• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    ON APPROXIMATE EFFICIENCY FOR NONSMOOTH ROBUST VECTOR OPTIMIZATION PROBLEMS?

    2020-08-02 05:29:30TadeuszANTCZAK

    Tadeusz ANTCZAK

    Faculty of Mathematics and Computer Science, University of Lód′z, Banacha 22, 90-238 Lódz, Poland E-mail: tadeusz.antczak@wmii.uni.lodz.pl

    Yogendra PANDEY

    Department of Mathematics, Satish Chandra College, Ballia 277001, India

    Vinay SINGH

    Department of Mathematics, National Institute of Technology, Aizawl-796012, Mizoram, India

    Shashi Kant MISHRA

    Department of Mathematics, Banaras Hindu University, Varanasi-221005, India

    Abstract In this article,we use the robust optimization approach(also called the worst-case approach) for finding ?-efficient solutions of the robust multiobjective optimization problem defined as a robust (worst-case) counterpart for the considered nonsmooth multiobjective programming problem with the uncertainty in both the objective and constraint functions.Namely, we establish both necessary and sufficient optimality conditions for a feasible solution to be an ?-efficient solution (an approximate efficient solution) of the considered robust multiobjective optimization problem. We also use a scalarizing method in proving these optimality conditions.

    Key words Robust optimization approach; robust multiobjective optimization; ?-efficient solution; ?-optimality conditions; scalarization

    1 Introduction

    Robust optimization methodology (the worst-case approach) is a powerful approach for examining and solving optimization problems under data uncertainty. In robust optimization,the data is uncertain but bounded, that is, the data is varying in a given uncertainty set,and we choose the best solution among the robust feasible ones; for detail, we refer to [1–6,9, 12, 20–26, 28, 29, 31, 32, 38]. Ben-Tal et al. [5] introduced the concept of the uncertain linear optimization problem and its robust counterpart, and discussed the computational issues. Also, Bertsimas et al. [6] characterized the robust counterpart of a linear mathematical programming problem with uncertainty set described by an arbitrary norm. Jeyakumar and Li[23, 24] presented basic theory and applications of an uncertain linear mathematical program problem. Jeyakumar and Li [25] derived a robust theorem of the alternative for parameterized convex inequality systems using conjugate analysis and introduced duality theory for convex mathematical programming problems in the face of data uncertainty via robust optimization.Jeyakumar et al. [26] considered a nonlinear optimization problem with face of data uncertainty and established robust KKT necessary and sufficient optimality conditions for a robust minimizer. Furthermore, Jeyakumar et al. [26] introduced robust duality theory for generalized convex mathematical programming problems in the face of data uncertainty within the framework of robust optimization and established robust strong duality between an uncertain nonlinear primal optimization problem and its uncertain Lagrangian dual.

    Robust optimization for solving multiobjective optimization problems with uncertain data is a current topic of research. Kuroiwa and Lee [31] defined three kind of robust efficient solutions and established necessary optimality theorems for weakly and properly robust efficient solutions for the considered uncertain multiobjective optimization problem. Recently, Chuong[9]established the necessary optimality conditions and,under the generalized convexity assumptions, sufficient optimality conditions for a (weakly) robust efficient solution of the considered uncertain multiobjective programming problem. Besides, robust multiobjective optimization with optimality defining by a partial order has been widely applied to solve many practical problems like internet routing, portfolio optimization, energy production scheduling in microgrids, transport, agriculture, industry-specific applications, health care applications, among others (see, for example, [3, 7, 8, 10, 14–17, 27, 30]).

    In recent years, many authors have established epsilon optimality conditions for several kind of optimization problems (see, for example, [13, 18, 19, 32–36, 39, 40]). Lee and Lee [32]established optimality theorems for epsilon solutions of the scalar robust convex optimization problem.

    In this article, we treat the robust approach for the considered uncertain multiobjective programming problem with the uncertainty in both objective and constraint functions which is the worst case approach for finding approximate efficient solutions for such multiobjective optimization problems. In this approach, for the considered uncertain multiobjective optimization problem, its robust (worst-case) counterpart is constructed as an associated robust multiobjective optimization problem. Then, motivated by works of Kuroiwa and Lee [31] and Lee and Lee[32],we derive the ?-efficiency theorem for the considered uncertain multiobjective programming problem by examining its robust (worst-case) counterpart. In other words, we prove necessary and sufficient optimality conditions for a feasible solution to be an ?-efficient solution (an approximate efficient solution) of the robust multiobjective optimization problem.Moreover, we use a scalarizing method in proving these optimality results. In this method,for the robust multiobjective optimization problem, its associated scalar optimization problem is constructed. Then, we prove the equivalence between an approximate efficient solution of the robust multiobjective optimization problem and an approximate solution of its associated scalar optimization problem constructed in the scalarizing method which is used in this article.The ?-efficiency theorem established in this article for the considered uncertain multiobjective programming problem is illustrated by an example of such a vector optimization problem with the uncertainty in both objective and constraint functions.

    2 Preliminaries

    The following convention for equalities and inequalities will be used throughout this article.

    For any vectors x=(x1,...,xn)T,y =(y1,...,yn)Tin Rn,we give the following definitions:

    (i) x=y if and only if xi=yifor all i=1,...,n;

    (ii) x

    (iii) x ≦y if and only if xi≦yifor all i=1,...,n;

    (iv) x ≤y if and only if x ≦y and

    In this section, we provide some definitions and results that we shall use in the sequel.

    The inner product in Rnis defined by 〈x,y〉:=xTy for all x,y ∈Rn: The set C is convex whenever λx+(1 ?λ)y ∈C for all x,y ∈C and any λ ∈[0,1]. The set C ?Rnis a cone if αC ?C for all α ≧0. The indicator function δC:Rn→R ∪{+∞} of a set C is defined by

    A function f :Rn→R∪{+∞}is said to be convex on Rnif the inequality f(λx+(1?λ)y)≦λf(y)is satisfied for all x,y ∈Rnand any λ ∈[0,1].The effective domain of f,denoted by domf,is defined by domf :={x ∈Rn:f(x)<+∞}.The epigraph of the function f :Rn→R∪{+∞},denoted by epif, is defined by

    Let f :Rn→R∪{+∞}be a convex function. The ?-subdifferential of f at∈domf is defined by

    Let f be a proper convex function on Rn. Its conjugate function f?: Rn→R ∪{+∞} is defined at x?∈Rnby

    Clearly, f?is a proper lower semicontinuous convex function and, moreover,

    Proposition 2.1([25, 32]) Let f1: Rn→R be a continuous convex function and f2:Rn→R ∪{+∞} be a proper lower semicontinuous convex function. Then,

    Proposition 2.2([25]) Let fi,i ∈I,(where I is an arbitrary index set)be a proper lower semicontinuous convex function on Rn.Furthermore,assume that there exists∈Rnsuch that. Then,

    Now, let gj(·,vj) : Rn×Rq→R,vj∈Vj?Rq,j = 1,...,m, be convex functions. Then,the set

    is called a robust characteristic cone.

    Proposition 2.3([25]) Let gj: Rn×Rq→R,j = 1,...,m, be a continuous function,such that for each vj∈Vj?Rq,gj(·,vj) is a convex function. Then, the following set

    is a cone.

    Proposition 2.4([25]) Let gj: Rn×Rq→R,j = 1,...,m, be continuous functions and, for each j = 1,...,m,Vj?Rqbe a convex set. Furthermore, assume that, for each vj∈Vj,gj(·,vj) is a convex function on Rnand, for each x ∈Rn,gj(x,·) is a concave function on Vj. Then,

    is a convex cone.

    Proposition 2.5([21, 32]) Let f : Rn→R ∪{+∞} be a proper lower semicontinuous convex function and∈domf. Then,

    Proposition 2.6([25]) Let gj:Rn×Rq→R,j =1,...,m,be continuous functions such that, for each vj∈Vj?Rq,gj(·,vj) be a convex function on Rn. Furthermore, assume that each set Vj,j = 1,...,m, is compact, and there exists∈Rnsuch that gj(,vj) < 0 for any vj∈Vj,j =1,...,m. Then, the set

    is closed.

    3 ?-Optimality Conditions

    In this section, we derive both necessary and sufficient optimality conditions for a feasible solution to be an ?-robust efficient solution of the considered uncertain multiobjective programming problem with the uncertainty in both objective and constraint functions by examining its robust (worst-case) counterpart, that is, its associated robust multiobjective programming problem.

    In this article, we consider an uncertain multiobjective programming problem defined as follows:

    where fi: Rn×Rp→R,i = 1,...,s, and gj: Rn×Rp→R,j = 1,...,m, are continuous functions and ui∈Ui,vj∈Vj, are uncertain parameters,that is,the data vectors uiand vjare not known exactly at the time when the solution has to be determined, Uiand Vjare convex compact subsets of Rpand Rq, respectively.

    Hence, the robust counterpart (RMP) of the uncertain multiobjective programming problem (UMP) is defined as the following multiobjective programming problem:

    Let us denote by S the set of all feasible solutions for the robust multiobjective programming problem (RMP), that is, S ={x ∈Rn:gj(x,vj)≦0,j =1,...,m,?vj∈Vj}.

    Definition 3.1A point x ∈Rnis a robust feasible solution of the considered uncertain robust multiobjective programming problem (RMP) if gj(x,vj)≦0,j =1,...,m,?vj∈Vj.

    Now,we give the definition of ?-efficiency(approximate efficiency)for the defined uncertain robust multiobjective programming problem (RMP) which is, at the same time, a robust ?efficient solution of the original uncertain multiobjective programming problem (UMP).

    Definition 3.2(?-efficient solution of (RMP)) Let ? ∈Rs,? ≧0 be given. A point∈S is said to be an ?-efficient solution of the robust multiobjective programming problem (RMP)(thus, a robust ?-efficient solution of the considered uncertain multiobjective programming problem (UMP)) if there is no a feasible solution x of (RMP) such that

    In this article, we shall assume that, for any ? ∈Rs,? ≧0, the set of ?-efficient solutions of the robust multiobjective programming problem (RMP) is nonempty.

    Now, for the considered uncertain robust multiobjective programming problem(RMP), we define its associated scalar optimization problem.

    Now, we give the definition of a γ-optimal solution of the scalar optimization problem(SMRP?).

    Definition 3.3Let γ be a given nonnegative real number. A feasible pointof the scalar optimization problem (SMRP?) is said to be a γ-optimal solution of the scalar optimization problem (SMRP?) if the inequality

    holds for all feasible solutions of the problem (SMRP?).

    Now, we prove the equivalency between the problems (RMP) and (SMRP?).

    Lemma 3.4Let ? ≧0 be given. Then,∈S is an ?-efficient solution of (RMP) if and only ifis a γ-optimal solution of (SMRP?), where

    ProofLetbe an ?-efficient solution of(RMP). Asis an ?-efficient solution of(RMP),by Definition 3.2, there is no other feasible solution x ∈S such that

    This means that there is no x ∈satisfying both (3.1) and (3.2). Then, by (3.1) and (3.2),it follows that the inequality

    is not fulfilled for any x ∈. Thus,by Definition 3.3,this means thatis a γ-optimal solution of (SMRP?).

    Now, we extend the result established by Jeyakumar and Li (Theorem 2.4 [25]) to the vectorial case.

    Lemma 3.5Let fi(·,ui),i = 1,...,s, be a convex and continuous function and gj:Rn×Rn→R,j =1,...,m, be a continuous function such that, for each vj∈Vj, where Vjis a compact subset of Rq,gj(·,vj) is a convex function. Furthermore, assume that S is nonempty.Then, exactly one of the following two statements holds:

    ProofAssume that condition (ii) is fulfilled. Then, the following relation

    By Proposition 2.2, it follows that

    Then, by Proposition 2.1, (3.7) is equivalent to

    Thus, by the definition of the epigraph, we have

    Hence, by the definition of the conjugate function, (3.9) is equivalent to

    By the definition of an indicator function, it follows that δA?(x) = 0 for any x ∈. Hence,(3.10) is equivalent to F(x) ≧0 for any x ∈. Thus, we have shown that the case when the condition (ii) is fulfilled is equivalent to the fact that the condition (i) is not satisfied. This completes the proof of this lemma.

    Now, we use Lemma 3.5 to prove the next result.

    Theorem 3.6Let fi: Rn×Rp→R,i=1,...,s, be continuous functions such that, for each ui,fi(·,ui) is a convex function. Also, let gj: Rn×Rq→R,j = 1,...,m, be continuous functions such that, for each vj∈Rq,gj(·,vj) is a convex function. Furthermore, assume that the set

    is closed and convex. Then,is a γ-optimal solution of (SMRP?), if and only if there existsuch that the following inequality

    ProofLetbe a γ-optimal solution of(SMRP?). Then,by Definition 3.2,it follows that the inequality F(x) ≧F()?γ holds for all x ∈, whereLet H(x)=F(x)?F()+γ. Then, by the above inequality, the inequality

    Using the definition of the function H, (3.13) can be re-written as follows:

    Thus, (3.14) gives

    Again by using the definition of the conjugate function, (3.15) yields

    By(3.12),it follows that the condition(i) in Lemma 3.5 is not satisfied. Hence,by Lemma 3.5,it follows that the condition (ii) is fulfilled, that is, the relation

    holds. By assumption, the set

    is closed and convex. Thus, (3.17) gives

    By (3.18), it follows that there existsuch that the relation

    holds. Then, there exist t?∈Rn,a ≧0,∈Rn,bi≧0,i = 1,...,s,∈Rn, and cj≧0,j =1,...,m, such that

    Hence, the above relation yields, respectively,

    and

    By (3.19), it follows that

    Combining(3.20)and(3.21),and by the definition of a conjugate function,then we obtain that the relation

    holds for any x ∈Rn. Then, (3.22) yields

    By the definition of a conjugate function for the functionswe have

    By the definition of the function F, it follows that the inequality

    Conversely, assume that there exist≧0,i = 1,...,s,∈Vj,j = 1,...,m, such that inequality (3.11) is fulfilled for any x ∈. As≧0,i = 1,...,s,≧0,j = 1,...,m, and x ∈, (3.11) implies that

    Using Lemma 3.4 and Lemma 3.5, we prove the following ?-efficiency theorem for the considered robust multiobjective programming problem (RMP).

    Theorem 3.7(?-efficiency theorem) Let fi: Rn×Rp→R,i = 1,...,s, be continuous functions such that, for each ui, every function fi(·,ui) is convex. Also, let gj: Rn×Rp→R,j =1,...,m, be continuous functions such that, for each vj∈Rq, every function gj(·,vj) is convex. Furthermore, assume that the setis closed and convex. Then, the following statements are equivalent:

    (i) x ∈S is an ?-efficient solution of the robust multiobjective programming problem(RMP);

    and

    ProofThe equivalency between the conditions (i) and (ii) follows by Theorem 3.6.

    Now, we prove the equivalency between the conditions (ii) and (iii).

    Now, we prove the equivalency of conditions (iii) and (iv).

    Let us assume that the condition (iii) is fulfilled, that is, (3.18) is satisfied. Hence, by(3.18), it follows that there existsuch that the relation

    holds. Then, by Proposition 2.5, it follows that there exist≧0,i=1,...,s,≧0,∈Vj,j =1,...,m, such that

    holds. The above relation implies equivalently that there existsuch that

    and

    Relation (3.25) is equivalent to the fact that there existj =1,...,m, such that

    Hence, (3.27) and (3.26) are precisely the condition (iv).

    Thus, the proof of this theorem is completed.

    In order to illustrate the results established in Theorem 3.7, we give the example of an uncertain multiobjective programming problem with the uncertainty in both objective and constraint functions.

    Example 3.8Consider the following uncertain multiobjective programming problem with the uncertainty in both objective and constraint functions defined as follows:

    where u1∈U1= [0,1],u2∈U2= [0,1],v1∈V1= [], v2∈V2= [?1,0] are uncertain parameters. Its robust counterpart,that is,the uncertain multiobjective programming problem(RMP1), is defined as follows:

    Condition (i)Note that the following inequalities, if at least one of them is strict,

    are not satisfied for any feasible solutionof the robust multiobjective programming problem (RMP1). Hence, by Definition 3.2,= (0,) is, in fact, an ?-efficient solution of(RMP1).

    Condition (ii)LetHence,. Note that the following inequalityis satisfied for all feasible solutions of the associated scalar optimization problem(SRMP?)defined for the robust multiobjective programming problem(RMP).Then,by Definition 3.3,=(0,12)is a γ-optimal solution of the problem (SRMP?).

    Condition (iii)By the definition of the conjugate function, we have

    Hence, by the definition of the epigraph of a function, we have

    By the definition of the epigraph of a function, Proposition 2.1, and (2.4), we have

    Now, we check the condition (iii). Thus, we have

    Condition (iv)In order to check condition (iv), we set

    Taking into account the calculated above ?-subdifferentials of the appropriate functions, we have

    Hence, it follows that

    Furthermore, note that also the second relation in condition (iv) is fulfilled at the considered case. Indeed, we haveThen, we have shown that condition (iv) is also fulfilled.

    4 Conclusions

    In this article, the robust approach is used for finding approximate efficient solutions of the considered multiobjective programming problem with the uncertainty in both objective and constraint functions. Namely, we study the ?-efficiency theorem for the considered uncertain convex multiobjective programming problem by examining its robust(worst-case)counterpart.In other words,we establish both necessary and sufficient optimality conditions for an ?-efficient solution of the robust multiobjective optimization problem. In proving this result, we also use a scalarizing method. Furthermore, the ?-efficiency theorem established in this article is illustrated by the example of a nondifferentiable multiobjective programming problem with the uncertainty in both objective and constraint functions.

    However, some interesting topics for further research remain. Also, it would be interesting to prove similar optimality results for other classes of uncertain multiobjective optimization problems. We shall investigate these questions in subsequent papers.

    日韩中字成人| 精品久久久久久久末码| 国产精品国产三级国产专区5o | 成人特级av手机在线观看| videossex国产| 久久精品久久久久久久性| 一级毛片aaaaaa免费看小| 人人妻人人澡人人爽人人夜夜 | 丰满人妻一区二区三区视频av| 女人十人毛片免费观看3o分钟| 国产人妻一区二区三区在| 国产精品麻豆人妻色哟哟久久 | 搡女人真爽免费视频火全软件| 国产视频首页在线观看| 麻豆乱淫一区二区| 久久这里有精品视频免费| 久久鲁丝午夜福利片| 亚洲成人av在线免费| 日韩欧美 国产精品| 日韩大片免费观看网站 | 伦精品一区二区三区| 亚洲激情五月婷婷啪啪| 欧美性猛交╳xxx乱大交人| 午夜精品在线福利| 国产精品野战在线观看| 亚洲精品自拍成人| 综合色丁香网| 亚洲人成网站在线播| 久久精品国产亚洲av涩爱| av播播在线观看一区| 丰满乱子伦码专区| 一个人观看的视频www高清免费观看| 搡老妇女老女人老熟妇| 高清日韩中文字幕在线| 国产av在哪里看| 亚洲精品,欧美精品| 亚洲国产欧洲综合997久久,| 一边摸一边抽搐一进一小说| 成人亚洲精品av一区二区| 国产激情偷乱视频一区二区| 全区人妻精品视频| 午夜a级毛片| 噜噜噜噜噜久久久久久91| 日韩欧美精品免费久久| 特大巨黑吊av在线直播| av在线亚洲专区| 日本黄大片高清| 少妇丰满av| 麻豆国产97在线/欧美| 欧美成人a在线观看| 狂野欧美白嫩少妇大欣赏| 内地一区二区视频在线| 少妇熟女aⅴ在线视频| 欧美一区二区精品小视频在线| 国产伦精品一区二区三区视频9| 久久久久久久午夜电影| 久99久视频精品免费| 亚洲人成网站高清观看| 成人午夜精彩视频在线观看| 国产伦精品一区二区三区四那| 熟女人妻精品中文字幕| 黄色欧美视频在线观看| 亚洲av二区三区四区| 91午夜精品亚洲一区二区三区| 99久久中文字幕三级久久日本| 丰满人妻一区二区三区视频av| 高清日韩中文字幕在线| 久久久久久久久大av| 一区二区三区四区激情视频| 中文字幕制服av| 久久久久久久久中文| 一级黄色大片毛片| 中文欧美无线码| 国产精品一区二区三区四区久久| 亚洲国产精品成人综合色| 久久精品国产亚洲网站| 99久久精品一区二区三区| 精品国产露脸久久av麻豆 | 日韩视频在线欧美| 亚洲欧美精品自产自拍| 久久久成人免费电影| 国产高清国产精品国产三级 | 亚洲国产欧美在线一区| 1024手机看黄色片| 午夜久久久久精精品| 内射极品少妇av片p| 国产高潮美女av| 看非洲黑人一级黄片| 一个人免费在线观看电影| 内地一区二区视频在线| 中文字幕av在线有码专区| 久久精品久久精品一区二区三区| 中文字幕制服av| 91精品一卡2卡3卡4卡| 蜜臀久久99精品久久宅男| 国产高清国产精品国产三级 | 日韩大片免费观看网站 | 色网站视频免费| av免费在线看不卡| 成人性生交大片免费视频hd| 高清视频免费观看一区二区 | 岛国在线免费视频观看| 观看免费一级毛片| 神马国产精品三级电影在线观看| 综合色丁香网| 久久人妻av系列| 寂寞人妻少妇视频99o| 日本猛色少妇xxxxx猛交久久| 国产精品一区www在线观看| 中文字幕av在线有码专区| 亚洲成人精品中文字幕电影| 男女视频在线观看网站免费| 国产精品久久久久久久久免| 99久久九九国产精品国产免费| 热99在线观看视频| 亚洲婷婷狠狠爱综合网| 一级av片app| 国产精品日韩av在线免费观看| 美女国产视频在线观看| 亚洲精品国产成人久久av| 免费观看的影片在线观看| 精品免费久久久久久久清纯| 亚洲av一区综合| 在线免费十八禁| 欧美变态另类bdsm刘玥| 成人国产麻豆网| 性色avwww在线观看| 国产成人精品一,二区| 午夜福利高清视频| 卡戴珊不雅视频在线播放| 亚洲国产精品国产精品| 亚洲精品乱码久久久久久按摩| 成人漫画全彩无遮挡| 特大巨黑吊av在线直播| 中文天堂在线官网| 欧美激情在线99| 尤物成人国产欧美一区二区三区| 日韩一本色道免费dvd| 欧美性猛交╳xxx乱大交人| 一卡2卡三卡四卡精品乱码亚洲| 精品酒店卫生间| 99久久精品热视频| 在线免费观看不下载黄p国产| 高清视频免费观看一区二区 | 国产伦精品一区二区三区视频9| 亚洲最大成人手机在线| 麻豆成人av视频| 看非洲黑人一级黄片| 亚洲国产最新在线播放| 成人午夜精彩视频在线观看| 久久婷婷人人爽人人干人人爱| 春色校园在线视频观看| 天堂av国产一区二区熟女人妻| 深夜a级毛片| 国产免费视频播放在线视频 | 国产av码专区亚洲av| 国产精品人妻久久久影院| 麻豆成人午夜福利视频| 欧美激情久久久久久爽电影| 麻豆国产97在线/欧美| 婷婷色av中文字幕| 国产午夜福利久久久久久| 九九热线精品视视频播放| 99久国产av精品| 国产老妇女一区| 日韩三级伦理在线观看| 亚洲精品影视一区二区三区av| 亚洲最大成人av| 三级男女做爰猛烈吃奶摸视频| 日韩精品青青久久久久久| 久久久久久大精品| 成人性生交大片免费视频hd| 99久久精品一区二区三区| 精品一区二区三区人妻视频| 99久久中文字幕三级久久日本| 免费搜索国产男女视频| 99热这里只有是精品50| 岛国毛片在线播放| 久久国产乱子免费精品| 国产精品女同一区二区软件| 三级国产精品片| 97超碰精品成人国产| 日韩一本色道免费dvd| 精品久久久久久久久av| 一本一本综合久久| 老师上课跳d突然被开到最大视频| 少妇高潮的动态图| 亚洲精品成人久久久久久| 免费看美女性在线毛片视频| 麻豆av噜噜一区二区三区| 少妇的逼水好多| ponron亚洲| 色哟哟·www| 熟女人妻精品中文字幕| 国产激情偷乱视频一区二区| 天天躁夜夜躁狠狠久久av| 久久人人爽人人爽人人片va| 亚洲欧美日韩高清专用| 亚洲三级黄色毛片| 国产精品一区二区性色av| 五月玫瑰六月丁香| 日本欧美国产在线视频| 国产亚洲av片在线观看秒播厂 | 日韩欧美国产在线观看| 欧美区成人在线视频| 久久精品国产自在天天线| 丝袜美腿在线中文| 九九爱精品视频在线观看| 亚洲经典国产精华液单| 久久久成人免费电影| 丰满少妇做爰视频| 日本一本二区三区精品| 亚洲第一区二区三区不卡| 亚州av有码| 99久国产av精品国产电影| 别揉我奶头 嗯啊视频| 成人亚洲欧美一区二区av| 26uuu在线亚洲综合色| a级毛片免费高清观看在线播放| 亚洲综合精品二区| 久久精品国产亚洲av天美| 亚洲成人av在线免费| 七月丁香在线播放| 综合色丁香网| 高清日韩中文字幕在线| 国语对白做爰xxxⅹ性视频网站| 麻豆乱淫一区二区| 在线播放无遮挡| 91aial.com中文字幕在线观看| 亚洲五月天丁香| av黄色大香蕉| 在线天堂最新版资源| 又粗又硬又长又爽又黄的视频| 国产精品熟女久久久久浪| av专区在线播放| 美女大奶头视频| 日韩高清综合在线| 国产精品电影一区二区三区| 国产一区二区三区av在线| 毛片一级片免费看久久久久| 亚洲在线自拍视频| 日本免费一区二区三区高清不卡| 舔av片在线| 99久久成人亚洲精品观看| 日韩欧美 国产精品| 成人特级av手机在线观看| 在线免费观看不下载黄p国产| 伦精品一区二区三区| 久久精品久久久久久噜噜老黄 | 免费av观看视频| 只有这里有精品99| 两个人视频免费观看高清| 国产高清三级在线| 男人舔奶头视频| 校园人妻丝袜中文字幕| 老师上课跳d突然被开到最大视频| 又粗又爽又猛毛片免费看| 国产精品女同一区二区软件| 中文字幕人妻熟人妻熟丝袜美| 欧美日韩一区二区视频在线观看视频在线 | 成人特级av手机在线观看| 欧美xxxx黑人xx丫x性爽| 欧美成人精品欧美一级黄| 日韩在线高清观看一区二区三区| 1000部很黄的大片| 国产精品人妻久久久影院| 成人无遮挡网站| 久久精品国产99精品国产亚洲性色| 欧美激情国产日韩精品一区| 欧美日本亚洲视频在线播放| 日韩成人av中文字幕在线观看| av女优亚洲男人天堂| 国产高清三级在线| av.在线天堂| 国产精品国产三级国产av玫瑰| 亚洲最大成人手机在线| 亚洲成色77777| 久久综合国产亚洲精品| 啦啦啦观看免费观看视频高清| 久久婷婷人人爽人人干人人爱| 欧美丝袜亚洲另类| 在线观看av片永久免费下载| 2021少妇久久久久久久久久久| 九草在线视频观看| 少妇的逼好多水| 中文字幕精品亚洲无线码一区| 高清午夜精品一区二区三区| 超碰av人人做人人爽久久| 欧美成人免费av一区二区三区| ponron亚洲| 禁无遮挡网站| 亚洲电影在线观看av| 少妇猛男粗大的猛烈进出视频 | 一级毛片aaaaaa免费看小| 色综合亚洲欧美另类图片| 91精品国产九色| 午夜福利在线在线| 99热全是精品| 男人舔奶头视频| 1024手机看黄色片| 熟女电影av网| 最近手机中文字幕大全| 在线天堂最新版资源| 一二三四中文在线观看免费高清| 亚洲在久久综合| 国产精品日韩av在线免费观看| 亚洲在线自拍视频| 国产精品人妻久久久影院| 国产av一区在线观看免费| 日韩中字成人| 中文字幕制服av| 国产精品人妻久久久影院| 在线播放国产精品三级| 午夜a级毛片| 久久精品国产99精品国产亚洲性色| 国产乱人视频| 久久精品夜色国产| 日韩人妻高清精品专区| 亚洲国产色片| ponron亚洲| 男人和女人高潮做爰伦理| av福利片在线观看| videossex国产| 国产精品.久久久| 免费观看性生交大片5| 91狼人影院| 男女国产视频网站| 97超碰精品成人国产| 国产亚洲午夜精品一区二区久久 | 久热久热在线精品观看| 亚洲av中文av极速乱| 18禁在线无遮挡免费观看视频| 在线免费观看不下载黄p国产| 亚洲av不卡在线观看| 亚洲无线观看免费| 婷婷六月久久综合丁香| 国产老妇伦熟女老妇高清| 尾随美女入室| 欧美bdsm另类| 一本久久精品| 高清在线视频一区二区三区 | 在线观看一区二区三区| 亚洲va在线va天堂va国产| 欧美一级a爱片免费观看看| 午夜福利高清视频| 激情 狠狠 欧美| 最近2019中文字幕mv第一页| 一个人看视频在线观看www免费| 久久久久久久久大av| 在线免费观看不下载黄p国产| 成人高潮视频无遮挡免费网站| 99久久中文字幕三级久久日本| 久久人人爽人人片av| 欧美不卡视频在线免费观看| 爱豆传媒免费全集在线观看| 国产 一区 欧美 日韩| 精品久久久久久电影网 | 成年女人永久免费观看视频| 久久久欧美国产精品| 亚洲欧洲日产国产| 国产精品人妻久久久久久| 国产亚洲最大av| 欧美变态另类bdsm刘玥| 激情 狠狠 欧美| 在线播放无遮挡| 久久久精品欧美日韩精品| 赤兔流量卡办理| 国国产精品蜜臀av免费| 国产又色又爽无遮挡免| 亚洲av一区综合| 精品少妇黑人巨大在线播放 | 全区人妻精品视频| 在现免费观看毛片| 欧美精品一区二区大全| 国产成人aa在线观看| 在线观看av片永久免费下载| 久久久久性生活片| 国产午夜精品久久久久久一区二区三区| 韩国高清视频一区二区三区| 精华霜和精华液先用哪个| 18禁动态无遮挡网站| 在线观看一区二区三区| 国产在视频线在精品| 在线观看一区二区三区| 中文字幕精品亚洲无线码一区| 一级黄色大片毛片| 91久久精品国产一区二区三区| 免费在线观看成人毛片| 三级经典国产精品| 久久人人爽人人爽人人片va| 麻豆乱淫一区二区| 欧美激情久久久久久爽电影| 亚洲av免费在线观看| 大香蕉97超碰在线| 日韩欧美在线乱码| 日产精品乱码卡一卡2卡三| 91久久精品国产一区二区成人| 国产色爽女视频免费观看| 91午夜精品亚洲一区二区三区| 最近视频中文字幕2019在线8| 欧美成人午夜免费资源| 国产精品福利在线免费观看| 特大巨黑吊av在线直播| 亚洲综合色惰| 青春草视频在线免费观看| 国产亚洲午夜精品一区二区久久 | 99热这里只有是精品50| 日本猛色少妇xxxxx猛交久久| 看片在线看免费视频| 色综合站精品国产| 欧美+日韩+精品| 亚洲av熟女| 国产白丝娇喘喷水9色精品| 国模一区二区三区四区视频| 国内精品美女久久久久久| 人妻少妇偷人精品九色| 久久99精品国语久久久| 一卡2卡三卡四卡精品乱码亚洲| 国产伦精品一区二区三区视频9| 亚洲天堂国产精品一区在线| 亚洲熟妇中文字幕五十中出| 18+在线观看网站| 我要搜黄色片| 人人妻人人澡人人爽人人夜夜 | 99热6这里只有精品| 2022亚洲国产成人精品| 亚洲国产成人一精品久久久| av免费在线看不卡| a级一级毛片免费在线观看| 舔av片在线| 成人高潮视频无遮挡免费网站| 性插视频无遮挡在线免费观看| 99视频精品全部免费 在线| 久久久色成人| 亚洲av中文av极速乱| 成人二区视频| 久久久久久伊人网av| 亚洲经典国产精华液单| 在线天堂最新版资源| 卡戴珊不雅视频在线播放| 国产午夜精品论理片| 国产精品日韩av在线免费观看| 联通29元200g的流量卡| 国内揄拍国产精品人妻在线| 免费在线观看成人毛片| ponron亚洲| 欧美色视频一区免费| 国产老妇女一区| 国产成人a∨麻豆精品| 免费看光身美女| 深爱激情五月婷婷| 欧美高清成人免费视频www| 99在线人妻在线中文字幕| 亚洲天堂国产精品一区在线| 久久这里只有精品中国| 极品教师在线视频| 人人妻人人澡欧美一区二区| 乱系列少妇在线播放| 国产精品一区二区性色av| 午夜福利视频1000在线观看| 一个人看的www免费观看视频| av视频在线观看入口| av在线亚洲专区| 久久久精品欧美日韩精品| 国产激情偷乱视频一区二区| 长腿黑丝高跟| 亚洲人成网站高清观看| 国产私拍福利视频在线观看| 高清毛片免费看| 免费播放大片免费观看视频在线观看 | 国产成人精品久久久久久| 视频中文字幕在线观看| 亚洲精华国产精华液的使用体验| АⅤ资源中文在线天堂| 欧美一级a爱片免费观看看| 丰满人妻一区二区三区视频av| 亚洲国产精品国产精品| 亚洲美女视频黄频| 99热这里只有精品一区| 日本色播在线视频| 国产精品伦人一区二区| 自拍偷自拍亚洲精品老妇| 99热6这里只有精品| 国内少妇人妻偷人精品xxx网站| 午夜福利视频1000在线观看| 国产白丝娇喘喷水9色精品| 综合色丁香网| 国产午夜精品一二区理论片| 亚洲国产欧洲综合997久久,| 在线播放国产精品三级| 日本色播在线视频| 韩国高清视频一区二区三区| 两个人的视频大全免费| 性色avwww在线观看| .国产精品久久| 久久6这里有精品| 欧美激情国产日韩精品一区| 听说在线观看完整版免费高清| 可以在线观看毛片的网站| 五月玫瑰六月丁香| 91久久精品国产一区二区成人| 国产精品99久久久久久久久| 成年女人永久免费观看视频| 麻豆av噜噜一区二区三区| 蜜臀久久99精品久久宅男| 国产老妇伦熟女老妇高清| 日本-黄色视频高清免费观看| 能在线免费看毛片的网站| 国产精品乱码一区二三区的特点| 国产黄a三级三级三级人| 久久人妻av系列| 久久久久久伊人网av| 色尼玛亚洲综合影院| 久久亚洲精品不卡| 老司机福利观看| 五月伊人婷婷丁香| 国产精品伦人一区二区| 国产一区二区三区av在线| 建设人人有责人人尽责人人享有的 | 亚洲av不卡在线观看| www日本黄色视频网| 中文字幕精品亚洲无线码一区| 神马国产精品三级电影在线观看| 三级毛片av免费| 日韩人妻高清精品专区| 国产精品熟女久久久久浪| 人体艺术视频欧美日本| 亚洲aⅴ乱码一区二区在线播放| 国产精品99久久久久久久久| 欧美性感艳星| av在线亚洲专区| 亚洲最大成人av| 亚洲欧美成人精品一区二区| 最近视频中文字幕2019在线8| 国产精品人妻久久久久久| 国产高清国产精品国产三级 | 汤姆久久久久久久影院中文字幕 | 最后的刺客免费高清国语| 人体艺术视频欧美日本| 亚洲va在线va天堂va国产| 国产熟女欧美一区二区| 国内精品宾馆在线| 91狼人影院| 午夜福利在线在线| 国产精品国产三级国产专区5o | 青青草视频在线视频观看| 看片在线看免费视频| 色综合站精品国产| 亚洲精华国产精华液的使用体验| 国产精品爽爽va在线观看网站| 精品久久久久久久人妻蜜臀av| 色综合站精品国产| 欧美高清成人免费视频www| 国产精品精品国产色婷婷| 国产成人一区二区在线| 最近中文字幕高清免费大全6| 国内精品一区二区在线观看| 两个人视频免费观看高清| 黄色一级大片看看| 国产成人a∨麻豆精品| 最近中文字幕2019免费版| 免费黄色在线免费观看| 色吧在线观看| 欧美+日韩+精品| 久久久久网色| 成人国产麻豆网| 国产成人午夜福利电影在线观看| 最近手机中文字幕大全| 免费观看a级毛片全部| 国产欧美日韩精品一区二区| 一个人观看的视频www高清免费观看| 91午夜精品亚洲一区二区三区| 级片在线观看| 一个人免费在线观看电影| 国产成人精品久久久久久| 欧美另类亚洲清纯唯美| 色综合站精品国产| 熟女电影av网| 亚洲欧美精品综合久久99| 床上黄色一级片| av天堂中文字幕网| 啦啦啦啦在线视频资源| 少妇的逼好多水| 国产精品国产高清国产av| 免费黄色在线免费观看| 亚洲精品乱久久久久久| 九草在线视频观看| 秋霞在线观看毛片| 国产精品人妻久久久影院| 色综合色国产| 青春草国产在线视频| 国产亚洲91精品色在线| 黄色日韩在线| 又爽又黄无遮挡网站| 久久人人爽人人爽人人片va| 黄色日韩在线| 欧美丝袜亚洲另类| 中文精品一卡2卡3卡4更新| 网址你懂的国产日韩在线| 久久99热6这里只有精品| 天堂√8在线中文| av福利片在线观看| 国产单亲对白刺激| 国产精品日韩av在线免费观看| 国产av码专区亚洲av| 午夜福利成人在线免费观看| 中文字幕熟女人妻在线| 日本黄色视频三级网站网址| 久久久久久久国产电影| 亚洲av电影在线观看一区二区三区 | 在线免费观看不下载黄p国产| 美女国产视频在线观看| 一个人看视频在线观看www免费| 插阴视频在线观看视频| 又粗又爽又猛毛片免费看|