Weak Sharp Minima in Set-Valued Optimization Problems

Abstract

We introduce the notion of a weak ψ-sharp minimizer for set-valued optimization problems. We present some sufficient and necessary conditions that a pair point is a weak ψ-sharp minimizer through the outer limit of set-valued map and develop the characterization of the weak ψ-sharp minimizer in terms of a generalized nonlinear scalarization function. These results extend the corresponding ones in Studniarski (2007).

1. Introduction

The notion of weak sharp minima in general mathematical program problems was first introduced by Ferris in [1]. It is a generalization of a sharp minimum in [2] to include the possibility of nonunique solution set. The study of weak sharp minima is motivated primarily by applications in convex and convex composite programming, where such minima commonly occur. Weak sharp minima plays an important role in the sensitivity analysis [3, 4] and convergence analysis of a wide range of optimization algorithms [5]. Recently, the study of weak sharp solution set covers real-valued optimization problems [5–8] and piecewise linear multiobjective optimization problems [9, 10].

In [11], Bednarczuk defined weak sharp Pareto minima of order 𝑚 for vector-valued mappings and used weak sharp Pareto minima to prove upper Hölderness and Hölder calmness of the solution set-valued mappings for parametric vector optimization problems. In [12], Studniarski gave the definition of weak 𝜓-sharp local Pareto minima in multiobjective optimization problems and presented necessary and sufficient conditions. In [13], Xu and Li established a sufficient and necessary condition for weak 𝜓-sharp local Pareto minima in vector optimization problems in infinite spaces, the approach is that they transformed weak 𝜓-sharp local Pareto minima of a vector-valued function to weak 𝜓-sharp local minima of a family of scalar functions. Most recently, Durea and Strugariu [14] introduced the definition of weak 𝜓-sharp local minima by an oriented distance function in set-valued optimization problems and established necessary optimality conditions in terms of Mordukhovich coderivative.

In the paper, motivated by the work in [15, 16], we also introduce the notion of weak 𝜓-sharp minima, which is different from one in [14], and establish some sufficient and necessary conditions through the outer limit of set-valued map. In particular, we develop the characterization of the weak 𝜓-sharp minimizer in terms of the generalized nonlinear scalarization function.

This paper is organized as follows. In Section 2, we recall some basic definitions and give the notion of the weak 𝜓-sharp local minimizer for set-valued optimization problems. In Section 3, we present some sufficient and necessary conditions through the outer limit of the set-valued map. In Section 4, we establish a characterization of weak 𝜓-sharp local minima in terms of the generalized nonlinear scalarization function.

2. Preliminary Results

Throughout this paper, let 𝑋,  𝑌 be real normed spaces. 𝐵(𝑥,𝛿) denotes the open ball with center 𝑥∈𝑋 and radius 𝛿>0, 𝒩(𝑥) is the family of all neighborhoods of 𝑥, and dist(𝑥,𝑊) is the distance from the point 𝑥 to the set 𝑊⊂𝑋. The symbols 𝑆^𝑐, cl𝑆, and int𝑆 denote, respectively, the complement, closure, and interior of 𝑆. Let 𝐷⊂𝑌 be a convex cone (containing 0) with nonempty interior int𝐷 and let 𝑌 be partially ordered by 𝐷.

Let 𝐹∶𝑋→2^𝑌 be a set-valued map. We denote the graph and domain of 𝐹, respectively, by Gr𝐹=𝑥,𝑦∈𝑋×𝑌∶𝑦∈𝐹(𝑥),Dom𝐹={𝑥∈𝑋∶𝐹(𝑥)≠∅}. (2.1) If 𝑆 is a subset of 𝑋, then 𝐹(𝑆)=∪_𝑥∈𝑆𝐹(𝑥) and the inverse set-valued map of 𝐹 is 𝐹⁻¹∶𝑌→2^𝑋 given by (𝑦,𝑥)∈Gr𝐹⁻¹ if and only if (𝑥,𝑦)∈Gr𝐹.

Definition 2.1. Suppose that 𝐷 is a closed convex pointed ((−𝐷)∩𝐷={0}) cone. A point 𝑦₀∈𝐴⊂𝑌 is called a strict efficient (resp., weak) point of 𝐴, denoted by 𝑦₀∈Str_𝐷𝐴 (resp., 𝑦₀∈𝑊Min_𝐷𝐴) if 𝐴−𝑦₀∩(−𝐷⧵{0})=∅resp.𝐴−𝑦₀∩(−int𝐷)=∅. (2.2)

Given a set-valued map 𝐹∶𝑋→2^𝑌 and a subset 𝑆 of 𝑋, the following abstract optimization is considered: min𝐹(𝑥),s.t.𝑥∈𝑆. (2.3)

Definition 2.2. Suppose that 𝐷 is a closed convex pointed cone. A point (𝑥₀,𝑦₀)∈Gr𝐹, with 𝑥₀∈𝑆, is said to be a local strict (resp., weak) minimizer of 𝐹 over 𝑆, written as (𝑥₀,𝑦₀)∈𝐿Str_𝐷(𝐹,𝑆)  (resp.,(𝑥₀,𝑦₀)∈𝐿𝑊Min_𝐷(𝐹,𝑆)), if there exists a neighborhood 𝑈 of 𝑥₀ in 𝑋 such that 𝑦₀∈Str_𝐷(𝐹(𝑈∩𝑆))resp.,𝑦₀∈𝑊min_𝐷(𝐹(,𝐹(𝑥)𝑈∩𝑆))thatis,∀𝑥∈𝑆∩𝑈,−𝑦₀∩(−𝐷⧵{0})=∅resp.∀𝑥∈𝑆∩𝑈,𝐹(𝑥)−𝑦₀∩(−int𝐷)=∅. (2.4)

We will say that (𝑥₀,𝑦₀) is a global strict (global weak) minimizers when 𝑈=𝑋. The set of all global strict minimizers (resp., weak minimizers) is denoted by Str_𝐷(𝐹,𝑆)  (resp.,𝑊Min_𝐷(𝐹,𝑆)).

Definition 2.3. Let 𝜓∶[0,+∞)→[0,+∞) be a nondecreasing function with the property 𝜓(𝑡)=0⇔𝑡=0 (such a family of functions is denoted by Ψ) and 𝑥₀∈𝑆. We say that a point pair (𝑥₀,𝑦₀)∈Gr𝐹∩(𝑆×𝑌) is a weak 𝜓-sharp local Pareto minimizer for (2.3), denoted by (𝑥₀,𝑦₀)∈𝑊𝑆𝐿(𝜓,𝐹,𝑆), if there exists a constant 𝛼>0 and 𝑈∈𝒩(𝑥₀) such that (𝐹(𝑥)+𝐷)∩𝐵𝑦₀,𝛼𝜓(dist(𝑥,𝑊))=∅,∀𝑥∈𝑆∩𝑈⧵𝑊, (2.5) where 𝑊=𝑥∈𝑆∶𝑦₀∈𝐹(𝑥)=𝑆∩𝐹⁻¹𝑦₀. (2.6) If we choose 𝑈=𝑋, we will say the point pair (𝑥₀,𝑦₀)∈Gr𝐹∩(𝑆×𝑌) is a weak 𝜓-sharp minimizer for (2.3), denoted by (𝑥₀,𝑦₀)∈𝑊𝑆(𝜓,𝐹,𝑆). In particular, let 𝜓_𝑚(𝑡)=𝑡^𝑚 for 𝑚=1,2,…. Then, we say (𝑥₀,𝑦₀)∈Gr𝐹∩(𝑆×𝑌) is a weak 𝜓-sharp local minimizer of order 𝑚 for (2.3) if (𝑥₀,𝑦₀)∈𝑊𝑆𝐿(𝜓_𝑚,𝐹,𝑆).

Obviously, condition (2.5) can be expressed in the following equivalent form: 𝐹(𝑥)⊂𝐵𝑦₀,𝛼𝜓(dist(𝑥,𝑊))−𝐷^𝑐,∀𝑥∈𝑆∩𝑈⧵𝑊. (2.7)

Remark 2.4. Clearly, if the map 𝐹 is a vector-valued function, the notion is equivalent to Definition 8.2.3 with 𝜓=𝜓_𝑚 in [17] and the weak 𝜓-sharp local minimizer for vector optimizations in [12].

Remark 2.5. In [14], the definition of weak 𝜓-sharp local minimizer for set-valued optimization is given by the oriented distance function ▵. However, we establish the definition by the map 𝐹. When the map 𝐹 is the real-valued function and the cone 𝐷=𝑅₊, our definition is equivalent to Definition  2.1 in [14].

3. Optimality Conditions for Weak 𝝍-Sharp Minimizer for Set-Valued Optimization

In this section, we present sufficient and necessary conditions that a point pair is a weak 𝜓-sharp local minimizer in set-valued optimization problems.

Theorem 3.1. Let 𝐹∶𝑋→2^𝑌, 𝑥₀∈𝑆, and 𝜓∈Ψ. Assume that 𝑊 defined in (2.6) is a closed set. Then, (𝑥₀,𝑦₀)∈𝑊𝑆𝐿(𝜓,𝐹,𝑆) if and only if 0∉limsup_{𝑥→𝑥0,𝑥∈𝑆}𝐹(𝑥)−𝑦₀+𝐷𝜓((.dist𝑥,𝑊)) (3.1)

Proof. Part “only if”: suppose that (3.1) is false, then there exist sequences 𝑥_𝑘∈𝑆⧵𝑊, 𝑦_𝑘∈𝐹(𝑥_𝑘), 𝑑_𝑘∈𝐷 such that 𝑥_𝑘→𝑥₀ and lim_𝑘→∞𝑦_𝑘+𝑑_𝑘−𝑦₀𝜓dist𝑥_𝑘,𝑊=0. (3.2) Hence, for any 𝜖>0, there is 𝑘₀=𝑘₀(𝜖) such that 𝑦_𝑘−𝑦₀+𝑑_𝑘<𝜖𝜓dist𝑥_𝑘,𝑊,∀𝑘≥𝑘₀. (3.3) Namely, 𝑦_𝑘+𝑑_𝑘∈𝐵(𝑦₀,𝜖𝜓(dist(𝑥_𝑘,𝑊))).
By assumption, there exist 𝛼>0 and 𝑈=𝐵(𝑥₀,𝛿) such that (2.5) holds. In particular, for 𝜖=min{𝛼,𝛿}, there exists 𝑘₀=𝑘₀(𝜖) such that for each 𝑘≥𝑘₀, we have that 𝑥_𝑘∈𝑆∩𝐵(𝑥₀,𝛿)⧵𝑊𝑦_𝑘+𝑑_𝑘∈𝐵𝑦₀,𝜖𝜓dist𝑥_𝑘,𝑊⊂𝐵𝑦₀,𝛼𝜓dist𝑥_𝑘,𝑊, (3.4) which is contradiction to (2.5).
Part “if”: suppose that the relation (2.5) is false, then for any 𝛿>0 and 𝛼>0, there exist 𝑥∈𝑆∩𝐵(𝑥₀,𝛿)⧵𝑊 and 𝑦∈𝐹(𝑥) such that 𝑦+𝐷∩𝐵𝑦₀,𝛼𝜓(dist(𝑥,𝑊))≠∅. (3.5) In particular, choosing 𝛼=𝛿=1/𝑘, there exist 𝑥_𝑘∈𝑆∩𝐵(𝑥₀,1/𝑘)⧵𝑊 and 𝑦_𝑘∈𝐹(𝑥_𝑘) and 𝑑_𝑘∈𝐷 such that 𝑦_𝑘+𝑑_𝑘∈𝐵𝑦₀,1𝑘𝜓dist𝑥_𝑘,𝑊, (3.6) that is, 𝑦_𝑘+𝑑_𝑘−𝑦₀𝜓dist𝑥_𝑘<1,𝑊𝑘. (3.7) Hence, for sufficiently large 𝑘, we have 𝑦_𝑘+𝑑_𝑘−𝑦₀𝜓dist𝑥_𝑘,𝑊→0, (3.8) which contradicts (3.1).

From Theorem 3.1, we easily obtain the following result.

Corollary 3.2. Let 𝐹∶𝑋→2^𝑌, 𝑥₀∈𝑆, and 𝜓∈Ψ. Assume that 𝑊 defined in (2.6) is a closed set. If (𝑥₀,𝑦₀)∈𝑊𝑆𝐿(𝜓,𝐹,𝑆), then, limsup_{𝑥→𝑥0,𝑥∈𝑆}𝐹(𝑥)−𝑦₀𝜓((∩()dist𝑥,𝑊))−𝐷=∅. (3.9)

Theorem 3.3. Let 𝑌=𝑅^𝑝, 𝐷=𝑅^𝑝₊=[0,+∞)^𝑝, and 𝑅^𝑝₊=[0,+∞]^𝑝. Let 𝐹∶𝑋→2^𝑌, 𝑥₀∈𝑆. Assume that 𝑊 defined in (2.6) is a closed set and 𝜓∈Ψ. Then, the following statements are equivalent:(i)(𝑥₀,𝑦₀)∉𝑊𝑆𝐿(𝜓,𝐹,𝑆),(ii)(limsup_{𝑥→𝑥0,𝑥∈𝑆}((𝐹(𝑥)−𝑦₀)/𝜓(dist(𝑥,𝑊))))∩(−𝑅^𝑝₊)≠∅.

Proof. (i⇒ii) By assumption and Theorem 3.1, there exist sequences 𝑥_𝑘∈𝑆⧵𝑊,  𝑦_𝑘∈𝐹(𝑥_𝑘), 𝑑_𝑘∈𝑅^𝑝₊ such that 𝑥_𝑘→𝑥₀ and lim_𝑘→∞𝑦_𝑘+𝑑_𝑘−𝑦₀𝜓dist𝑥_𝑘,𝑊=0. (3.10) Let 𝑏_𝑘=𝑎_𝑘+𝑐_𝑘, where 𝑎_𝑘=𝑦_𝑘−𝑦₀𝜓dist𝑥_𝑘,𝑊,𝑐_𝑘=𝑑_𝑘𝜓dist𝑥_𝑘,𝑊∈𝑅^𝑝₊. (3.11)
Consider the first component of the vector 𝑐_𝑘=(𝑐¹_𝑘,𝑐²_𝑘,…,𝑐^𝑝_𝑘). Let 𝑐¹=limsup_𝑘→∞𝑐¹_𝑘. Then, there is an infinite set 𝐾₁⊂𝒩 such that lim_{𝐾1∋𝑘→∞}𝑐¹_𝑘=𝑐¹. We have 𝑐¹≥0 (it can be taken +∞), since 𝑐¹_𝑘≥0. Now, let us consider the second component of sequence (𝑐_𝑘)_𝑘∈𝐾1. Let 𝑐²=limsup_{𝐾1∋𝑘→∞}𝑐²_𝑘. Hence, there exists an infinite set 𝐾₂⊂𝐾₁ such that lim_{𝐾2∋𝑘→∞}𝑐²_𝑘=𝑐². We still have 𝑐²≥0 (it can be taken +∞). So, we have lim_{𝐾2∋𝑘→∞}(𝑐¹_𝑘,𝑐²_𝑘)=(𝑐¹,𝑐²). Continuing the process, we obtain a vector 𝑐=(𝑐¹,𝑐²,…,𝑐^𝑝)∈𝑅^𝑝₊ and an infinite set 𝐾_𝑝⊂𝒩 such that 𝑐=lim_{𝐾𝑝∋𝑘→∞}𝑐_𝑘.
Since 𝑏_𝑘=𝑎_𝑘+𝑐_𝑘, taking the limit on both sides of the equation, we have 0=lim_{𝐾𝑝∋𝑘→∞}𝑏_𝑘=lim_{𝐾𝑝∋𝑘→∞}𝑎_𝑘+lim_{𝐾𝑝∋𝑘→∞}𝑐_𝑘. (3.12) Therefore, 𝑑∶=lim_{𝐾𝑝∋𝑘→∞}𝑎_𝑘=−𝑐≤0. Namely, limsup_{𝑥→𝑥0,𝑥∈𝑆}𝐹(𝑥)−𝑦₀𝜓((∩−dist𝑥,𝑊))𝑅^𝑝₊≠∅. (3.13)
(i⇐ii) If −𝑑∈𝑅^𝑝₊, by Theorem 3.1, the result is true. So, we suppose that some components of 𝑑 are −∞. Reordering to 𝑑, let 𝑑=(𝑑¹,𝑑²,…,𝑑^𝑛,𝑑^𝑛+1,…,𝑑^𝑝) with 𝑑^𝑖=−∞ for 𝑖=1,2,…,𝑛 and 𝑑^𝑖∈(−∞,0] for 𝑖>𝑛, with 𝑛≥1. Hence, from relation (3.13), we see that there exist 𝑥_𝑘∈𝑆⧵𝑊,𝑥_𝑘→𝑥₀,𝑦_𝑘∈𝐹(𝑥_𝑘) such that lim_𝑘→∞𝑦^𝑖_𝑘−𝑦^𝑖₀𝜓dist𝑥_𝑘,𝑊=−∞,𝑖=1,2,…,𝑛. (3.14) Since, for sufficiently large 𝑘 and for 𝑖=1,2,…,𝑛,  𝑑^𝑖_𝑘∶=−(𝑦^𝑖_𝑘−𝑦^𝑖₀)>0. Let 𝑑_𝑘=𝑑¹_𝑘,…,𝑑^𝑛_𝑘,−𝜓dist𝑥_𝑘,𝑊𝑑^𝑛+1,…,−𝜓dist𝑥_𝑘,𝑊𝑑^𝑝∈𝑅^𝑝₊. (3.15) Clearly, one has lim_𝑘→∞𝑦_𝑘−𝑦₀+𝑑_𝑘𝜓dist𝑥_𝑘,𝑊=0. (3.16) Namely, 0∈limsup_{𝑥→𝑥0,𝑥∈𝑆}𝐹(𝑥)−𝑦₀+𝑅^𝑝₊𝜓((.dist𝑥,𝑊)) (3.17) By Theorem 3.1, we derive the result.

4. Scalarization

Scalarization is one of the most important procedures in vector optimization. In this section, we apply a generalized nonlinear scalarization function introduced by Hernández and Rodríguez-Marín in [18] to discuss the weak 𝜓-sharp minimizer in set-valued optimization problems.

Let 𝐷 be a proper closed convex cone and int𝐷≠∅. Let 𝑒∈int𝐷 be a fixed point.

Definition 4.1 (see [18]). The generalized nonlinear scalarization function 𝐺∶2^𝑌→𝑅∪{−∞} is defined by 𝐺(𝐴)=inf{𝑡∈𝑅∶te∈𝐴+𝐷}. (4.1) A nonempty set 𝐴⊂𝑌 is said to be 𝐷-proper if 𝐴+𝐷≠𝑌.

Next, we present several properties about the generalized nonlinear scalarization function 𝐺.

Lemma 4.2 (see [18]). 𝐴 is 𝐷-proper if and only if 𝐺(𝐴)>−∞.

Lemma 4.3 (see [16]). Let 𝑟∈𝑅 and 𝐴 be a nonempty subset of 𝑌. Then, 𝐺(𝐴)>𝑟⟺𝑟𝑒∉cl(𝐴+𝐷). (4.2)

Lemma 4.4 (see [18]). Let 𝐴⊂𝑌 and 𝑦₀∈𝐴. If 𝑦₀∈Str_𝐷𝐴, then 𝐺(𝐴−𝑦₀)=0.

Given a set-valued map 𝐹∶𝑋→2^𝑌 and (𝑥₀,𝑦₀)∈Gr𝐹. Define 𝐹∶𝑋→2^𝑌 by 𝐹(𝑥)=𝐹(𝑥)−𝑦₀. (4.3)

Now, we consider weak 𝜓-sharp local minimizer for a set-valued map 𝐹 through the weak sharp local minimizer of the scalarization function 𝐺∘𝐹∶𝑋→𝑅∪{−∞}.

Theorem 4.5. Let 𝑥₀∈𝑆 and (𝑥₀,𝑦₀)∈Gr𝐹. Suppose that 𝑊 defined in (2.6) is a closed set and 𝑦₀∈Str_𝐷𝐹(𝑥₀). Then, 𝑥₀,𝑦₀∈𝑊𝑆𝐿𝜓,𝐹,𝑆⟺𝑥₀∈𝑊𝑆𝐿𝜓,𝐺∘𝐹,𝑆. (4.4)

Proof. Part “only if”: assume that (𝑥₀,𝑦₀)∈𝑊𝑆𝐿(𝜓,𝐹,𝑆), there exist 𝛼>0 and 𝑈∈𝒩(𝑥₀) such that 𝐹(𝑥)−𝑦₀+𝐷∩𝐵0,𝛼𝜓(dist(𝑥,𝑊))=∅,∀𝑥∈𝑆∩𝑈⧵𝑊. (4.5) Since 𝐵(0,𝛼𝜓(dist(𝑥,𝑊)) is an open set, 𝐵0,𝛼𝜓(dist(𝑥,𝑊))⊂cl𝐹(𝑥)−𝑦₀+𝐷^𝑐. (4.6) Note that, when 𝑊 is a closed set, 𝛼4‖𝑒‖𝜓(dist(𝑥,𝑊))𝑒∈𝐵0,𝛼𝜓(dist(𝑥,𝑊)). (4.7) Hence, 𝛼4‖𝑒‖𝜓(dist(𝑥,𝑊))𝑒∉cl𝐹(𝑥)−𝑦₀+𝐷. (4.8) By Lemma 4.3, we have 𝐺𝐹(𝑥)−𝑦₀>𝛼4‖𝑒‖𝜓(dist(𝑥,𝑊)). (4.9) On the other hand, since 𝑦₀∈Str𝐹(𝑥₀), in terms of Lemma 4.4, we get 𝐺𝐹𝑥₀−𝑦₀=0. (4.10) This relation, together with (4.9), yields 𝐺𝐹(𝑥)−𝑦₀>𝐺𝐹𝑥₀−𝑦₀+𝛼4‖𝑒‖𝜓(dist(𝑥,𝑊)),∀𝑥∈𝑆∩𝑈⧵𝑊. (4.11) Namely, 𝐺∘𝐹(𝑥)>𝐺∘𝐹𝑥₀+𝛼4‖𝑒‖𝜓(dist(𝑥,𝑊)),∀𝑥∈𝑆∩𝑈⧵𝑊, (4.12) that is, 𝑥₀∈𝑊𝑆𝐿(𝜓,𝐺∘𝐹,𝑆).
Part “if”: by assumption, there exist 𝛽>0 and 𝑈∈𝒩(𝑥₀) such that 𝐺𝐹(𝑥)>𝐺𝐹𝑥₀+𝛽𝜓(dist(𝑥,𝑊)),∀𝑥∈𝑆∩𝑈⧵𝑊. (4.13) Since 𝑦₀∈Str_𝐷𝐹(𝑥₀), by applying Lemma 4.4, we get 𝐺(𝐹(𝑥₀)−𝑦₀)=0. Thus, we have 𝐺𝐹(𝑥)−𝑦₀>𝛽𝜓(dist(𝑥,𝑊)),∀𝑥∈𝑆∩𝑈⧵𝑊. (4.14) Once more using Lemma 4.3, one has 𝛽𝜓(dist(𝑥,𝑊))𝑒∉cl𝐹(𝑥)−𝑦₀+𝐷. (4.15) Furthermore, 𝛽𝜓(dist(𝑥,𝑊))𝑒∉𝐹(𝑥)−𝑦₀+𝐷, (4.16) which implies that 𝛽𝜓(dist(𝑥,𝑊))𝑒−𝐷∩𝐹(𝑥)−𝑦₀+𝐷=∅,∀𝑥∈𝑆∩𝑈⧵𝑊. (4.17) Since 𝑒∈int𝐷, there exists a number 𝜖>0 such that 𝐵(0,𝜖)⊂𝑒−𝐷. Moreover, 𝐵(0,𝜆𝜖)⊂𝜆𝑒−𝐷,∀𝜆>0. (4.18) Hence, from (4.18), we obtain 𝐵0,𝜖𝛽𝜓(dist(𝑥,𝑊))⊂𝛽𝜓(dist(𝑥,𝑊))𝑒−𝐷. (4.19) Combining it with relation (4.17), we deduce that 𝐵0,𝜖𝛽𝜓(dist(𝑥,𝑊))∩𝐹(𝑥)−𝑦₀+𝐷=∅,∀𝑥∈𝑆∩𝑈⧵𝑊. (4.20) By the definition of weak 𝜓-sharp local minimum, we have (𝑥₀,𝑦₀)∈𝑊𝑆𝐿(𝜓,𝐹,𝑆).

In Theorem 4.5, if the map 𝐹 is a vector-valued function and the function 𝐺 becomes the nonlinear scalarization function 𝑔, we easily obtain the following result, which is Theorem  3.4 in [13].

Corollary 4.6. Let 𝑥₀∈𝑆, 𝑓∶𝑋→𝑌, and 𝑓(𝑥)=𝑓(𝑥)−𝑓(𝑥₀). Then, 𝑥₀∈𝑊𝑆𝐿𝜓,𝑓,𝑆⟺𝑥₀∈𝑊𝑆𝐿𝜓,𝑔∘𝑓,𝑆. (4.21)

Acknowledgments

This research was partially supported by Heilongjiang Department of Education Science and Technology Research Project (Grant no.: 12521457) and the National Natural Science Foundation of China (no. 11071267).