COMPLEXITYComplexity1099-05261076-2787Hindawi10.1155/2020/98496369849636Research ArticleOptimality Conditions for a Nonsmooth Uncertain Multiobjective Programming ProblemHanWenyanhttps://orcid.org/0000-0003-4729-7748YuGuolinGongTiantianHafsteinSigurdur F.Institute of Applied MathematicsNorth Minzu UniversityYinchuan 750021Chinanun.edu.cn202025720202020130420200107202025720202020Copyright © 2020 Wenyan Han et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This note is devoted to the investigation of optimality conditions for robust approximate quasi weak efficient solutions to a nonsmooth uncertain multiobjective programming problem (NUMP). Firstly, under the extended nonsmooth Mangasarian–Fromovitz constrained qualification assumption, the optimality necessary conditions of robust approximate quasi weak efficient solutions are given by using an alternative theorem. Secondly, a class of generalized convex functions is introduced to the problem (NUMP), which is called the pseudoquasi-type-I function, and its existence is illustrated by a concrete example. Finally, under the pseudopseudo-type-I generalized convexity hypothesis, the optimality sufficient conditions for robust approximate quasi weak efficient solutions to the problem (NUMP) are established.

National Natural Science Foundation of China11861002Natural Science Foundation of Ningxia ProvinceNZ17112North Minzu UniversityZDZX201804
1. Introduction

It is well known that multiobjective programming problems are widely used in the fields of portfolio, resource allocation, and information transfer. In practical problems, most of the objectives or constraints to the optimization model are nonsmooth and are affected by various factors with uncertain information. Therefore, it is a very valuable work to study the nonsmooth uncertain optimization problems. Robust optimization is one of the effective methods to deal with uncertain optimization problems. The robust method is committed to ensuring the worst-case solution which is immunized against the data uncertainty to optimization problems, and for its more details, the reader is referred to . In this paper, the optimality conditions to the nonsmooth uncertain multiobjective programming problem (NUMP) are described by using the robust optimization method.

Convexity and its generalization play an important role in mathematical programming, especially in establishing optimality sufficient conditions of optimization problems. Chuong and Kim  presented the generalized convex-affine function based on the Mordukhovich subdifferential for a class of nonsmooth multiobjective fractional programming problems. For objective and constraint functions f,g of a class of nonsmooth robust multiobjective programming problems, the concept that f,g is a generalized convexity of degree n is introduced in literature . Inspired by the generalized convexity in the above literatures, this paper introduces a kind of generalized convexities based on the Clarke subdifferential, which is called the f,g-pseudoquasi-type-I function, and under its assumption, it proves the optimality conditions of the problem (NUMP).

As we all know, the (weak) efficient solution of multiobjective optimization problems usually does not exist in the noncompact case, but the approximate solution exists under very mild conditions. In addition, most of the solutions obtained by the numerical algorithm are approximate solutions in the real world. Therefore, there exist the important theoretical value and practical significance to study the approximate solution of the optimization problem. Recently, Lee and Jiao  dealt with the optimality conditions of the robust approximate solution for an uncertain convex optimization problem involving a kind of constraint qualifications, which is called closed convex cone constrained qualification; Sun  established the optimality conditions of the robust optimal solution under the constrained qualification with respect to the subdifferential of the convex function; Sun and Li  discussed the optimality conditions of the robust approximate weak efficient solution under the hypothesis of closed convex cone constrained qualification. It is worth mentioning that the approximate weak efficient solution is a special case of the approximate quasi weak efficient solution. The purpose of this paper is to study optimality conditions of the robust approximate quasi weak efficient solution for the problem (NUMP). The convexities and constrained qualification are different from those of mentioned literatures, and we adopt the newly introduced f,g-pseudopseudo-type-I convexity and extended nonsmooth Mangasarian–Fromovitz constrained qualification (see ).

The content of this paper is arranged as follows: In Section 2, some basic concepts and lemmas which will be used in subsequent sections are proposed. The concept of the f,g-pseudoquasi-type-I generalized convexity with respect to the Clarke subdifferential is introduced, and an example is given to illustrate its existence. The main results are present in Section 3, in which the optimality conditions of the robust approximate quasi weak efficient solution to the problem (NUMP) are proven.

2. Preliminaries

This paper considers the following nonsmooth multiobjective programming problem (NMP):(1)NMPminfx=f1x,f2x,,flx,s.t.xC,gjx0,j=1,,m,where C is a nonempty subset of n dimension Euclid space n and fi,gj:n,i=1,,l,j=1,,m, are the Lipschitz functions. The feasible set of the problem (NMP) is denoted as(2)0xC:gjx0,j=1,,m.

When the constraint set of problem (NMP) contains uncertain data, the corresponding nonsmooth uncertain multiobjective programming problem (NUMP) is expressed as(3)NUMPminfx=f1x,f2x,,flx,s.t.xC,gjx,vj0,j=1,,m,where vjVjq is the uncertain parameter, Vj is the compact convex set, and gj:n×q,j=1,,m, are the Lipschitz functions with respect to the first variable. We denote v=v1,,vmV=V1,,Vm. The feasible set of problem (NUMP) is defined by(4)vxC:gjx,vj0,j=1,,m.

The optimality conditions of problem (NUMP) will be studied by the robust optimization method in this note. For this purpose, we consider the following robust counterpart (see ) of the problem (NUMP):(5)NRMPminfx=fx,f2x,,fx,s.t.xC,gjx,vj0,vjVj,j=1,,m.

The robust counterpart problem is called as the nonsmooth robust multiobjective programming problem (NRMP), and the robust feasible set of problem (NRMP) is given by(6)vVv.

Let(7)ψjxmaxgjx,vj:vjVj,j=1,,m,  xn.

For a given x¯n, we divide J=1,,m into two index sets, J=J1x¯J2x¯, where(8)J1x¯=jJ:ψjx¯=0,J2x¯=JJ1x¯.

For any jJ1x¯, let(9)Vjx¯vjVj:gjx¯,vj=ψjx¯.

Let B be a closed unit ball in n. For any x,yn, use x,y to represent the inner product between x and y. Set(10)+n=xn:xi0,i=1,,n,++n=xn:xi>0,J1x¯.

It is said that φ:n is a convex function, if for any x,yn,λ0,1,(11)φλx+1λyλφx+1λφy.

If φ is a convex function, then is said to be a concave function. For any xn, if(12)limsupyxφyφx,then φ is termed to be a upper semicontinuous function. Let Xn be a nonempty open subset. It is said that φ:X is a Lipschitz function, if there exists L>0, such that(13)φx1φx2Lx1x2,x1,x2X.

Let dn, and the directional derivative (see ) of φ at x¯X in the direction d is given by(14)φx¯;d=limt0+φx¯+tdφx¯t,and the Clarke generalized directional derivative (see ) of φ at x¯ in the direction d is defined by(15)φ°x¯;d=limsupyx¯t0+φy+tdφyt,

If(16)φ°x¯;d=φx¯;d,dn,then φ is called to be regular at x¯. The Clarke subdifferential (see ) of φ at x¯ is denoted as(17)φx¯ξn:φ°x¯;dξ,d,dn.

Lemma 1 (see [<xref ref-type="bibr" rid="B7">7</xref>]).

Let φ:Xn be a Lipschitz function, then the following conclusions hold:

φx is a nonempty compact convex set in X, and for any ξφx, one has ξL (L is Lipschitz constant of φ).

dφ°x;d is convex, and

(18)φ°x;dLd.

For any dn, we have

(19)φ°x;d=maxξ,d:ξφx.

For a given function g:U×V0n×q, suppose g satisfies the following assumptions (see ):

gx,v0 is upper semicontinuous in x,v0U×V0.

gx,v0 is a Lipschitz function with respect to the first variable xU, that is, g,v0 is a Lipschitz function for any v0V0.

g is regular with respect to the first variable xU, that is,

(20)gx°x,v0;=gxx,v0;,

where gx°x,v0; is a Clarke generalized directional derivative with respect to the first variable x and gxx,v0; is the directional derivative with respect to the first variable x.

The Clarke subdifferential xgx,v0 with respect to the first variable x is weak upper semicontinuous in x,v0U×V0.

Let(21)ψxmaxgx,v0:v0V0,xn.

If gx,v0 satisfies Assumptions (i)–(iv), then ψx is a Lipschitz function (see ). For a given x¯n, set(22)V0x¯v0V0:gx¯,v0=ψx¯.

Remark 1.

(see ). If Assumptions (i)–(iv) are fufilled, V0 is a convex subset and gx, is a concave on V0; then,(23)ψx=ξ:v0V0xs.t.ξxgx,v0,xn.

Lemma 2 (see [<xref ref-type="bibr" rid="B9">9</xref>]).

Let φ,ϕ:Xn be a Lipschitz function, x¯X. Then,(24)φ+ϕx¯φx¯+ϕx¯.

If φ,ϕ are regular at x¯, then φ+ϕ is regular at x¯, and φ+ϕx¯=φx¯+ϕx¯.

Lemma 3 (alternative theorem, see [<xref ref-type="bibr" rid="B10">10</xref>]).

Let An be a convex set, and ϕ1x,,ϕmx are convex on A, if the following system of inequalities(25)ϕx<0,i=1,,m,xA,has no solution on A, then there exist λ1,,λm0, not all zero, such that(26)i=1mλiϕix0,xA.

Definition 1 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

Let ε=ε1,,εl+l,x¯.

It is said that x¯ is a robust εquasi weak efficient solution of the problem (NUMP), iff x¯ is an εquasi weak efficient solution of the problem (NRMP), that is,

(27)fxfx¯+εxx¯++l,x.

It is called that x¯ is a robust εquasi efficient solution of the problem (NUMP), iff x¯ is an εquasi efficient solution of the problem (NRMP), that is,

(28)fxfx¯+εxx¯+l0,x.

The following generalized convexity is introduced for the objective and constraint functions f,g to the problem (NUMP).

Definition 2.

It is said that f,g is a pseudoquasi-type-I function at x¯C, if for any xC, ξifix¯,i=1,,l,bB,ηjxgjx¯,vj, and vjVjx¯,j=1,,m, there exists un, such that(29)ξi,u+εib,u0fixfix¯+εixx¯0,i=1,,l,(30)gjx,vjgjx¯,vjηj,u0,j=1,,m.

If equation (29) takes a strictly inequality, i.e.,(31)ξi,u+εib,u0fixfix¯+εixx¯>0,i=1,,l,then f,g is called a strictly pseudoquasi-type-I function at x¯C.

The following is an example to illustrate the existence of the pseudoquasi-type-I function.

Example 1.

Let fi:,i=1,2, be given by(32)f1x=32x,if x0,x,if x<0,f2x=43x,if x0,x,if x<0,and g:×V be defined as(33)gx,v=vx2,x,vV0,gx,0=23x,if x<0,x,if x0,where V=0,1. Taking x¯=0,ε=ε1,ε2=1,1, by simple calculation, we can obtain that f1x¯=1,3/2,f2x¯=1,4/3, xgx¯,v=0,vV0,xgx¯,0=1,2/3, and Vx¯=0,12. For any x,ξifix¯,i=1,2,bB,ηxgx¯,0, and vVx¯, there exists 0u, such that(34)ξ1,u+ε1b,u=ξ1,u+b,u0f1xf1x¯+ε1xx¯=f1x+x0(35)ξ2,u+ε2b,u=ξ2,u+b,u0f2xf2x¯+ε2xx¯=f2x+x0(36)gx,vgx¯,v0u0,vV0,(37)gx,0gx¯,0ηu0,v=0.

Hence, f,g is a pseudoquasi-type-I function at x¯=0.

3. Optimality Conditions

In this section, we begin with establishing the optimality necessary conditions for a robust εquasi weak efficient solution to the problem (NUMP) by using the alternative theorem (Lemma 3).

Theorem 1.

In the problem (NUMP), if x¯ is a robust εquasi weak efficient solution of the problem (NUMP), then there exists not all the zero real values λ¯i0,i=1,,l,μ¯j0,jJ1x¯, such that(38)i=1lλ¯ifi°x¯;d+jJ1x¯μ¯jψj°x¯;d+i=1lλ¯iεix¯°x¯;d0,dn.

Proof.

Firstly, we claim that the following system of inequalities:(39)fi°x¯,d+εix¯°x¯,d<0,i=1,,l,ψj°x¯,d<0,jJ1x¯,has no solution dn. Otherwise, there exists dn, such that(40)fi°x¯,d+εix¯°x¯,d<0,i=1,,l,ψj°x¯,d<0,jJ1x¯.

Since(41)limsupt0+fix¯+tdfix¯t=infδ¯i1>0sup0<t<δ¯i1fix¯+tdx¯,dt,(42)limsupt0+εix¯+tdx¯εix¯x¯t=infδ¯i2>0sup0<t<δ¯i2εix¯+tdx¯εix¯x¯t,(43)i=1,,l,we get(44)limsupt0+fix¯+td+εix¯+tdx¯fix¯εix¯x¯tlimsupt0+fix¯+tdfix¯t+limsupt0+εix¯+tdx¯εix¯x¯t=infδ¯i1>0sup0<t<δ¯i1fix¯+tdfix¯t+infδ¯i2>0sup0<t<δ¯i2εix¯+tdx¯εix¯x¯tinfδ¯i1>0ϵ>0sup0<t<δ¯i1h<ϵfix¯+h+tdfix¯+ht+infδ¯i2>0ϵ>0sup0<t<δ¯i2h<ϵεix¯+h+tdx¯εix¯+hx¯t=limsuph0, t0+fix¯+h+tdfix¯+ht+lim suph0, t0+εix¯+h+tdx¯εix¯+hx¯t=fi°x¯,d+εix¯°x¯,d<0,i=1,,l.

Therefore, there exists δ¯i, for any 0<t<δ¯i, and we arrive at(45)fix¯+tdfix¯+εix¯+tdx¯<0,i=1,,l.

On the other hand, for any jJ1x¯,(46)limsupt0+ψjx¯+tdψjx¯t=infδ˜j1>0sup0<t<δ˜j1ψjx¯+tdψjx¯tinfδ˜j1>0ε>0sup0<t<δ˜j1h<εψjx¯+h+tdψjx¯+ht=limsuph0,t0+ψjx¯+h+tdψjx¯+ht=ψj°x¯;d<0.

Hence, there exists δ˜j,jJ1x¯ such that ψjx¯+td<ψjx¯=0, for any 0<t<δ˜j. In addition, for any jJ2x¯, one has ψjx¯<0. Noticing that ψj is a Lipschitz function, we know that there exists δ^j>0, for any t0,δ^j:(47)ψjx¯+td<0,jJ2x¯.

Let δ=minδ¯,δ˜,δ^, where δ¯=mini1,,lδ¯i,δ˜=minjJ1x¯δ˜j, and δ^=minjJ2x¯δ^j. Then, for any t0,δ, it yields that(48)x¯+td,fix¯+tdfix¯+εix¯+tdx¯<0,i=1,,l,which contradicts to the fact that x¯ is a robust εquasi weak efficient solution of the problem (NUMP).

We conclude from Lemma 1 (ii) that(49)dfi°x¯;d+εix¯°x¯;d,i=1,,l,(50)ψj°x¯;d,jJ1x¯are convex functions. Again by Lemma 3, it can be known that there exists not all zero real values λ¯i0,i=1,,l,μ¯j0,jJ1x¯, such that equation (38) holds.

Next, we will examine the optimality necessary conditions of the robust εquasi weak efficient solution to the problem (NUMP). For this purpose, we need to introduce the following extended nonsmooth Mangasarian–Fromovitz constrained qualification.

Definition 3.

In the problem (NRMP), let x¯. If(51)dns.t.gjx°x¯,vj;d<0,vjVjx¯,jJ1x¯,then it is called that the problem (NRMP) satisfies extended the nonsmooth Mangasarian–Fromovitz constrained qualification at x¯.

Theorem 2.

In the problem (NUMP), suppose that gj,j=1,,m, satisfy Assumptions (i)–(iv), and for any xn, gjx, is a concave function on Vj. If x¯ is a robust εquasi weak efficient solution to the problem (NUMP), then there exists λ¯,μ¯,v¯+l×+m×V and λ¯,μ¯0, where λ¯=λ¯1,,λ¯l, μ¯=μ¯1,,μ¯m, v¯=v¯1,,v¯m, and v¯jVjx¯,j=1,,m, such that(52)0i=1lλ¯ifix¯+j=1mμ¯jxgjx¯,v¯j+i=1lλ¯iεiB,(53)μ¯jgjx¯,v¯j=0,j=1,,m.

In addition, if the problem (NRMP) satisfies the extended nonsmooth Mangasarian–Fromovitz constrained qualification at x¯, then there exists λ¯+l\0 and μ¯+m,v¯jVjx¯,j=1,,m, such that equations (52) and (53) hold.

Proof.

Since x¯ is a robust εquasi weak efficient solution to the problem (NUMP), it yields from Theorem 2 that there exists not all zero real values λ¯i0,i=1,,l,μ¯j0,jJ1x¯, such that(54)i=1lλ¯ifi°x¯;d+jJ1x¯μ¯jψj°x¯;d+i=1lλ¯iεix¯°,x¯;d0,dn.

According to Lemma 1 (iii), we know that(55)i=1lλ¯imaxξi,d:ξifix¯+jJ1x¯μ¯jmaxηj,d:ηjψjx¯+i=1lλ¯imaxζi,d:ζiεiB0,dn.

This means that(56)maxξifix¯ηjψjx¯ζiεiBi=1lλ¯iξi+jJ1x¯μ¯jηj+i=1lλ¯iζi,d0,dB,this is equivalent to(57)infdBmaxξifix¯ηjψjx¯ζiεiBi=1lλ¯iξi+jJ1x¯μ¯jηj+i=1lλ¯iζi,d0.

Noticing that fix¯,εiB,i=1,,l,ψjx¯,jJ1x¯, are the nonempty compact convex set in n (by Lemma 1 (i)). Hence, it follows from lop-sided minimax theorem  that there exists ξ¯ifix¯,ζ¯iεiB,i=1,,l,η¯jψjx¯,jJ1x¯, such that(58)infdBi=1lλ¯iξ¯i+jJ1x¯μ¯jη¯i+i=1lλ¯iζi¯,d=maxξifix¯ηjψjx¯ζiεiBinfdBi=1lλ¯iξi+jJ1x¯μ¯jηj+i=1lλ¯iζi,d0.

Again because(59)infdBi=1lλ¯iξ¯i+jJ1x¯μ¯jη¯j+i=1lλ¯iζ¯i,d=i=1lλ¯iξ¯i+jJ1x¯μ¯jη¯j+i=1lλ¯iζ¯i0,we arrive at(60)i=1lλ¯iξ¯i+jJ1x¯μ¯jη¯j+i=1lλ¯iζ¯i=0.

Therefore,(61)0i=1lλ¯ifix¯+jJ1x¯μ¯jψjx¯+i=1lλ¯iεiB.

In addition, for any jJ2x¯, let μ¯j=0. Then, from the above equation, we get that(62)0i=1lλ¯ifix¯+j=1mμ¯jψjx¯+i=1lλ¯iεiB,(63)μ¯jψjx¯=0,j=1,,m.

It yields from Remark 1 that there exists v¯jVjx¯,j=1,,m, such that (52) and (53) are true.

On the other hand, if the problem (NRMP) satisfies the extended nonsmooth Mangasarian–Fromovitz constrained qualification at x¯ and λ¯=0, then there exists μ¯+m, v¯jVjx¯,jJ1x¯, such that(64)0jJ1x¯μ¯jxgjx¯,v¯j=xjJ1x¯μ¯jgjx¯,v¯j,and the above equation holds according to Lemma 2. Again by Lemma 1 (iii), we have(65)jJ1x¯μ¯jgjx°x¯,vj;d0,dn,which contradicts to Definition 3. Hence, this leads to λ¯+l\0. □

It is said that equations (52) and (53) are robust optimality necessary conditions of the problem (NUMP). We present the following Theorem 3 which is an optimality sufficient condition for the robust ε-quasi weak efficient solution to the problem (NUMP).

Theorem 3.

In the problem (NUMP), supposing that x¯,λ¯,μ¯,v¯×+l\0×+m×V satisfies the robust optimality necessary conditions.

If f,g is a pseudoquasi-type-I function at x¯, then x¯ is a robust ε-quasi weak efficient solution of the problem (NUMP).

If f,g is a strictly pseudoquasi-type-I function at x¯, then x¯ is a robust ε-quasi efficient solution of the problem (NUMP).

Proof.

By the given conditions, it follows that the robust optimality necessary conditions hold at x¯,λ¯,μ¯,v¯×+l\0×+m×V, that is,(66)0i=1lλ¯ifix¯+j=1mμ¯jxgjx¯,v¯j+i=1lλ¯iεiB,(67)μ¯jgjx¯,v¯j=0,j=1,,m.

Therefore, there exists ξ¯ifix¯,bB,η¯jxgjx¯,v¯j,j=1,,m, such that(68)i=1lλ¯iξ¯i+j=1mμ¯jη¯j+i=1lλ¯iεib=0.

Let us prove conclusion (i). If x¯ is not a robust ε-quasi weak efficient solution of the problem (NUMP), then there exists x^, such that(69)fx^fx¯+εx^x¯++l,that is,(70)fx^fx¯+εx^x¯<0,i=1,,l.

On the other hand, in equation (66), if μ¯j0, then gjx¯,v¯j=0,j=1,,m. Again since x^, then(71)gjx^,v¯j0.

Hence,(72)gjx^,v¯j0=gjx^,v¯j.

Because f,g is a pseudoquasi-type-I function at x¯, combining equation (70) with equation (72), we conclude that for ξ¯ifix¯,i=1,,l,bB,η¯jxgjx¯,v¯j,v¯jVjx¯,j=1,,m, there exists un, such that(73)ξ¯i,u+εib,u<0,i=1,,l,(74)η¯j,u0,μj0.

Noticing that λ¯+l\0, it holds that(75)i=1lλ¯iξ¯i+εib,u+j=1mμ¯jη¯j,u<0,which contradicts to equation (74). Therefore, x¯ is a robust ε-quasi weak efficient solution of the problem (NUMP).

By the similar arguments, we can prove (ii).

Finally, as the end of this article, we give a concrete example to verify Theorem 3.

Example 2.

Consider the following nonsmooth robust multiobjective programming (NRMP)0 problem:(76)NRMP0minf=f1x,f2x,s.t.gx,v0,vV=0,1,where(77)f1x=52x,if x0,x,if x<0,f2x=53x,if x0,x,if x<0,and g:×V is given by(78)gx,v=vx2,x,vV0,gx,0=13x,if x<0,x,if x0.

Taking x¯=0 and ε=ε1,ε2=1,1, it is obvious to get that x¯=0 is a robust ε-quasi weak efficient solution of the problem (NRMP)0. It is easy to know that(79)f1x¯=1,52,f2x¯=1,53,(80)xgx¯,v=0,vV0,(81)xgx¯,0=1,13,Vx¯=0,1.

For any x,ξifix¯,i=1,2,bB,ηxgx¯,0, and vVx¯, there exists 0u, such that(82)ξ1,u+ε1b,u=ξ1,u+b,u0f1xf1x¯+ε1xx¯=f1x+x0,(83)ξ2,u+ε2b,u=ξ2,u+b,u0f2xf2x¯+ε2xx¯=f2x+x0,(84)gx,vgx¯,v0u0,vV0,(85)gx,0gx¯,0ηu0,v=0.

It is clear that f,g is a pseudoquasi-type-I function at x¯. Equations (58) and (59) hold for λ1=1,λ2=0,μ=0, and b=1. It yields from Theorem 3 that x¯ is a ε-quasi weak efficient solution of the problem (NRMP)0.

4. Conclusions

The optimality conditions of the robust approximate quasi weak efficient solution to a nonsmooth uncertain multiobjective programming problem (NUMP) are studied by using the robust optimization method in this note. Firstly, we have introduced f,g-pseudoquasi-type-I functions to the problem (NUMP), and an example is presented to illustrate its existence. Secondly, under the assumptions that the problem (NUMP) satisfies the extended nonsmooth Mangasarian–Fromovitz constrained qualification and pseudoquasi-type-I convexity, optimality conditions of the robust ε-quasi weak efficient solution are proved.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This research was supported by the Natural Science Foundation of China, under Grant no. 11861002; Natural Science Foundation of Ningxia, under Grant no. NZ17112; Key Project of North Minzu University, under Grant no. ZDZX201804; Nonlinear analysis and financial optimization research center of North Minzu University.

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