On higher-order adjacent derivative of perturbation map in parametric vector optimization
- Le Thanh Tung^{1}Email author
https://doi.org/10.1186/s13660-016-1059-3
© Tung 2016
Received: 31 December 2015
Accepted: 1 April 2016
Published: 8 April 2016
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.
Keywords
MSC
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
- (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 \(A\subseteq\operatorname{WMin}_{C}A+\operatorname{int}C\cup\{ 0\}\).
- (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
Example 2.2
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
- (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 thatwhere \(B_{Y}\) stands for the closed unit ball in Y.$$F(x_{2})\subset F(x_{1})+\lambda\|x_{1}-x_{2} \|^{\alpha}B_{Y}, \quad\forall x_{1},x_{2} \in U, $$
- (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$$F(x)\cap V\subset\{y_{0}\}+\lambda\|x-x_{0} \|^{m}B_{Y},\quad \forall x\in U. $$
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
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 \(u\in X\).
Definition 2.3
(see [31])
Remark 2.2
(see [31])
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
Proof
Note that the opposite inclusion of (1) may not hold. The following example illustrates the case.
Example 3.1
Proposition 3.2
- (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.
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 \(s_{n}:=\sqrt[m]{\|c_{n}\|}>0\).
Proposition 3.3
- (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.
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
- (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.
Proposition 3.5
- (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.
The following example illustrates that we cannot replace K̃ by C in the conclusion of Proposition 3.5.
Example 3.2
4 Higher-order adjacent derivatives of perturbations maps
- (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
- (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
- (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
Now, we investigate the reverse conclusion in Proposition 4.1.
Proposition 4.2
- (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.
Proof
The following example shows that the assumption (iii) in Proposition 4.2 cannot be dropped.
Example 4.1
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
- (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.
Proof
The proof is similar to that of Proposition 4.2. □
Proposition 4.4
Proof
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
Proof
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.
Corollary 4.1
- (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 \(((u_{0},y_{0}),x_{0})\).
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
- (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 \((u_{0},x_{0})\).
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
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.
Theorem 4.2
- (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 \((u_{0},x_{0})\).
Proof
It follows from Proposition 4.1(ii), Proposition 4.3, and Proposition 4.5 that the conclusion is obtained. □
Theorem 4.3
- (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 \((u_{0},x_{0})\).
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.
Declarations
Acknowledgements
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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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