, the proper perturbation map, and the weak perturbation map via the adjacent derivative of constraint map and the mthorder Fréchet derivative of the objective map. Some examples are provided to ensure the need of the assumptions and illustrate the results. When \(m=1\), the results become the firstorder sensitivity analysis using adjacent derivatives and also may be new.
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The author would like to thank Professor Alexander Zalavski and the handling editors for the help in the processing of the paper. The author is very grateful to the anonymous referees for the useful suggestions and remarks which helped to improve the contents of this article.
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Tung, L.T. On higherorder adjacent derivative of perturbation map in parametric vector optimization. J Inequal Appl 2016, 112 (2016). https://doi.org/10.1186/s1366001610593
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DOI: https://doi.org/10.1186/s1366001610593
MSC
 90C46
 49J52
 46G05
 90C26
 90C29
Keywords
 higherorder adjacent derivative
 parameterized vector optimization problem
 perturbation map
 proper perturbation map
 weak perturbation map
 higherorder sensitivity analysis