On higher-order adjacent derivative of perturbation map in parametric vector optimization

Abstract

This paper deals with higher-order sensitivity analysis in terms of the higher-order adjacent derivative for nonsmooth vector optimization. The relations between the higher-order 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 higher-order adjacent derivative of the perturbation map/the proper perturbation map/the weak perturbation map, and the higher-order adjacent derivative of a feasible map in objective space are considered. Finally, the formulas for estimating the higher-order adjacent derivative of the perturbation map, the proper perturbation map, the weak perturbation map via the adjacent derivative of the constraint map, and the higher-order Fréchet derivative of the objective map are also obtained.

1 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 first-order contingent derivative, some results concerning the behavior of perturbation maps were obtained. The TP-derivative 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 proto-differentiability 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 proto-differentiability of the efficient solution maps were obtained for generalized equations, a general model including optimization problems. Some developments were obtained in [15, 16]. A second-order sensitivity analysis via the second-order contingent derivatives were considered in [17, 18]. In [19], the second-order proto-differentiability of a multifunction was proposed to discuss the second-order sensitivity properties for generalized perturbation maps. The second-order radial-asymptotic derivative, introduced in [20], was used in qualification conditions to consider the second-order proto-differentiability of the efficient solution map and the efficient frontier map of a parameterized vector optimization problem in [21]. Some results in higher-order sensitivity analysis using a higher-order adjacent derivative in [22] and a higher-order contingent derivative in [23] of perturbation maps in a parameterized vector optimization were given. Using higher-order variational sets, presented in [24], some results in higher-order sensitivity analysis were obtained in [25].

Unlike higher-order contingent derivatives based on encounter information, the mth-order 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 set-valued maps in [27]. Another kind of mth-order 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 mth-order derivatives in [28] were generalized to set-valued maps in [30–32] to establish higher-order optimality conditions. In [31], the higher-order sensitivity was consider by using the mth-order contingent-type 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 mth-order adjacent derivatives of perturbation maps of parameterized vector optimization problems. Moreover, the proper perturbation maps and the case that the objective function is higher-order Fréchet differentiable in constraint vector optimization have not been considered yet. Motivated by the above observations, in this paper, by making use of the mth-order adjacent derivatives of set-valued 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 mth-order adjacent derivatives of a set-valued map and those of its profile map are discussed. The obtained results are employed in Section 4 to investigate the relationships between the mth-order adjacent derivatives of the perturbation map/the proper perturbation map/the weak perturbation map and the mth-order adjacent derivative of the feasible map in the objective space. In Section 4, the formulas for estimating the mth-order adjacent derivatives of the perturbation map, the proper perturbation map, and the weak perturbation map via the adjacent derivative of constraint map and the mth-order Fréchet derivative of the objective map are also given.

2 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.,

$$\begin{aligned} &\operatorname{dom} F := \bigl\{ x \in X\mid F(x) \neq\emptyset\bigr\} ,\qquad \operatorname{gr} F := \bigl\{ (x, y) \in X\times Y\mid y \in F(x)\bigr\} , \\ &\operatorname{epi}F := \bigl\{ (x, y) \in X\times Y\mid y \in F(x) + C\bigr\} . \end{aligned}$$

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

  $$(A\cap U-a_{0})\cap\bigl(-C \setminus\{0\}\bigr)= \emptyset. $$

• (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

  $$(A\cap U-a_{0})\cap(-\operatorname{int}C)= \emptyset. $$

• (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

  $$(A\cap U-a_{0})\cap(-K)= \{0\}. $$

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 higher-order 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 mth-order radial-contingent 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 mth-order contingent-type 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 mth-order 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 mth-order 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

$$F(x)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \{ 0\}, & \mbox{if } x\le0,\\ \{-x^{2}\}, & \mbox{if } x\in\mathcal{I},\\ \emptyset, & \mbox{otherwise}. \end{array}\displaystyle \right . $$

Then, for $(x_{0},y_{0})=(0,0)\in\operatorname{gr}F$, we can check that

$$D^{2}F(x_{0},y_{0}) (2)=\{-4\}. $$

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,

$$D^{b2}F(x_{0},y_{0}) (2)=\emptyset. $$

Hence,

$$D^{2}F(x_{0},y_{0}) (2)\nsubseteq D^{b2}F(x_{0},y_{0}) (2). $$

Example 2.2

Let $F:\mathbb{R}\rightrightarrows\mathbb{R}^{2}$ be defined by

$$F(x)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \{(y_{1},y_{2})\in\mathbb{R}^{2}:y_{1}\le x^{4},y_{2}\le0\}, & \mbox{if } x< 0,\\ \{(y_{1},y_{2})\in\mathbb{R}^{2}:y_{1}\le0, y_{2}\ge2x^{4}\}, & \mbox{if } x\ge0. \end{array}\displaystyle \right . $$

Then, for $(x_{0},y_{0})=(0,(0,0))$,

$$D^{b2}F(x_{0},y_{0}) (x)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \{(y_{1},y_{2})\in\mathbb{R}^{2}:y_{1}\le0,y_{2}\le0\}, & \mbox{if } x< 0,\\ \{(y_{1},y_{2})\in\mathbb{R}^{2}:y_{1}\le0,y_{2}\in\mathbb{R}\}, & \mbox{if }x=0,\\ \{(y_{1},y_{2})\in\mathbb{R}^{2}:y_{1}\le0, y_{2}\ge0\}, & \mbox{if } x> 0, \end{array}\displaystyle \right . $$

and

$$D^{l2}F(x_{0},y_{0}) (x)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \{(y_{1},y_{2})\in\mathbb{R}^{2}:y_{1}\le0,y_{2}\le0\}, & \mbox{if } x< 0,\\ \{(y_{1},y_{2})\in\mathbb{R}^{2}:y_{1}\le0,y_{2}=0\}, & \mbox{if } x=0,\\ \{(y_{1},y_{2})\in\mathbb{R}^{2}:y_{1}\le0, y_{2}\ge0\}, & \mbox{if } x> 0. \end{array}\displaystyle \right . $$

Hence,

$$D^{b2}F(x_{0},y_{0}) (0)\nsubseteq D^{l2}F(\overline{x},\overline{y}) (0). $$

Definition 2.1

(see [31])

For $u\in X$, $F:X\rightrightarrows Y$ is called mth-order u-directionally 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 set-valued 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 pseudo-Hölder 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

  $$F(x)\cap V\subset\{y_{0}\}+\lambda\|x-x_{0} \|^{m}B_{Y},\quad \forall x\in U. $$

When $m=1$, the word ‘Hölder’ is replaced by ‘Lipschitz’. If $V=Y$, then ‘locally pseudo-Hölder 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 $F(x)\subset F(x_{0})+\lambda\|x-x_{0}\|B_{Y}$, $\forall x\in U$. 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 mth-order u-directionally 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

$$y_{0}+t_{n}^{m}y_{n}\in y_{0}+\lambda\|t_{n}x_{n}\|^{m}B_{Y}. $$

Consequently,

$$y_{n}\in\lambda\|x_{n}\|^{m}B_{Y}. $$

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 mth-order u-directionally 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 vector-valued map. f is said to be mth-order 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

$$f(x)=f(x_{0})+d^{m} f(x_{0}) (x-x_{0},\ldots,x-x_{0})+o\bigl(\|x-x_{0} \|^{m}\bigr) \quad (m \mbox{ times } x-x_{0}), $$

where $o(\|x-x_{0}\|^{m})$ satisfies $o(\|x-x_{0}\|^{m})/\|x-x_{0}\|^{m}\to0$ when $x\to x_{0}$. $d^{m}f(x_{0})$ is called the mth-order Fréchet derivative. f is said mth-order Fréchet differentiable on X if f is mth-order Fréchet differentiable at any $x\in X$. If $d^{m} f(\cdot)$ is continuous at $x_{0}$ then f is said to be mth-order 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

$$\begin{aligned} \bigl\{ d^{m}f(x_{0}) (u,u,\ldots,u)\bigr\} &=D^{lm}f\bigl(x_{0},f(x_{0})\bigr) (u)= D^{bm}f\bigl(x_{0},f(x_{0})\bigr) (u) \\ &=D^{m}f\bigl(x_{0},f(x_{0})\bigr) (u)=D_{S}^{m}f\bigl(x_{0},f(x_{0}) \bigr) (u). \end{aligned}$$

3 Higher-order adjacent derivatives of set-valued maps

In this section, the relations between higher-order adjacent derivative of a set-valued 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 $u\in X$,

$$ D^{bm}F(x_{0},y_{0}) (u)+C \subseteq D^{bm}(F+C) (x_{0},y_{0}) (u). $$

(1)

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,

$$y_{0}+t_{n}^{m}\overline{v}_{n}=y_{0}+t_{n}^{m}(v_{n}+c) \in F(x_{0}+t_{n}u_{n})+C=(F+C) (x_{0}+t_{n}u_{n}). $$

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

$$F(x)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \{-1,x^{3}\},&\mbox{if } x \ge0,\\ \{0\},&\mbox{if } x< 0. \end{array}\displaystyle \right . $$

Let $(x_{0},y_{0})=(0,0)\in\operatorname{gr}F$. Then

$$D^{b3}F(0,0) (u)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \{u^{3}\},&\mbox{if } u \ge0,\\ \{0\},&\mbox{if } u < 0, \end{array}\displaystyle \right . \qquad D^{b3}(F+C) (0,0) (u)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \mathbb{R},&\mbox{if } u \ge0,\\ \mathbb{R}_{+},&\mbox{if } u < 0. \end{array}\displaystyle \right . $$

Hence, for all u,

$$D^{b3}(F+C) (x_{0},y_{0}) (u)\nsubseteq D^{b3}F(x_{0},y_{0}) (u)+C. $$

Proposition 3.2

Suppose that either of the following conditions is satisfied:

• (i) for any $u\in X$, F is mth-order u-directionally 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 $u\in X$,

$$D^{bm}F(x_{0},y_{0}) (u)+C = D^{bm}(F+C) (x_{0},y_{0}) (u). $$

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 u-directionally 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

$$y_{0}+(s_{n})^{m}\bigl[(t_{n}/s_{n})^{m}v_{n}-c_{n}/s_{n}^{m} \bigr]\in F\bigl(x_{0}+s_{n}\bigl[(t_{n}/s_{n})u_{n} \bigr]\bigr), $$

$(t_{n}/s_{n})^{m}y_{n}-c_{n}/s_{n}^{m}\to-\overline{c}$, 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

$$y_{0}+(t_{n})^{m}\bigl[v_{n}-(s_{n}/t_{n})^{m} \bigl(c_{n}/s_{n}^{m}\bigr)\bigr]\in F(x_{0}+t_{n}u_{n}), $$

$v_{n}-(s_{n}/t_{n})^{m}(c_{n}/s_{n}^{m})\to v-\alpha^{m}\overline{c}$, 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 $u\in X$,

$$ D^{bm}(F+C) (x_{0},y_{0}) (u) \subseteq\operatorname{Min}_{C}D^{bm}(F+C) (x_{0},y_{0}) (u)+C. $$

(2)

We claim that, for any $u\in X$,

$$ \operatorname{Min}_{C}D^{bm}(F+C) (x_{0},y_{0}) (u)\subseteq D^{bm}F(x_{0},y_{0}) (u). $$

(3)

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

$$y_{0}+t_{n}^{m}(v_{n}- \overline{c}_{n})=y_{0}+t_{n}^{m}(v_{n}-c_{n})+t_{n}^{m}(c_{n}- \overline {c}_{n})\in F(x_{0}+t_{n}u_{n}) +C=(F+C) (x_{0}+t_{n}u_{n}). $$

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

$$D^{bm}(F+C) (x_{0},y_{0}) (u)\subseteq D^{bm}F(x_{0},y_{0}) (u)+C, \quad \forall u\in X. $$

The proof is complete. □

Proposition 3.3

Suppose that either of the following conditions holds:

• (i) for any $u\in X$, F is mth-order u-directionally 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 $u\in X$,

$$\operatorname{Min}_{C}D^{bm}F(x_{0},y_{0}) (u) = \operatorname{Min}_{C}D^{bm}(F+C) (x_{0},y_{0}) (u). $$

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 mth-order u-directionally 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 $u\in X$,

$$\operatorname{PrMin}_{C}D^{bm}F(x_{0},y_{0}) (u) = \operatorname{PrMin}_{C}D^{bm}(F+C) (x_{0},y_{0}) (u). $$

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 mth-order u-directionally 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 $u\in X$,

$$\operatorname{WMin}_{C}D^{bm}F(x_{0},y_{0}) (u) = \operatorname{WMin}_{C}D^{bm}(F+\widetilde {K}) (x_{0},y_{0}) (u). $$

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

$$D^{2b}(F+C) (x_{0},y_{0}) (u)=\bigl\{ (y_{1},y_{2})\in\mathbb{R}^{2}_{+}: y_{1}\ge u^{2}, y_{2}\ge0\bigr\} . $$

Hence, $\operatorname{WMin}_{C} D^{2b}F(x_{0},y_{0})(u)=\{(u^{2},0)\}$ and

$$\begin{aligned} &\operatorname{WMin}_{C}D^{2b}(F+C) (x_{0},y_{0}) (u) \\ &\quad=\bigl\{ (y_{1},y_{2})\in\mathbb{R}^{2}_{+}\mid y_{1} \ge u^{2}, y_{2} =0\bigr\} \cup\bigl\{ (y_{1},y_{2})\in\mathbb{R}^{2}_{+}\mid y_{1}= u^{2}, y_{2} \ge0\bigr\} . \end{aligned}$$

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

$$\operatorname{WMin}_{C} D^{2b}(F+C) (x_{0},y_{0}) (u) \nsubseteq\operatorname{WMin}_{C} D^{2b}F(x_{0},y_{0}) (u). $$

4 Higher-order adjacent derivatives of perturbations maps

In this section, we consider the following parameterized vector optimization problem:

$$\operatorname{Min}_{K}f(x,u)=\bigl(f_{1}(x,u),f_{2}(x,u), \ldots,f_{q}(x,u)\bigr), \quad \mbox{s.t. } x\in X(u)\subseteq \mathbb{R}^{l}, $$

where x is a l-dimensional decision variable, u is a p-dimensional 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 set-valued 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.,

$$F(u):=\bigl\{ y\in\mathbb{R}^{q} \mid y=f(x,u) \mbox{ for some } x\in X(u)\bigr\} . $$

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:

$$\mathcal{F}(u):=\operatorname{Min}_{K}F(u), \qquad \mathcal{W}(u):= \operatorname{WMin}_{K}F(u),\qquad \mathcal{P}(u):= \operatorname{PrMin}_{K}F(u). $$

For $u_{0}\in\mathbb{R}^{p}$ and a closed convex cone $\widetilde {K}\subseteq\mathbb{R}^{q}$,

• (i) F is said to be K-dominated 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 K-dominated 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 K̃-dominated 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 K-dominatedness 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 K-dominated by $\mathcal{F}$ near $u_{0}$, then for any $u_{0}\in\mathbb{R}^{p}$, $y_{0}\in\mathcal{F}(u_{0})$, and $u\in U$,

$$D^{bm}(\mathcal{F}+K) (u_{0},y_{0}) (u)=D^{bm}(F+K) (u_{0},y_{0}) (u). $$

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 K-dominated 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 K̃-dominated by $\mathcal{W}$ near $u_{0}$.

4.1 Higher-order adjacent derivatives of perturbation maps without constraints

Now, the relations between the higher-order adjacent derivative of feasible map and the higher-order adjacent derivative of perturbation/ weak perturbation maps are investigated in this subsection.

Proposition 4.1

Assume that F is mth-order u-directionally compact at $(u_{0},y_{0})$ for any $u\in\mathbb{R}^{p}$.

• (i) If F is K-dominated 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 K-dominated 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 K̃-dominated 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 mth-order u-directionally compact at $(u_{0},y_{0})$ implies that $\mathcal{W}$ is mth-order u-directionally compact at $(u_{0},y_{0})$. Therefore, one has

$$\begin{aligned} \operatorname{WMin}_{K} D^{bm}F(u_{0},y_{0}) (u) =&\operatorname{WMin}_{K} D^{bm}(F+\widetilde{K}) (u_{0},y_{0}) (u) \\ =&\operatorname{WMin}_{K} D^{bm}(\mathcal{W}+ \widetilde{K}) (u_{0},y_{0}) (u) \\ =&\operatorname{WMin}_{K} D^{bm}\mathcal{W}(u_{0},y_{0}) (u) \\ \subseteq& D^{bm}\mathcal{W}(u_{0},y_{0}) (u). \end{aligned}$$

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 K-dominated 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 single-point set.

Then, for any $u\in\mathbb{R}^{p}$,

$$D^{bm}\mathcal{F}(u_{0},y_{0}) (u)\subseteq \operatorname{Min}_{K} D^{bm}F(u_{0},y_{0}) (u). $$

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

$$ y_{0}+t_{n}^{m}v_{n}\in \mathcal{F}(u_{0}+t_{n}u_{n})\subseteq F(u_{0}+t_{n}u_{n}), \quad\forall n. $$

(4)

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

$$ y_{0}+t_{n}^{m} \overline{v}_{n}\in F(u_{0}+t_{n} \overline{u}_{n}),\quad\forall n. $$

(5)

Since F is K-dominated by $\mathcal{F}$ near $u_{0}$, there exists $U_{1}\in\mathcal{U}(u_{0})$ such that, for all $u\in U_{1}$,

$$ F(u)\subseteq\mathcal{F}(u)+K. $$

(6)

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

$$ F(u_{1})\subseteq F(u_{2})+L \|u_{1}-u_{2}\|^{m}B_{\mathbb{R}^{q}}. $$

(7)

Naturally, since $t_{n}\downarrow0$, there exists $N>0$ such that

$$ u_{0}+t_{n}\overline{u}_{n}, u_{0}+t_{n}u_{n}\in U\cap U_{1}\cap U_{2}, \quad\forall n>N. $$

(8)

Therefore, from (5), (7), (8), and (6), there exists $b_{n}\in B_{\mathbb{R}^{q}}$ such that, for all n large enough,

$$ y_{0}+t_{n}^{m}\bigl( \overline{v}_{n}-L\|\overline{u}_{n}-u_{n} \|^{m}b_{n}\bigr)\in F(u_{0}+t_{n}u_{n}) \subseteq\mathcal{F}(u_{0}+t_{n}u_{n})+K. $$

(9)

Thus, it follows from (4), (9), and assumption (iii) that

$$ y_{0}+t_{n}^{m}\bigl( \overline{v}_{n}-L\|\overline{u}_{n}-u_{n}\| ^{m}b_{n}\bigr)-\bigl(y_{0}+t_{n}^{m}v_{n} \bigr)=t_{n}^{m}\bigl(\overline{v}_{n}-L\| \overline{u}_{n}-u_{n}\| ^{m}b_{n}-v_{n} \bigr)\in K. $$

(10)

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

$$F(u)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} K, & \mbox{if } u=0,\\ K\cup\{(y_{1},y_{2})\in\mathbb{R}^{2}\mid y_{1}=u^{2}, y_{2}\ge-\sqrt{1+u^{2}}\},& \mbox{if } u\neq0. \end{array}\displaystyle \right . $$

Then

$$\mathcal{F}(u)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \{(0,0)\}, & \mbox{if } u=0,\\ \{ (0,0), (u^{2}, -\sqrt{1+u^{2}})\},& \mbox{if } u\neq0. \end{array}\displaystyle \right . $$

Hence, we can check that $\mathcal{F}(u)$ is not a single-point set near $u_{0}$, F is K-dominated 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 $u\in\mathbb{R}$,

$$\begin{aligned}& D^{b2}F(u_{0},y_{0}) (u)=K\cup\bigl\{ (y_{1},y_{2})\in\mathbb{R}^{2}\mid y_{1}=u^{2} \bigr\} , \\& D^{b2}\mathcal{F}(u_{0},y_{0}) (u)=\bigl\{ (0,0) \bigr\} , \end{aligned}$$

and then

$$\operatorname{Min}_{K}D^{b2}F(u_{0},y_{0}) (0)=\emptyset. $$

Therefore,

$$D^{b2}\mathcal{F}(u_{0},y_{0}) (0)\nsubseteq \operatorname{Min}_{K} D^{b2}F(u_{0},y_{0}) (0). $$

Remark 4.2

Similar properties to Proposition 4.2 for higher-order contingent-type derivatives of perturbation maps have not yet been investigated in [31]. With some suitable modifications, we can obtain similar properties for higher-order contingent-type 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 K-dominated 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 single-point set.

Then, for any $u\in\mathbb{R}^{p}$,

$$D^{bm}\mathcal{P}(u_{0},y_{0}) (u)\subseteq \operatorname{PrMin}_{K} D^{bm}F(u_{0},y_{0}) (u). $$

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 $u\in\mathbb{R}^{p}$,

$$D^{bm}\mathcal{W}(u_{0},y_{0}) (u)\subseteq \operatorname{WMin}_{K} D^{bm}F(u_{0},y_{0}) (u). $$

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

$$ y_{0}+t_{n}^{m}v_{n}\in \mathcal{W}(u_{0}+t_{n}u_{n})\subseteq F(u_{0}+t_{n}u_{n}),\quad\forall n. $$

(11)

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

$$ y_{0}+t_{n}^{m} \overline{v}_{n}\in F(u_{0}+t_{n} \overline{u}_{n}),\quad\forall n. $$

(12)

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

$$ F(u_{1})\subseteq F(u_{2})+L \|u_{1}-u_{2}\|^{m}B_{\mathbb{R}^{q}}. $$

(13)

Naturally, since $t_{n}\downarrow0$, there exists $N>0$ such that

$$ u_{0}+t_{n}\overline{u}_{n}, u_{0}+t_{n}u_{n}\in U,\quad \forall n>N. $$

(14)

Therefore, from (12), (13), and (14), there exists $b_{n}\in B_{\mathbb{R}^{q}}$ such that, for all n large enough,

$$ y_{0}+t_{n}^{m}\bigl( \overline{v}_{n}-L\|\overline{u}_{n}-u_{n} \|^{m}b_{n}\bigr)\in F(u_{0}+t_{n}u_{n}). $$

(15)

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,

$$y_{0}+t_{n}^{m}v_{n}- \bigl(y_{0}+t_{n}^{m}\bigl(\overline{v}_{n}-L \|\overline{u}_{n}-u_{n}\| ^{m}b_{n} \bigr) \bigr)\in\operatorname{int}K, $$

which contradicts with (11). The conclusion is obtained. □

4.2 Higher-order adjacent derivatives of perturbation maps with constraints

In this section, the formulas for estimating higher-order adjacent derivative of perturbation map/ weak perturbation map via adjacent derivative of constraint map together with higher-order 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 mth-order Fréchet continuously differentiable at $(x_{0},u_{0})$. Then, for any $u\in\mathbb{R}^{p}$,

$$\begin{aligned} &\bigl\{ y\in\mathbb{R}^{q}\mid\exists x\in D^{b}X(x_{0},u_{0}) (u), y=d^{m}f(x_{0},u_{0}) \bigl((x,u), \ldots,(x,u)\bigr)\bigr\} \\ &\quad \subseteq D^{bm}F(u_{0},y_{0}) (u). \end{aligned}$$

(16)

Moreover, if X is (first-order) u-directionally 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

$$ f(x_{0}+t_{n}x_{n},u_{0}+t_{n}u_{n}) \in F(u_{0}+t_{n}u_{n}),\quad\forall n. $$

(17)

It follows from the mth-order Fréchet continuously differentiability of f and (17) that we have

$$f(x_{0},u_{0})+t_{n}^{m}d^{m}f(x_{0},u_{0}) \bigl((x_{n},u_{n}),\ldots,(x_{n},u_{n}) \bigr)+o\bigl(t_{n}^{m}\bigl\| (x_{n},u_{n}) \bigr\| ^{m}\bigr)\in F(u_{0}+t_{n}u_{n}). $$

Consequently,

$$ y_{0}+t_{n}^{m} \biggl(d^{m}f(x_{0},u_{0}) \bigl((x_{n},u_{n}), \ldots,(x_{n},u_{n})\bigr)+\frac{o(t_{n}^{m}\| (x_{n},u_{n})\|^{m})}{t_{n}^{m}} \biggr)\in F(u_{0}+t_{n}u_{n}). $$

(18)

It follows from (18) and

$$d^{m}f(x_{0},u_{0}) \bigl((x_{n},u_{n}), \ldots,(x_{n},u_{n})\bigr)+\frac{o(t_{n}^{m}\|(x_{n},u_{n})\| ^{m})}{t_{n}^{m}}\to d^{m}f(x_{0},u_{0}) \bigl((x,u),\ldots,(x,u) \bigr) $$

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

$$ y_{0}+t_{n}^{m}y_{n}=f(x_{n},u_{0}+t_{n}u_{n}), \quad\forall n. $$

(19)

Setting $\widetilde{x}_{n}:=\frac{x_{n}-x_{0}}{t_{n}}$, we have

$$ x_{0}+t_{n}\widetilde{x}_{n}\in X(u_{0}+t_{n}u_{n}) $$

(20)

and

$$ y_{0}+t_{n}^{m}y_{n}=f(x_{0}+t_{n} \widetilde{x}_{n},u_{0}+t_{n}u_{n}), \quad\forall n. $$

(21)

Since X is first-order u-directionally 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 mth-order Fréchet continuously differentiability of f at $(x_{0},u_{0})$ that one has

$$y_{0}+t_{n}^{m}y_{n}=f(x_{0},u_{0})+t_{n}^{m}d^{m}f(x_{0},u_{0}) \bigl((\widetilde{x}_{n},u_{n}),\ldots,( \widetilde{x}_{n},u_{n})\bigr)+o\bigl(t_{n}^{m} \bigl\| (\widetilde {x}_{n},u_{n})\bigr\| ^{m}\bigr). $$

It implies that

$$y_{n}=d^{m}f(x_{0},u_{0}) \bigl(( \widetilde{x}_{n},u_{n}),\ldots,(\widetilde{x}_{n},u_{n}) \bigr)+\frac {o(t_{n}^{m}\|(\widetilde{x}_{n},u_{n})\|^{m})}{t_{n}^{m}}. $$

Letting $n\to\infty$, we have

$$y=d^{m}f(x_{0},u_{0}) \bigl((\widetilde{x},u), \ldots,(\widetilde{x},u)\bigr). $$

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 u-directionally 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 $u\in\mathbb{R}$,

$$\begin{aligned}& D^{b}X(u_{0},x_{0}) (u)=\mathbb{R}_{+},\qquad D^{b2}F(u_{0},y_{0}) (u)=\mathbb{R}_{+}, \\& d^{2}f(x,u)= \begin{bmatrix} 12x^{2} & 0\\ 0& 0 \end{bmatrix}, \end{aligned}$$

and then

$$d^{2}f(x_{0},u_{0})= \begin{bmatrix} 0 & 0\\ 0& 0 \end{bmatrix}. $$

Hence, for any $u\in\mathbb{R}$,

$$\bigl\{ y\in\mathbb{R}\mid x\in D^{b}X(x_{0},u_{0}) (u), y= d^{2}f(x_{0},u_{0}) \bigl((x,u),(x,u) \bigr)\bigr\} =\{0\} \subsetneqq D^{b2}F(u_{0},y_{0}) (u). $$

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 mth-order 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 pseudo-Lipschitz at $((u_{0},y_{0}),x_{0})$.

Then, for any $u\in\mathbb{R}^{p}$,

$$\begin{aligned} &D^{bm}F(u_{0},y_{0}) (u) = \bigl\{ y\in\mathbb{R}^{q}\mid\exists x\in D^{b}X(x_{0},u_{0}) (u), y=d^{m}f(x_{0},u_{0}) \bigl((x,u), \ldots,(x,u)\bigr)\bigr\} . \end{aligned}$$

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 mth-order u-directionally compact at $(u_{0},y_{0})$ for any $u\in\mathbb{R}^{p}$;

• (ii) F is K-dominated 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 single-point set;

• (v) f is mth-order Fréchet continuously differentiable at $(x_{0},u_{0})$;

• (vi) X is first-order u-directionally compact at $(u_{0},x_{0})$.

Then, for any $u\in\mathbb{R}^{p}$,

$$\begin{aligned} &D^{bm}\mathcal{F}(u_{0},y_{0}) (u)\\ &\quad= \operatorname{Min}_{K} D^{bm}F(u_{0},y_{0}) (u)\\ &\quad=\operatorname{Min}_{K}\bigl\{ y\in\mathbb{R}^{q}\mid\exists x \in D^{b}X(x_{0},u_{0}) (u), y=d^{m}f(x_{0},u_{0}) \bigl((x,u),\ldots,(x,u)\bigr)\bigr\} . \end{aligned}$$

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

$$\begin{aligned}& F(u)=\bigl\{ y\in\mathbb{R}\mid u^{4}\le y\le4u^{4}\bigr\} , \\& \mathcal{F}(u)=\bigl\{ u^{4}\bigr\} . \end{aligned}$$

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 first-order u-directionally 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 $u\in\mathbb{R}$,

$$\begin{aligned}& D^{b}X(u_{0},x_{0}) (u)=\{0 \},\qquad D^{b2}F(u_{0},y_{0}) (u)=\{0\}, \\& D^{b2}\mathcal{F}(u_{0},y_{0}) (u)=\{0\}, \\& d^{2}f(x,u)= \begin{bmatrix} 2 & 0\\ 0& 0 \end{bmatrix}, \end{aligned}$$

and then

$$d^{2}f(x_{0},u_{0}) \bigl((x,u),(x,u) \bigr)=2x^{2}. $$

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

$$d^{2}f(x_{0},u_{0}) \bigl((x,u),(x,u)\bigr)=0. $$

Hence, for any $u\in\mathbb{R}$,

$$\begin{aligned} &D^{b2}\mathcal{F}(u_{0},y_{0}) (u) \\ &\quad=\operatorname{Min}_{K}D^{b2}F(u_{0},y_{0}) (u) \\ &\quad= \operatorname{Min}_{K}\bigl\{ y\in\mathbb{R}\mid x\in D^{b}X(x_{0},u_{0}) (u), y= d^{2}f(x_{0},u_{0}) \bigl((x,u),(x,u)\bigr)\bigr\} . \end{aligned}$$

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 mth-order u-directionally compact at $(u_{0},y_{0})$ for any $u\in\mathbb{R}^{p}$;

• (ii) F is K-dominated 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 single-point set;

• (v) f is mth-order Fréchet continuously differentiable at $(x_{0},u_{0})$;

• (vi) X is first-order u-directionally compact at $(u_{0},x_{0})$.

Then, for any $u\in\mathbb{R}^{p}$,

$$\begin{aligned} &D^{bm}\mathcal{P}(u_{0},y_{0}) (u) \\ &\quad= \operatorname{PrMin}_{K} D^{bm}F(u_{0},y_{0}) (u) \\ &\quad=\operatorname{PrMin}_{K}\bigl\{ y\in\mathbb{R}^{q}\mid \exists x\in D^{b}X(x_{0},u_{0}) (u), y=d^{m}f(x_{0},u_{0}) \bigl((x,u),\ldots,(x,u) \bigr)\bigr\} . \end{aligned}$$

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 mth-order u-directionally 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 K̃-dominated by $\mathcal{W}$ near $u_{0}$;

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

• (iv) f is mth-order Fréchet continuously differentiable at $(x_{0},u_{0})$;

• (v) X is u-directionally compact at $(u_{0},x_{0})$.

Then, for any $u\in\mathbb{R}^{p}$,

$$\begin{aligned} &D^{bm}\mathcal{W}(u_{0},y_{0}) (u) \\ &\quad= \operatorname{WMin}_{K} D^{bm}F(u_{0},y_{0}) (u) \\ &\quad=\operatorname{WMin}_{K}\bigl\{ y\in\mathbb{R}^{q}\mid \exists x\in D^{b}X(x_{0},u_{0}) (u), y=d^{m}f(x_{0},u_{0}) \bigl((x,u),\ldots,(x,u) \bigr)\bigr\} . \end{aligned}$$

Proof

It follows from Proposition 4.1(iii), Proposition 4.4, and Proposition 4.5 that the conclusion is given. □

5 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 proto-differentiability 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 proto-differentiability assumption, the adjacent derivatives can be used. Based on the above observation, the mth-order adjacent derivatives was employed to consider higher-order sensitivity analysis for nonsmooth vector optimization in this paper. First of all, we considered the relationships between the mth-order adjacent derivatives of the perturbation map/the proper perturbation map/the weak perturbation map, and the mth-order adjacent derivative of feasible map in objective space. Then the above relations were used to establish the formulas for estimating the mth-order adjacent derivatives of the perturbation map, the proper perturbation map, and the weak perturbation map via the adjacent derivative of constraint map and the mth-order 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 first-order sensitivity analysis using adjacent derivatives and also may be new.