• <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 ?

    Two-Stage Robust Optimization Under Decision Dependent Uncertainty

    2022-07-18 06:17:26YunfanZhangFengLiuYifanSuYueChenZhaojianWangandJoCatal
    IEEE/CAA Journal of Automatica Sinica 2022年7期

    Yunfan Zhang, Feng Liu,,, Yifan Su, Yue Chen,,,Zhaojian Wang,,, and Jo?o P. S. Catal?o,,

    Abstract—In the conventional robust optimization (RO)context, the uncertainty is regarded as residing in a predetermined and fixed uncertainty set. In many applications, however,uncertainties are affected by decisions, making the current RO framework inapplicable. This paper investigates a class of twostage RO problems that involve decision-dependent uncertainties.We introduce a class of polyhedral uncertainty sets whose righthand-side vector has a dependency on the here-and-now decisions and seek to derive the exact optimal wait-and-see decisions for the second-stage problem. A novel iterative algorithm based on the Benders dual decomposition is proposed where advanced optimality cuts and feasibility cuts are designed to incorporate the uncertainty-decision coupling. The computational tractability,robust feasibility and optimality, and convergence performance of the proposed algorithm are guaranteed with theoretical proof.Four motivating application examples that feature the decisiondependent uncertainties are provided. Finally, the proposed solution methodology is verified by conducting case studies on the pre-disaster highway investment problem.

    I. INTRODUCTION

    A. Background

    UNCERTAINTIES widely exist in real-world decisionmaking problems and various mathematical programming techniques, including scenario-based or chance-constrained stochastic programs, robust optimization (RO) [1] and distributionally robust optimization [2], have been developed to fit in different characteristics of the applications [3]. RO seeks a risk-averse solution by explicitly considering the worst-case effect of all possible realizations of the uncertain parameter within a pre-determined uncertainty set. It appeals especially when the decision maker has no knowledge of the probability distributions of the uncertain parameters, or when the feasibility of the system over the entire uncertainty set is prioritized. Due to its salient advantages on modeling capability, feasibility, and computational tractability [4], RO has gained increasing popularity over the recent decades and encompassed a wide variety of applications including process scheduling [4]–[6], power system planning and scheduling[7]–[10], and network optimization [3], [11], [12], etc.

    In the context of RO, the uncertainty sets are usually assumed to be a priori and fixed. The underlying assumption is that the decision maker’s strategies would not alter the range of uncertainty realization. In many real-world decisionmaking problems, however, uncertainties can depend on the strategies chosen by the decision makers and are assumed as endogenous. For example, in a batch-process scheduling problem, the processing time or the production yield of a task is endogenously uncertain since it retains a physical meaning only when the optimizer decides to operate the associated task in a given period [4]–[6], [13]. Another example is the demand response program on buildings’ electricity consumption [8]. For buildings participating in the program, the reserve demand requested from the system operator is uncertain and of endogenous nature due to its dependency on the reserve capacity provided by the building in the day-ahead market. The uncertainties affected by decisions are called decision-dependent uncertainties (DDUs) or endogenous uncertainties. In this paper, we use these two terms interchangeably without distinction. Also, we refer to decisionindependent uncertainties (DIUs) or exogenous uncertainties as those not altered by decisions. Consideration of DDUs in RO problems can provide considerably less conservative solutions, attributed to the fact that DDUs can be proactively controlled by the optimizer. However, differently from DIUs,the presence of DDUs brings great challenges to solving RO problems, mainly due to the mutual influences between uncertainties and decisions. In this paper, we propose a convergence-guaranteed algorithm to efficiently solve the two-stage RO problems with DDUs.

    B. Literature Review

    Regarding RO under DDUs, existing works can be categorized into two classes, based on whether there are recourse or wait-and-see decisions after the revelation of uncertainty realization:

    1)Static RO Under DDUs:In static RO, which is also called single-stage RO, all the decisions are here-and-now,i.e., to be determined before observing the uncertainty realization. In [14] and [15], variable budgeted static RO is studied, where the budget parameter that regulates the conservatism of the uncertainty set is modeled as an affine function with respect to the decision variables. In [16],polyhedral and conic uncertainty sets are introduced where decisions control the upper bounds of the uncertain variables.In [3] and [4], uncertainties are governed by binary-valued materialization indicator variables. The materialization indicator attains the values of 1 if the corresponding uncertain parameter retains a physical meaning in the problem and 0 otherwise. The DDU-involved RO models in the above works maintain the advantage of computational tractability of conventional RO problems, since a robust counterpart of mixed-integer linear programming (MILP) formulation can be derived by applying the strong duality theory and McCormick envelopes convex relaxation.

    2)Adaptive RO Under DDUs:Adaptive robust optimization(ARO), or called adjustable RO [17], is established in a twostage or a more general multi-stage setting where recourse or wait-and-see decisions are incorporated in response to actual uncertainty realization. The computational complexity of ARO lies in the fact that multi-level programming problems with more than two decision levels are non-deterministic polynomial-time (NP)-hard [18]. To overcome this intrinsic computational burden as well as the challenges raised by the coupling relationship between the here-and-now decisions and the uncertainties, affine decision rules parameterized in the uncertainty realizations and the here-and-now decisions are postulated on the wait-and-see decisions in [5], [19]–[21].Differently from the aforementioned approximate affine policies, the decision rules in [22] are generated with the aid of multi-parametric programming technique, rendering the exact solution to the ARO problem; however, computation burden of solving the second-stage problem parametrically as an explicit function of the here-and-now decisions and the uncertainty is transferred off-line but not eliminated. In the line of primal cut algorithms [23], [24], [9] proposes an improved column-and-constraint generation (C&CG) algorithm based on a worst-case scenario mapping technique. Its application, however, is limited to a high-dimensional rectangle decision-dependent uncertainty set (DDUS).

    Compared with conventional ARO problems, the challenges of solving DDUs-incorporated ARO problems reside in the fact that the existing cutting plane algorithms [23]–[26], with constraints generated by whether primal or dual information of the second-stage problem, fail to warrant finite convergence to the global optimum when the uncertainty set depends on the here-and-now decision variable which changes with iterations. This is because in that case the optimality cuts and feasibility cuts generated with concrete uncertain parameter values may become invalid and ruin the optimality of the solution, or even render an infeasible master problem which falsely implies that the original ARO problem is not robust feasible. Moreover, since the vertices of the uncertainty set change with the here-and-now decision, it is difficult to justify the finite convergence of these cutting plane algorithms by completely enumerating the vertices of the polyhedral uncertainty set. To the best of the authors’ knowledge, the solution approach for ARO under DDUs with a generic decision-dependency and without any assumption on approximation policies has not been addressed.

    C. Contribution and Organization

    The main contributions of this paper are twofold:

    1)From a Modeling Perspective:This paper addresses a generic two-stage robust optimization model with a class of polyhedral DDUSs whose right-hand-side vector has a dependency on the here-and-now decisions. Differently from[3], [4], [6], [14]–[16] that contemplate the static RO involving endogenous uncertainty, our two-stage RO grants more feasibility by the implementation of wait-and-see decisions after the revelation of uncertainty realization.Compared with [5], [20], [21] that are established in a twostage or multi-stage setting and postulate affine decision rules on the wait-and-see decisions, our model seeks to derive the exact optimal solution to the second-stage problem. In view of the DDUS, we extend the polyhedral uncertainty sets with reduced decision-dependency considered in [6], [8], [9], [15],[16] into a generic form where both the shape and size of the uncertainty set may be altered by decisions. Due to the aforementioned improved modeling capability, our model can considerably eradicate the conservatism effects of RO by proactively controlling the level of uncertainties and inherently capture the trade-off between the robustness and conservatism of the solution.

    2)From a Solution Standpoint:To solve the two-stage RO model with DDUs, we propose a novel iterative solution algorithm involving one master problem and two subproblems, based on the Benders dual decomposition.Compared with the C&CG algorithm [23], [24] which is widely applied to conventional two-stage RO problems, the proposed algorithm overcomes the challenges stemming from endogenous worst-case uncertainty scenarios by using only dual information of the second-stage problem. Also differently from classical Benders dual cutting-plane algorithms[25]–[27], advanced optimality cuts and feasibility cuts are designed to accommodate the coupling between uncertain parameters and decision variables. Performance of the proposed algorithm, including convergence and optimality, is guaranteed with a strict proof. To implement the algorithm,we derive the robust counterpart of the master problem and the reformulations of the sub-problems, maintaining the computational tractability of the proposed solution algorithm.

    The remainder of the paper is organized as follows. The remainder of this section introduces the notation. The mathematical formulation of the two-stage RO with DDUs is provided in Section II. Section III proposes the solution approach based on modified Benders decomposition. Section IV addresses some implementation issues of the proposed algorithm, including the derivation of a tractable robust counterpart of the master problem which is a static RO and the reformulation of the bi-level sub-problems into MILPs. In Section V, four examples are provided as the applications of the proposed model and solution methodology. And we present a case study on the pre-disaster highway network investment problem. We conclude the paper in Section VI.

    Notation:Rnis the set ofn-dimensional real vectors and Rm×nis the set ofm×nreal matrices. Z ( Z+) denotes the set of(positive) integers. [n]?{1,...,n} denotes the set of integers from 1 tonand [0]??.Fora vectorx∈Rn(amatrixA∈Rm×n),xT(AT)denotes its transpose. Weuse1and0 to denote vectors of ones and zeros, respectively. Forx,y∈Rn,we denote theinnerproductxTy=xiyiwherexi,yistand for thei-th entries ofxandy, respectively. The Hadamard product is defined asx?y=(x1y1,...,xnyn)T, i.e., the elementwise product of two equally-sized vectors or matrices. For a polytopeX?Rn, vert(X) is the set of the vertices ofX. ?a」 is the largest integer less than or equal toa. We define the difference of setXandYasXY?{x:x∈X,x?Y}.

    II. PROBLEM FORMULATION

    A. Two-Stage RO With DDU

    The two-stage RO with DDU (TRO-DDU) problem is formulated as

    where

    The here-and-now decisionx∈RnR×ZnZ is a mixed-integer variablevectorwitha dimensionofnx=nR+nZ.X?RnR×ZnZis the feasibleregionofx.f(x):RnR×ZnZ →R1is the costfunction withrespecttox.TheDDU parameteris denoted byw∈Rnwand the DDUSofwisW(x):RnR×ZnZ2Rnw,which is essentially a set-valued map parameterized byxwhereG∈Rr×nwis a constant matrix,g∈Rris a constant vector andh(x):RnR×ZnZ →Rris a vector-valued function with respect tox. The wait-and-see decision vector isy∈Rny.We consider the case that the second-stage decision problem isalinearprogram(LP) andc∈Rnydenotesthecost coefficientofy.Y(x,w):RnR×ZnZ×Rnw2Rny,whichis essentially a set-valued map parameterized byxandw,denotes the feasible region ofywith a specific form in (1d)whereA∈Rm×nx,B∈Rm×ny,C∈Rm×nwandb∈Rmare constant parameters. The constraint setXR?RnR×ZnZ defined in (1e) is the set ofxthat are robust feasible. A hereand-now decisionxis called robust feasible [1], [26] if for any realization of the uncertainwwithinW(x) there exists at least one feasible wait-and-see decisionylying withinY(x,w). The TRO-DDU problem (1) aims at minimizing the total cost of the two stages, which isf(x)+cTy, under the worst-case uncertainty realization. The following assumptions are made on problem (1).

    Assumption 1:1)f(x) is a convex function; 2)Xis a convex set; 3)X∩XR≠?; 4)Xis bounded,W(x) is bounded for anyx∈XandY(x,w) is bounded for anyx∈Xandw∈W(x).

    Under Assumption 1-1) and 1-2), the nominal problem of(1) is a convex minimization problem. Assumption 1-3) and 1-4) imply that the TRO-DDU problem (1) is feasible and bounded, thus there exists the optimal solution to (1).

    The TRO-DDU problem distinguishes itself from the existing literature by the consideration of the DDUwand the DDUSW(x). Regarding the generality of the DDU modeling in the TRO-DDU problem (1), we have the following remark.

    Remark 1(Decision-Dependent Uncertainty Set):The polyhedral DDUSW(x) in (1c), whose right-hand-side vector has a dependency onx, covers the formulations proposed in the existing literature [8], [9], [14]–[16], [20], [21]. It provides generic modeling capability since both the shape and size of the uncertainty set can be altered byx. Moreover,W(x) in (1c)applies readily with DIUs by settingh(x)=0. Though the coefficient matrixGis fixed in (1c), the solution strategy proposed in Sections III and IV can be easily extended to the cases in [3]–[5] wherexaffects the coefficient matrix through certain binary-valued functions with respect tox.

    Regarding the difficulty in solving TRO-DDU problem (1),we have the following remark.

    Remark 2:The well-known C&CG algorithm is no longer applicable to TRO-DDU problem (1) since finite convergence to the global optimum is not warranted in the presence of DDUs.

    1)Failure in Robust Feasibility and Optimality:The cutting planes in the C&CG algorithm with recourse decision variables in the primal space are generated with concrete values of the uncertain parameter for each identified scenario;and these cuts can become invalid when the uncertainty set depends on the here-and-now decisionxwhich varies with iterations. Forexample,given ahere-and-now decisionx1,the identified worst-case uncertaintyrealizationw1∈W(x1)would nevercometrueifanotherhere-and-nowdecisionx2is adopted andw1?W(x2);andinthat case, thefeasibilityand optimality cuts generated according tow1no longer necessarily perform a relaxation to problem (1) and may ruin the optimality of the solution, or even render an infeasible master problem.

    2)Failure in Finite Convergence:The finite convergence of C&CG algorithm is justified by a complete enumeration of the vertices of the uncertainty set. However, in problem (1), the vertices set of polytopeW(x) changes withxwhich varies with iterations, thus the C&CG algorithm no longer necessarily terminates within a finite number of iterations.

    III. SOLUTION METHOD

    A. Equivalent Transformation

    1)Robust Optimality:Looking at the inner-level problem in

    (1), we denote the following max-min bi-level optimization byS(x):

    By noting the duality of the inner problem inS(x),S(x) can be equivalently transformed into the following bi-linear maximization problem:

    whereu∈Rmis the dual variable that corresponds with the constraintsAx+By+Cw≤b. Then, problem (1) is equivalent to

    Introducing a supplementary variable α ∈R1, then problem(4) can be further rewritten in an epigraph form as

    2)Robust Feasibility:The robust feasibility of decisionx,i.e., whetherxlies inXR, can be examined by solving the following relaxed bi-level problem:

    wheres∈Rmis the slack variable vector introduced to relax the feasible region ofydefined in (1d). IfR(x)≤0,xis robust feasible, i.e.,x∈XR.ElseifR(x)>0, theremustexista realizationw?∈W(x)suchthat nofeasiblewait-and-see decisionyis available. It is useful to write the dual of the inner minimization problem inR(x) . ThenR(x) is equivalent to the single-level bi-linear optimization problem as follows:

    where π ∈Rmis the dual variable vector corresponding with constraints (6c). Sincex∈XRif and only ifR(x)≤0, problem(5) can be further written as

    From the above transformation, we derive problem (8)which is the surrogate model of TRO-DDU problem (1).

    B. Modified Benders Decomposition Algorithm

    Next, we have the overall Algorithm 1 to solve problem (8)where the master problem (MP) involved is formulated as

    Regarding the difference between the proposed modified Benders dual decomposition (Algorithm 1) and the existing algorithms such as the C&CG algorithm and classic Benders dual cutting-plane algorithms, we give a remark as below.

    Algorithm 1 Modified Benders Dual Decomposition Algorithm for TRO-DDU Problem (1)

    Remark 3(Advanced Optimality and Feasibility Cuts):Constraints (9c) and (9d), which are appended to the master problem with iterations, are called optimality cuts and feasibility cuts, respectively. They are designed to have the following salient features to adapt to the DDUS: 1) Concrete worst-case uncertainty, i.e., the solution toR(xk) orS(xk)denoted bywk, is not involved in the optimality cuts or the feasibility cuts. This is different from the C&CG algorithm, as it incorporates the coupling relationship betweenxandw;2) Dual information of sub-problemsR(xk) andS(xk), i.e.,πkanduk, are utilized to generate the feasibility cuts and optimality cuts, inspired by the Benders dual decomposition.However, these cuts are designed to be no longer hyperplanes,but a set of static robust constraints, to comprise a cluster of endogenous worst-case uncertainty realizations.

    Regarding the extensive adaptability of Algorithm 1, we have the following remark.

    Remark 4(Adaptability to DDU Formulations):Algorithm 1 is designed to be applicable to a wide range of DDU formulations. Note that the explicit formulation of DDUSW(x)is not involved in Algorithm 1, nor in the theoretical justification of its performance. Thus, Algorithm 1 does not preclude the implementation of more DDUS structures besides polyhedrons, such as ellipsoidal and conic sets.However, considering the difficulty in solving master problem(9) and sub-problemsR(x) andS(x), here we focus on a class of polyhedral uncertainty sets with decision dependency in the right-hand-side vector, as formulated in (1c).

    We justify the convergence and optimality of Algorithm 1 by the following theorem.

    Theorem 1:Letqbe the number of vertices ofUdefined in(3b) andpbe the number of vertices of Π defined in (7c). Letx?denote the optimal solution to the TRO-DDU problem (1).Then, Algorithm1terminateswithinO(p+q)iterationsand outputsthe solutionxksuch thatxk∈X∩XRand|f(xk)+S(xk)?f(x?)?S(x?)|<ε.

    Theorem 1 indicates that the proposed modified Benders dual decomposition algorithm is finitely convergent to the optimum. Proof of Theorem 1 can be found in Appendix.

    IV. ROBUST COUNTERPART AND BIG-M REFORMULATION

    In this section, we show that the implementation of Algorithm 1 is computationally tractable, by deriving the robust counterpart of the master problem (9) and reformulations of sub-problemsR(xk) andS(xk).

    In Algorithm 1, the master problem (9) involves static robust constraints (9c) and (9d). Without loss of generality,we illustrate how to deal with the robust constraint (9c) in the master problem by substituting it with its robust counterpart.

    The following robust constraint with respect to (α,x) with given:

    A. Robust Counterpart of the Master Problem

    is equivalent to

    wherewesubstitutewbyfordenotingvariablein constraintformed by.Next,thefollowing twocasesare discussed:

    1)A Bi-Linear TermλTh(x)can be Precisely Linearized Through the Big-M Method Whereλ ∈Rris an r-Dimensional Variable and x is annx-Dimensional Variable:To illustrate this, we takex∈{0,1}nxandh(x)=Hxas an example. Duality of the inner-level problem in (11) is deployed, thus (11) can be reformulated as

    where λj∈Rris the dual variable corresponding with constraint (11b). Note that constraint (12) is equivalent to

    bydroppingthe minimizationoperation.Sincexisbinary,the bi-lineartermxT HTλjcanbeexactlylinearizedthroughthe big-Mmethodby introducingsupplementaryvariableηj∈Rnxand a largeenoughpositivenumberM,whereηj=x?(HTλj).Then, constraint (13) has the following equivalent formulation:

    which constitutes the robust counterpart of constraint (10).The robust counterpart of feasibility cuts (9d) can be derived in a similar way. Thus the master problem (9) has the following MILP robust counterpart:

    where the convex functionf(x) is substituted by its piecewise linearization form as in (15c) and the optimality cuts and the feasibility cuts are substituted by their robust counterparts.∈Rnx,∈R1(i∈[I])areconstant parameters for the piecewise linearizationoff(x).

    2)Otherwise:If a bi-linear term λTh(x) cannot be exactly linearized by the big-M method, we deploy the Karush-Kuhn-Tucker (KKT) conditions of the inner-level problem in (11) as follows:

    where λj∈Rris the dual variable corresponding with constraints (11b), and (16b) denotes the complementary relaxation condition. The nonlinearity of complementary condition (16b) can be eliminated through the big-M method by introducing the binary supplementary variablezj∈{0,1}r.Thus, the optimality cut (11) has the following equivalent robust counterpart:

    where the complementary relaxation condition (16b) is substituted by (17c) and (17d). The robust counterpart of the feasibility cut (9d) can be derived similarly. Thus the master problem (9) has the following robust counterpart which is a mixed-integer programming problem.

    B. Reformulation of Sub-Problems

    In Algorithm 1, the robust feasibility examination subproblemR(x) in (7) and the robust optimality sub-problemS(x)in (3) both have bi-linear objective, imposing difficulties on the solving. Next, we provide linear surrogate formulations ofS(x) andR(x). Since they have similar structure, without loss of generality we only focus onR(x).

    R(x) in (7) can be equivalently rewritten into

    where the here-and-now decisionxis given by solving the master problem. We deploy the KKT condition of the innerlevel LP problem in (19) as follows:

    where ζ ∈Rris the dual variable corresponding with constraintw∈W(x) in (19). The complementary constraints(20a) can be linearlized by introducing binary supplementary variablez∈{0,1}r. Then, the complementary constraints (20a)can be substituted with its equivalent linear formulations, like what we have done to (16b). Moreover, since strong duality holds, we substitute the ? πTCwin (19) by (g+h(x))Tζ. Then,sub-problemR(x) is equivalent to the following MILP problem:

    V. APPLICATIONS

    The proposed model and solution algorithm cater for a variety of application problems. In this section, we provide four motivating DDU-featured examples that can be formulated into the proposed TRO-DDU model (1). For the first three applications, we focus on the endogeneity of the uncertainties by constructing the DDUS and do not go into the details of problem formulation. For the last application, predisaster network investment, detailed problem statements and numerical case studies are provided with an out-of-sample analysis.

    A. Application 1: Batch Scheduling Problem

    This case is from [13] (Section 8) where stochastic programming formulations with endogenous uncertainties are established. Here we reform the characterization of DDUs to fall in the scope of RO. Consider a chemical process network as shown in Fig. 1.

    Fig. 1. Illustration of Application 1.

    Here, chemical A is produced in Process 3 from chemical B while chemical B can be produced in Process 2 from chemical D or produced in Process 1 from chemical C. If needed,chemicals A, B, C, and D can be purchased from market. Now there is a demand for chemical A that must be satisfied and decisions on which specific processes should be operated are made to maximize the net profit. The per unit yields of the three processes, denoted by θi,tfor each processi∈I={1,2,3} and timet∈T, are uncertain. The endogeneity stems from the fact that θi,tretains a physical meaning only when processiis operated at time periodt. Letxi,t∈{0,1}denote the binary decision of operating processiat timet,then the DDUS forθis constructed as

    B. Application 2: Shortest Path Over an Uncertain Network

    The second case comes from [16] (Example 2), considering to find the shortest path from the origin node A to the destination node B over a network with uncertain arc lengths,as shown in Fig. 2.

    Fig. 2. Illustration of Application 2.

    Let E denote the arcset of the graph and the uncertain length for an arce∈E is denoted byde. The uncertain arc lengthsdlie in the DDUS as follows:

    C. Application 3: Frequency Reserve Provision

    The third application comes from [8]. Consider a smart building participating in the demand response program on electricity consumption. The building is asked by the grid operator to adjust its electricity consumption within a prespecified reserve capacity. The building is rewarded for providing this reserve service and the payment depends on both the size of the reserve capacity and the actually deployed reserve energy.

    Consider a symmetric reserve capacityRt(Xt)=[?Xt,Xt]whereXt≥0 is the size of reserve capacity at timet. For the smart building, the operator’s real-time reserve deployment requestrtis endogenously uncertain at the planning stage. The endogeneity stems from the fact that the range of reserve deployment that the smart building admits is fundamentally affected by its decisions on the reserve capacity. Specifically,rt∈Rt(Xt) andXtis the building’s day-ahead decision. It is assumed that providing reserve capacity of sizeXtis rewarded byXtandofferingreserveenergyofsizertis compensatedbyrt.Thefrequencyreserveprovision problem is formulated to maximize the building’s profit while always satisfying inner technical and comfort constraints.

    D. Application 4: Pre-Disaster Network Investment

    In this subsection, a pre-disaster highway network investment problem is provided, including its problem formulation and a case study. This application is motivated by the work of [28] where the decision maker aims to strengthen the highway system in advance to prevent link failures due to earthquakes. By contrast, we model the problem into a twostage RO with DDUs for the case that link failure probabilities are not known.

    The goal is to guarantee the existence of a path between a given origin-destination (O-D) pair after the earthquake and concurrently minimize the post-disaster travel cost from the origin node to the destination node and the costs of necessary pre-disaster investment. Endogenous uncertainties arise from the fact that the post-disaster state of each link, either functional or non-functional, is uncertain but can be altered by pre-disaster reinforcement.

    1)Problem Formulation:The high-way network is modeled into a directed graphN=(V,E) with node setVand arc setE.To characterize the functionality of the highway system after the disaster, two nodes in the graphNare specified:Ddenotes the destination node which is the district with the highest expected damage in the earthquake, andOdenotes the origin node which usually refers to the district with the most support resources.

    We use binary-valued variablexe∈{0,1}1to denote the investment decision on linke∈E.xetakes the values of 1 if thereisaninvestmentonlinkeand0otherwise.Letwe∈{0,1}1denotethedecision-dependentuncertainpostdisaster state of linke∈E.weattains the value of 1 if the linkeis not functional after the occurrence of the disaster and the value of 0 otherwise. Strengthened links have guaranteed functionality after the disaster and the non-strengthened links are subject to random failures. Thus the realization ofweis restricted by

    where ψ ∈[0,1] is the robustness budget to reduce conservativeness. Note that the constraint matrix of (24)satisfies total unimodularity, thus the binary-valuedwecan be relaxed to be continuous without any compromise [3]. Thus,the budgeted DDUS forweis constructed as

    To verify the post-disaster connectivity fromOtoDas well as find the path fromOtoDwith minimal traversal costs after the disaster, the following second-stage optimization problem is considered:

    whereY(w)?R|E|is defined through the following constraints:

    andce∈R1denotes the length of linke. The objective is to minimize the traversal cost fromOtoDin the network. The wait-and-see decision variableye∈{0,1}1takes the values of 1 if the path fromOtoDgoes through linkeand 0 otherwise.Note that in (27), the binary-valuedyeis substituted by its continuous relaxation (27a) since the constraint matrix ofysatisfies total unimodularity property. There exists a path fromOtoDafter the occurrence of the disaster if and only if (26)has at least one feasible solution and the optimal solution to(26) implies the path with least traversal cost fromOtoDin the network.

    Next, the mathematical formulation of the two-stage robust pre-disaster highway network investment problem is given

    whereae∈R1denotes the investment cost of linke.W(x) andY(w)are defined in (25) and (27), respectively. The objective is to minimize the traversal costs between the O-D pair under the worst-case uncertainty realization as well as the necessary investment cost. Constraint (28c) guarantees the existence of at least one path fromOtoDafter the disaster under any realization ofwin the uncertainty setW(x). The proposed modified Benders dual decomposition algorithm is applied to solve problem(∑28).Notethat givenavariableλ∈R1, thebilinear item λ?ψe∈E(1?xe)」 canbepreciselylinearized,thus the robust counterpart of the master problem in Algorithm 1 can be formulated into an MILP problem.

    2)Basic Results:The computational results are based on a network with 8 nodes and 9 links, as demonstrated in Fig. 3.The relevant data are from [28]. The traversal costs (lengths)and investment costs of links are provided in Table I. The O-D pair is chosen as (1,6), i.e.,O=1,D=6. For the nominal 9-link network, there are 4 possible paths fromOtoD, as listed in Table II. If all links remain functional, the shortest path fromOtoDis Path 1 (1-3-5-9) with a length of 13.52. The program runs on an Intel Core-i5 1.6-GHz computer and is coded with YALMIP. CPLEX 12.6.0 is utilized as the solver.

    The proposed modified Benders dual decomposition algorithm is applied with an initialization ofUB0=4000,LB0=0, ε=0.01, ψ=0.3. The algorithm reaches convergence ofUB=LB=1100.65 after 8 rounds of iterations and the evolution process ofUBkandLBkis shown in Fig. 4. The optimal robust investment (the here-and-now decision) is on Link 3, Link 8, and Link 9. For all possible network realizations under this pre-disaster investment scheme, the worst-case is the failure on Link 5 after the earthquake. In that case, the shortest post-disaster path fromOtoD(the waitand-see decision) is Path 3 with a length of 20.65. The correctness of the computational results can be verified by enumeration.

    Fig. 3. The 8-node 9-link network.

    Link e 1 2 3 4 5 6 7 8 9 ce 6.41 8.09 1.97 6.35 2.87 4.11 2.27 3.91 2.27 ae 500 620 160 780 260 220 500 120 800

    TABLE II PATHS FOR O-D PAIR (1,6)

    Fig. 4. Evolution of the U Bk and L Bk .

    3)Comparative Study:To emphasize the necessity and superiority of the proposed algorithm for DDU-involved twostage RO problems, existing cutting plane algorithms that are theoretically applicable to only the case of DIUs are adopted to solve the same problem in (28). In both the classical Benders decomposition algorithm and the C&CG algorithm,the master problem becomes infeasible at the second round,then the solution procedure interrupts without convergence.This is because the worst-case uncertainty realization identified on the first round is a failure in Link 9, under which there is no path fromOtoD. This situation can be eliminated by strengthening Link 9; however, the invalid feasibility cutgenerated with this concrete scenario is appended to the master problem and the consequent infeasibility falsely implies that there is no feasible solution to the original problem (28). The observations in this comparative case study verify the statements in Remark 2.

    TABLE III IMPACT OF ROBUSTNESS PARAMETER ψ

    4)Sensitivity Analysis:We present the impact of robustness budgetψo(hù)n the optimal solution to problem (28). In this case,7 robustness parametersψfrom 0 to 0.6 with a gradient of 0.1 are introduced and the results are displayed in Table III. It is observed that, asψincreases, the optimal objective value including investment costs and worst-case traversal costs increases accordingly. This is because the decision maker has to enhance more links to hedge against the increasing uncertainties on link failures. However, the optimal predisaster investment saturates whenψis greater than or equal to 0.5. This is due to the fact that no matter how largeψis,there always exists a path between the O-D pair if links 1,3,5,9 (Path 1) are all reinforced.

    5)Out-of-Sample Analysis:Finally, an out-of-sample analysis is carried out to verify the performance of the robust investment decision. Steps of the out-of-sample assessment areasfollows [29]:a)Consideringallpossible statesofthe 9 links,wegenerate29=512scenarios, namelywo∈{0,1}9,o∈[512]. b) Givenx?derived by solving (28), for each scenarioo, the following relaxed second-stage problem is solved:

    where non-negative slack variablesso+,so?are introduced to the flow conservation constraints (29d)–(29f) and penalized in the objective (29a). The penalty cost coefficientP∈R1is set as 5000 in this case. Problem (29) is a modified version of the second-stage problem (26) with a given here-and-now decisionx?andthegiven uncertaintyrealizationwo. Denote the solutionto(29)byyo?,so+?,so??.c)Atlast,the average sampled second-stage costCavand the average sampled infeasibility levelsavare computed as follows:

    Table IV shows the out-of-sample assessment results for the robust optimal solution under different values of the robustness budgetψ. It is observed thatCavandsavdecrease with the increasingψ, indicating that a biggerψleads to a more robust solution that can hedge against a higher level of uncertainties, but also accordingly generates higher investment cost in the first stage. Comparison with the deterministic model is also provided in Table IV . We can see that disregarding uncertainty would give rise to significantly high second-stage costs (due to the penalty on the slack variablesand)andinfeasibilitylevel.By choosingaproper r obustnessbudgetψ,the decisionmaker can achievethetradeoff between the first-stage investment costs and the secondstage feasibility level.

    TABLE IV RESULTS FROM THE OUT-OF-SAMPLE ANALYSIS

    VI. CONCLUSIONS

    In this paper, a novel two-stage RO model with DDUs is proposed. We introduce a class of polyhedral DDUSs whose right-hand-side vectors are in a dependency of the here-andnow decision. Solution methodology for the problem is designed based on modified Benders decomposition, robust counterpart derivation, and linearization techniques. Computational tractability and convergence performance of the proposed algorithm is guaranteed by a strict proof.

    Case studies on the pre-disaster highway network investment problem verify that the proposed DDU-incorporated two-stage RO model is an amenable framework for addressing decision-making problems under endogenous uncertainties.The optimality and feasibility of the robust solution are validated by enumeration in this case. Furthermore, the computational studies elucidate that the DDU-involved twostage RO model inherently captures the trade-off between uncertainty mitigation (line investment) and the corresponding expenses (line investment costs) and provides less conservative robust results.

    Though we focus on polyhedral DDUS whose right-handside vector has a general correlation with the here-and-now decision, the proposed solution framework based on the modified Benders decomposition in Section III is also applicable to DDUSs with ellipsoidal or conic structure. The limitation of the work in this paper lies in the fact that the duality-based Benders decomposition is conditional upon the second-stage problem which is an LP. Also, uncertain parameters that are of discrete or binary nature would prohibit us from formulating the robust counterpart of the master problem as described in Section IV. These issues remain to be addressed in our future study. In addition, as an extension of this work, future study could combine the optimality and feasibility cuts (9c) and (9d) with the so-called Pareto cuts[30], or extend the robust counterpart (17) to incorporate primal cuts with recourse decision variables in the primal space, to further improve the convergence rate in practice.

    APPENDIX

    Before we give the proof of Theorem 1, a crucial lemma is provided as follows.

    Lemma 2:Letx?denote the optimal solution to TRO-DDU problem (1). For anyk∈Z+:

    Since the TRO-DDU problem (1), problem (8) and problem(31) are equivalent,f(x?)+S(x?) is also the optimal objective value of (31). Also note that the master problem (9) is a relaxation to minimization problem (31) since∈Ufor allj∈[ku] and∈Π for allj∈[kπ]. Thus, the optimal objective value of problem (9), which is denoted byLBk, is less than or equaltotheoptimal objective value of problem (31), i.e.,LBk≤f(x?)+S(x?).

    Recall the updating rule of the upper bound which isUBk=UBk?1whenR(xk)>0 andUBk=f(xk)+S(xk) whenR(xk)=0 . Next, we proveUBk≥f(x?)+S(x?) by induction.First of all,UB0=M>f(x?)+S(x?). Suppose for the sake of induction thatUBk?1≥f(x?)+S(x?) wherek∈Z+. Then, ifUBk=UBk?1, we haveUBk≥f(x?)+S(x?) directly. Else ifUBk=f(xk)+S(xk) whenR(xk)=0, then we haveUBk=f(xk)+S(xk)≥f(x?)+S(x?) sincexkis a feasible solution to the TRO-DDU problem (1) (by recalling thatxk∈XRif and only ifR(xk)=0 ) andf(x?)+S(x?) is the optimal objective value of TRO-DDU problem (1).

    Assertion b):∈vert(Π) canbeeasily verified bynoting thattheoptimalsolutionofabi-linear programwith polyhedral feasible region can be achieved at one of the vertices of the polytopes [31].

    which contradicts with (33).

    Assertion c):Similarly to the proof of Lemma 2–b),∈vert(U)can be verified by noting that the optimal solution of bi-linear programming over a polytope can be achieved at one of its vertices.Suppose there existj1,j2∈[ku] andj1≠j2such that=. We assume thatis the optimum toS(xk j1) andis the optimum toS(xkj2). Without loss of generality, we assume thatj1

    then, we have

    which is equivalent to

    Recalling the updating rule of lower bound, we have

    wherei) comes from the relationship in (38),ii) comes from the fact that=,iii) comes from the fact thatis the optimum toS(xk j2), andiv) comes from the updating rule of the upper bound. Thus, we haveUBk j2 ≤LBk j2. According to Lemma 2-a),LBk j2 ≤f(x?)+S(x?)≤UBk j2. Thus, we haveLBk j2 =UBk j2 =f(x?)+S(x?)and the Algorithm 1 terminates at iterationkj2. This completes the proof of the statement in Lemma 2-c). ■

    The proof of Theorem 1 is given as follows.

    Proof of Theorem 1:First, we prove that Algorithm 1 terminates within a finite round of iterations through a complete enumeration of the vertices of the polyhedral feasible regions of the dual multipliers. According to Lemmas 2-b) and 2-c), no vertex of Π orUwill be appended twice to the master problem. Thus, the termination condition must be met within O (p+q) iterations.

    Next, we prove the feasibilityofsolutionxk.Sincexkis ge nerated through master problem(9),xk∈X. Moreover,since Algorithm 1 terminates withR(xk)=0 , wehavexk∈XRbyrecallingthatx∈XRif and only ifR(x)=0. Thus,xk∈X∩XR.

    Finally, we prove the optimality of solutionxk. According to Lemma 2-a),LBk≤f(x?)+S(x?)≤UBk. Together with the fact that the Algorithm 1 terminates with |UBk?LBk|<ε andR(xk)=0 , we have |UBk?f(x?)?S(x?)|<ε. Recalling thatUBk=f(xk)+S(xk), we have|f(xk)+S(xk)?f(x?)?S(x?)|<ε, which justifies the optimality ofxk.■

    www.自偷自拍.com| 国产精品久久久人人做人人爽| 男女床上黄色一级片免费看| av网站免费在线观看视频| 欧美日韩一级在线毛片| 亚洲男人的天堂狠狠| 一级毛片高清免费大全| 国产亚洲av高清不卡| 一级a爱片免费观看的视频| 久久精品影院6| 久久天躁狠狠躁夜夜2o2o| 亚洲一区高清亚洲精品| 好看av亚洲va欧美ⅴa在| 国产精品1区2区在线观看.| 亚洲成人精品中文字幕电影| 一夜夜www| 久久久精品国产亚洲av高清涩受| 9色porny在线观看| 黄网站色视频无遮挡免费观看| 巨乳人妻的诱惑在线观看| 精品国产美女av久久久久小说| 黄色视频不卡| 国产极品粉嫩免费观看在线| 最近最新中文字幕大全电影3 | 老熟妇乱子伦视频在线观看| 久久国产精品男人的天堂亚洲| 亚洲性夜色夜夜综合| 69精品国产乱码久久久| 啦啦啦韩国在线观看视频| 精品不卡国产一区二区三区| 丰满的人妻完整版| a在线观看视频网站| 亚洲avbb在线观看| 手机成人av网站| 久久国产乱子伦精品免费另类| 久久久国产精品麻豆| a级毛片在线看网站| 十分钟在线观看高清视频www| 久久国产亚洲av麻豆专区| 在线观看www视频免费| 日本 av在线| 久久精品国产亚洲av高清一级| 日本撒尿小便嘘嘘汇集6| 一级毛片精品| 日本免费一区二区三区高清不卡 | 国内久久婷婷六月综合欲色啪| 午夜福利影视在线免费观看| 午夜免费成人在线视频| 俄罗斯特黄特色一大片| 免费不卡黄色视频| 亚洲最大成人中文| 啦啦啦免费观看视频1| 丝袜美腿诱惑在线| 精品久久蜜臀av无| 国产亚洲欧美精品永久| 欧美人与性动交α欧美精品济南到| 18禁观看日本| 9热在线视频观看99| 成人国产综合亚洲| 久久香蕉精品热| 黄网站色视频无遮挡免费观看| 久久精品91蜜桃| 亚洲美女黄片视频| 女人高潮潮喷娇喘18禁视频| 亚洲 国产 在线| 精品国产超薄肉色丝袜足j| 老汉色∧v一级毛片| 欧美日本亚洲视频在线播放| 丁香欧美五月| www.999成人在线观看| 欧美日韩亚洲综合一区二区三区_| 国产精品 国内视频| 国产欧美日韩一区二区精品| 国产成人欧美在线观看| 天堂影院成人在线观看| 亚洲av片天天在线观看| 国产精品1区2区在线观看.| 午夜福利高清视频| 成在线人永久免费视频| 欧美一级毛片孕妇| 国产视频一区二区在线看| 国产一卡二卡三卡精品| 久久久久九九精品影院| 久久 成人 亚洲| 好男人电影高清在线观看| 黄色视频不卡| 久久精品国产99精品国产亚洲性色 | 国产精品亚洲美女久久久| 亚洲少妇的诱惑av| 欧美日韩亚洲综合一区二区三区_| 亚洲成国产人片在线观看| 久久精品亚洲精品国产色婷小说| 国产一区在线观看成人免费| 一级,二级,三级黄色视频| 亚洲男人的天堂狠狠| 女同久久另类99精品国产91| 色在线成人网| 免费少妇av软件| 国产精品,欧美在线| 国产成人精品在线电影| 电影成人av| 老司机午夜十八禁免费视频| 欧美国产日韩亚洲一区| 精品久久久久久久人妻蜜臀av | 久久精品国产亚洲av香蕉五月| 国产三级黄色录像| 亚洲精华国产精华精| 很黄的视频免费| 亚洲欧美日韩高清在线视频| 乱人伦中国视频| 国产精品av久久久久免费| 99久久精品国产亚洲精品| 国产亚洲精品av在线| 亚洲精品一区av在线观看| 亚洲九九香蕉| √禁漫天堂资源中文www| 麻豆成人av在线观看| 欧美午夜高清在线| 女警被强在线播放| 欧美日韩瑟瑟在线播放| 国产精品二区激情视频| 很黄的视频免费| 亚洲自偷自拍图片 自拍| 国产又爽黄色视频| 中亚洲国语对白在线视频| 淫妇啪啪啪对白视频| 成人av一区二区三区在线看| 香蕉丝袜av| 91麻豆av在线| 一区二区三区激情视频| 亚洲性夜色夜夜综合| 久久久久国产一级毛片高清牌| 在线十欧美十亚洲十日本专区| 一a级毛片在线观看| a级毛片在线看网站| 国产成人av激情在线播放| 亚洲成人久久性| 亚洲av日韩精品久久久久久密| 最新在线观看一区二区三区| 一级黄色大片毛片| 亚洲成av片中文字幕在线观看| 男人操女人黄网站| 无限看片的www在线观看| 首页视频小说图片口味搜索| 成人精品一区二区免费| 一区二区日韩欧美中文字幕| tocl精华| 精品久久久精品久久久| 欧美黑人精品巨大| 亚洲人成电影免费在线| 亚洲最大成人中文| 变态另类丝袜制服| 亚洲男人的天堂狠狠| 成人特级黄色片久久久久久久| 长腿黑丝高跟| aaaaa片日本免费| 免费在线观看视频国产中文字幕亚洲| 国产精品二区激情视频| 亚洲第一欧美日韩一区二区三区| 国产亚洲精品av在线| 欧美在线黄色| 一区在线观看完整版| 99久久99久久久精品蜜桃| 亚洲国产精品sss在线观看| 一级,二级,三级黄色视频| 国产高清videossex| 精品国产亚洲在线| 19禁男女啪啪无遮挡网站| 亚洲成a人片在线一区二区| 欧美精品啪啪一区二区三区| 长腿黑丝高跟| 首页视频小说图片口味搜索| 欧美亚洲日本最大视频资源| 99国产精品一区二区蜜桃av| 日日夜夜操网爽| 久久欧美精品欧美久久欧美| 人成视频在线观看免费观看| 99精品在免费线老司机午夜| 一区福利在线观看| 一进一出抽搐动态| 成年版毛片免费区| 国产亚洲欧美在线一区二区| 多毛熟女@视频| 亚洲精品在线美女| 日韩视频一区二区在线观看| 成人三级做爰电影| 夜夜看夜夜爽夜夜摸| 久久精品人人爽人人爽视色| 日本在线视频免费播放| 成人av一区二区三区在线看| a级毛片在线看网站| 看片在线看免费视频| 91成人精品电影| 视频在线观看一区二区三区| 国产亚洲精品av在线| 国产高清激情床上av| 麻豆国产av国片精品| 亚洲国产日韩欧美精品在线观看 | 亚洲国产欧美一区二区综合| 成人欧美大片| 久久精品aⅴ一区二区三区四区| 91国产中文字幕| 在线国产一区二区在线| 亚洲,欧美精品.| 无人区码免费观看不卡| 国产精品99久久99久久久不卡| 国产精品二区激情视频| 国产精品 欧美亚洲| 91字幕亚洲| 高清黄色对白视频在线免费看| 日本在线视频免费播放| 好男人电影高清在线观看| 啪啪无遮挡十八禁网站| 色播在线永久视频| 人人妻人人爽人人添夜夜欢视频| 亚洲第一欧美日韩一区二区三区| 两性午夜刺激爽爽歪歪视频在线观看 | 亚洲av日韩精品久久久久久密| 午夜精品久久久久久毛片777| 正在播放国产对白刺激| 久热爱精品视频在线9| 午夜a级毛片| cao死你这个sao货| 国产激情久久老熟女| 欧美不卡视频在线免费观看 | 男男h啪啪无遮挡| 悠悠久久av| 在线天堂中文资源库| 69精品国产乱码久久久| 啦啦啦韩国在线观看视频| 免费看a级黄色片| 久久精品91无色码中文字幕| 精品国产国语对白av| 黄色女人牲交| 亚洲av电影不卡..在线观看| 黑人巨大精品欧美一区二区蜜桃| 18禁裸乳无遮挡免费网站照片 | 亚洲欧美日韩另类电影网站| 91av网站免费观看| 亚洲 欧美一区二区三区| 欧美av亚洲av综合av国产av| 久久久国产成人免费| 天天一区二区日本电影三级 | 一本综合久久免费| 亚洲av成人av| 亚洲天堂国产精品一区在线| 一级a爱片免费观看的视频| 在线观看免费午夜福利视频| 久久国产精品影院| 亚洲一码二码三码区别大吗| 中文字幕精品免费在线观看视频| 亚洲男人天堂网一区| 亚洲免费av在线视频| 亚洲中文字幕日韩| 免费女性裸体啪啪无遮挡网站| 亚洲情色 制服丝袜| 久久国产乱子伦精品免费另类| 午夜精品在线福利| 亚洲三区欧美一区| 99国产精品一区二区蜜桃av| 午夜福利影视在线免费观看| 亚洲狠狠婷婷综合久久图片| 国内精品久久久久精免费| 一a级毛片在线观看| 女警被强在线播放| 夜夜看夜夜爽夜夜摸| 51午夜福利影视在线观看| 啦啦啦韩国在线观看视频| 丰满的人妻完整版| 亚洲男人的天堂狠狠| 久久久久亚洲av毛片大全| 一区二区三区精品91| 老司机深夜福利视频在线观看| 国产一区二区三区综合在线观看| 亚洲第一电影网av| 巨乳人妻的诱惑在线观看| 精品国产一区二区久久| 亚洲精品国产区一区二| 一级,二级,三级黄色视频| 午夜福利影视在线免费观看| 美女高潮到喷水免费观看| 亚洲欧洲精品一区二区精品久久久| 日本五十路高清| 91国产中文字幕| 老熟妇乱子伦视频在线观看| 满18在线观看网站| 国产亚洲精品一区二区www| 99热只有精品国产| 色尼玛亚洲综合影院| 久久久国产欧美日韩av| 热99re8久久精品国产| 女同久久另类99精品国产91| 电影成人av| 成人av一区二区三区在线看| 一夜夜www| 精品高清国产在线一区| 老汉色av国产亚洲站长工具| 两个人看的免费小视频| 久久婷婷人人爽人人干人人爱 | 久久精品91蜜桃| 中文字幕人妻丝袜一区二区| 十八禁人妻一区二区| 夜夜躁狠狠躁天天躁| 日韩欧美国产一区二区入口| 亚洲九九香蕉| 国产成人欧美在线观看| 欧美精品亚洲一区二区| 欧美黑人欧美精品刺激| 欧美日韩福利视频一区二区| 亚洲av成人一区二区三| 波多野结衣av一区二区av| 国产精品久久久久久人妻精品电影| 成人18禁在线播放| 999精品在线视频| 国产一区二区三区在线臀色熟女| 免费久久久久久久精品成人欧美视频| 一区二区三区激情视频| 黄色视频,在线免费观看| 一级毛片精品| 亚洲男人天堂网一区| 多毛熟女@视频| 巨乳人妻的诱惑在线观看| 性少妇av在线| 村上凉子中文字幕在线| 免费在线观看亚洲国产| 日韩欧美国产一区二区入口| 老司机午夜十八禁免费视频| 亚洲精品在线观看二区| 精品熟女少妇八av免费久了| 久久久久久人人人人人| 国产国语露脸激情在线看| 一区二区三区激情视频| 禁无遮挡网站| 亚洲人成网站在线播放欧美日韩| 一区二区三区精品91| 国产精品99久久99久久久不卡| 18禁美女被吸乳视频| 久久久久久久久中文| 久久久久久国产a免费观看| 国内精品久久久久精免费| 1024香蕉在线观看| 在线播放国产精品三级| 国产欧美日韩一区二区三| 精品日产1卡2卡| 亚洲精品一区av在线观看| 亚洲欧美激情综合另类| 最好的美女福利视频网| 日本撒尿小便嘘嘘汇集6| 男人舔女人的私密视频| 极品人妻少妇av视频| 成人av一区二区三区在线看| 亚洲人成电影免费在线| 国产高清视频在线播放一区| 久久久国产成人免费| 国产99白浆流出| 午夜免费激情av| 中文亚洲av片在线观看爽| 啪啪无遮挡十八禁网站| 九色亚洲精品在线播放| 久久久久九九精品影院| 亚洲 欧美 日韩 在线 免费| 男人的好看免费观看在线视频 | 不卡av一区二区三区| 天天添夜夜摸| 无限看片的www在线观看| 中文字幕另类日韩欧美亚洲嫩草| 好看av亚洲va欧美ⅴa在| 免费高清在线观看日韩| 中文字幕精品免费在线观看视频| 51午夜福利影视在线观看| 又黄又粗又硬又大视频| 69av精品久久久久久| 熟妇人妻久久中文字幕3abv| 国产亚洲精品久久久久5区| netflix在线观看网站| 久久久国产精品麻豆| 看片在线看免费视频| 国产精品九九99| 免费观看精品视频网站| 久久久久久国产a免费观看| 中文字幕av电影在线播放| 亚洲久久久国产精品| 精品一区二区三区av网在线观看| 黄色视频不卡| 精品高清国产在线一区| 久久中文字幕人妻熟女| 岛国在线观看网站| 免费搜索国产男女视频| 精品少妇一区二区三区视频日本电影| 如日韩欧美国产精品一区二区三区| 午夜两性在线视频| www.www免费av| 久久香蕉精品热| 成人手机av| 不卡一级毛片| 久久久国产成人免费| 久久久久久人人人人人| 操美女的视频在线观看| 无遮挡黄片免费观看| 夜夜夜夜夜久久久久| 国产亚洲精品第一综合不卡| 自拍欧美九色日韩亚洲蝌蚪91| 国产精品久久久av美女十八| 中文字幕色久视频| 亚洲中文字幕一区二区三区有码在线看 | 人人妻人人爽人人添夜夜欢视频| 在线av久久热| 亚洲国产高清在线一区二区三 | 亚洲精品美女久久久久99蜜臀| 欧美一级毛片孕妇| 久久久久久人人人人人| 1024视频免费在线观看| 好看av亚洲va欧美ⅴa在| 男女床上黄色一级片免费看| 色综合欧美亚洲国产小说| 精品电影一区二区在线| 母亲3免费完整高清在线观看| 午夜日韩欧美国产| 日韩欧美三级三区| 国内精品久久久久精免费| 国产黄a三级三级三级人| 麻豆一二三区av精品| 叶爱在线成人免费视频播放| 日韩国内少妇激情av| 久久久久久久精品吃奶| 国产亚洲av高清不卡| 真人一进一出gif抽搐免费| 欧美日韩黄片免| 真人一进一出gif抽搐免费| 成人国产一区最新在线观看| 免费不卡黄色视频| 一a级毛片在线观看| 午夜久久久久精精品| 日本精品一区二区三区蜜桃| 日日夜夜操网爽| 日韩视频一区二区在线观看| 又黄又粗又硬又大视频| 亚洲国产欧美一区二区综合| 丝袜在线中文字幕| 亚洲欧美激情综合另类| 午夜福利欧美成人| 九色国产91popny在线| 天堂√8在线中文| 真人做人爱边吃奶动态| 久久精品国产亚洲av高清一级| 亚洲专区国产一区二区| 在线观看日韩欧美| 国产野战对白在线观看| 亚洲国产中文字幕在线视频| 99热只有精品国产| 一级毛片高清免费大全| 午夜免费成人在线视频| 女人精品久久久久毛片| 久久亚洲精品不卡| 国产日韩一区二区三区精品不卡| 亚洲无线在线观看| 少妇熟女aⅴ在线视频| 一级,二级,三级黄色视频| 亚洲精品中文字幕在线视频| 亚洲人成伊人成综合网2020| 日韩 欧美 亚洲 中文字幕| 91大片在线观看| 91精品三级在线观看| 一级毛片高清免费大全| а√天堂www在线а√下载| 黄色成人免费大全| 午夜精品在线福利| 精品人妻1区二区| 成熟少妇高潮喷水视频| 欧美黄色片欧美黄色片| 在线国产一区二区在线| 精品国产超薄肉色丝袜足j| 久99久视频精品免费| 午夜免费成人在线视频| 亚洲男人的天堂狠狠| 免费在线观看完整版高清| 日日摸夜夜添夜夜添小说| 男女之事视频高清在线观看| 成人特级黄色片久久久久久久| 一进一出抽搐动态| 久久狼人影院| 亚洲在线自拍视频| 日韩欧美免费精品| 中文亚洲av片在线观看爽| 黄色片一级片一级黄色片| 国产亚洲精品第一综合不卡| 国产精品永久免费网站| 日韩三级视频一区二区三区| 黑丝袜美女国产一区| 咕卡用的链子| 啪啪无遮挡十八禁网站| 亚洲精品中文字幕在线视频| 久久精品国产99精品国产亚洲性色 | 精品免费久久久久久久清纯| 国产视频一区二区在线看| 在线观看免费午夜福利视频| 国产又色又爽无遮挡免费看| 50天的宝宝边吃奶边哭怎么回事| 久久中文字幕人妻熟女| 亚洲人成网站在线播放欧美日韩| 18禁黄网站禁片午夜丰满| 国产在线观看jvid| 欧美一级a爱片免费观看看 | 国产精品日韩av在线免费观看 | 亚洲成av人片免费观看| 嫩草影视91久久| 老司机午夜福利在线观看视频| 在线免费观看的www视频| 色综合婷婷激情| 免费人成视频x8x8入口观看| 欧美一区二区精品小视频在线| 欧美中文日本在线观看视频| 国产免费av片在线观看野外av| 亚洲色图综合在线观看| 亚洲av第一区精品v没综合| 久久香蕉精品热| 热99re8久久精品国产| 国产主播在线观看一区二区| 国产激情欧美一区二区| 制服丝袜大香蕉在线| 国产精品亚洲一级av第二区| 黄片大片在线免费观看| 在线观看www视频免费| 后天国语完整版免费观看| 涩涩av久久男人的天堂| 丰满人妻熟妇乱又伦精品不卡| avwww免费| √禁漫天堂资源中文www| АⅤ资源中文在线天堂| 亚洲最大成人中文| 99精品久久久久人妻精品| 丁香欧美五月| 我的亚洲天堂| 性色av乱码一区二区三区2| 亚洲色图av天堂| 99精品久久久久人妻精品| 日本欧美视频一区| 国产精品电影一区二区三区| 亚洲一码二码三码区别大吗| 免费在线观看影片大全网站| 欧美另类亚洲清纯唯美| 19禁男女啪啪无遮挡网站| 少妇熟女aⅴ在线视频| 欧美人与性动交α欧美精品济南到| 欧美av亚洲av综合av国产av| 国产在线精品亚洲第一网站| 日韩欧美免费精品| 激情视频va一区二区三区| 日本vs欧美在线观看视频| 他把我摸到了高潮在线观看| 香蕉久久夜色| 久久性视频一级片| 人人妻人人澡欧美一区二区 | 极品教师在线免费播放| 亚洲熟女毛片儿| 老司机午夜十八禁免费视频| 黄片小视频在线播放| 黑人操中国人逼视频| 亚洲五月天丁香| 日本五十路高清| 亚洲天堂国产精品一区在线| 大陆偷拍与自拍| 亚洲熟女毛片儿| 可以免费在线观看a视频的电影网站| 88av欧美| 欧美精品啪啪一区二区三区| 国产精品美女特级片免费视频播放器 | 精品一区二区三区四区五区乱码| 午夜福利欧美成人| 日日爽夜夜爽网站| 啪啪无遮挡十八禁网站| 欧美+亚洲+日韩+国产| 高清在线国产一区| 免费看美女性在线毛片视频| 999久久久精品免费观看国产| 女人精品久久久久毛片| 午夜福利,免费看| 国产免费男女视频| 青草久久国产| 在线视频色国产色| 一级毛片精品| 18美女黄网站色大片免费观看| 欧美+亚洲+日韩+国产| 久久久久精品国产欧美久久久| 天天添夜夜摸| 精品国产亚洲在线| 婷婷丁香在线五月| 亚洲无线在线观看| 午夜两性在线视频| 可以免费在线观看a视频的电影网站| 亚洲精品粉嫩美女一区| 一边摸一边抽搐一进一出视频| 一进一出抽搐动态| 国产精品免费一区二区三区在线| 久久精品亚洲精品国产色婷小说| 男人操女人黄网站| 久久精品国产亚洲av香蕉五月| 久久精品国产综合久久久| 禁无遮挡网站| 男女下面进入的视频免费午夜 | 九色亚洲精品在线播放| aaaaa片日本免费| 午夜福利,免费看| 欧美不卡视频在线免费观看 | 日韩av在线大香蕉| 黑人巨大精品欧美一区二区蜜桃| 久久热在线av| 免费搜索国产男女视频| 亚洲精品一区av在线观看| 啦啦啦观看免费观看视频高清 | 伊人久久大香线蕉亚洲五| 国产日韩一区二区三区精品不卡| 男女下面插进去视频免费观看| 韩国精品一区二区三区| 精品一区二区三区av网在线观看|