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Game Theory in Wireless and CommunicationNetworks Theory Models and ApplicationsLecture 12 VariationalMathematical Programs Inequalities with Equilibrium.

MathematicalConstraints Programsand with Equilibriumits Application inConstraints.

Solving MPEC andStackelberg Equilibrium Programswith Equilibrium Constraints EPEC Zhu Han Dusit Niyato Walid Saad and Tamer BasarThanks for Xiao Tang Xi an Jiaotong.

UniversityOverview of Lecture NotesIntroduction to Game Theory Lecture 1 book 1 Non cooperative Games Lecture 1 Chapter 3 book 1 Bayesian Games Lecture 2 Chapter 4 book 1.

Differential Games Lecture 3 Chapter 5 book 1 Evolutionary Games Lecture 4 Chapter 6 book 1 Cooperative Games Lecture 5 Chapter 7 book 1 Auction Theory Lecture 6 Chapter 8 book 1 Matching Game Lecture 7 Chapter 2 book 2.

Contract Theory Lecture 8 Chapter 3 book 2 Learning in Game Lecture 9 Chapter 6 book 2 Stochastic Game Lecture 10 Chapter 4 book 2 Game with Bounded Rationality Lecture 11 Chapter 5 book 2 Equilibrium Programming with Equilibrium Constraint Lecture 12 Chapter 7 book 2.

Zero Determinant Strategy Lecture 13 Chapter 8 book 2 Mean Field Game Lecture 14 UCLA course book 2 Network Economy Lecture 15 Dr Jianwei Huang book 2 Introduction Stackelberg Game.

Variational Inequalities Definition Examples Definition Examples.

Definition Examples Stackelberg Game Definition and features Modeling sequential decision makings .

Solution by backward induction Leader s ProblemDecision making Backward induction Follower chooses the Leader makes decisions byaction by reacting to considering follower s.

leader s action reactionFollower s Problem07 27 2020 4 43 Basics on Variational Inequality Basics on Variational Inequality.

Basics on Variational Inequality Basics on Variational Inequality Example IterativeWater Filling IterativeWater Filling IterativeWater Filling.

IterativeWater Filling IterativeWater Filling IterativeWater Filling Numerical result07 27 2020 16 43.

Basics on Quasi VIXi an Jiaotong University University of07 27 2020 17 43 Energy Efficient Multi Channel PowerAllocation.

07 27 2020 18 43 Energy Efficient Multi Channel PowerAllocation07 27 2020 19 43 Energy Efficient Multi Channel Power.

Allocation07 27 2020 20 43 Energy Efficient Multi Channel PowerAllocation07 27 2020 21 43.

Energy Efficient Multi Channel PowerAllocation07 27 2020 22 43 Introduction Stackelberg Game.

Variational Inequalities Definition Power Control with Aggregated InterferenceConstraints Definition.

Examples Definition Examples 23 43 MPEC Problem Definition.

A mathematical optimization with part of theconstraints being in the form of complementarityconditionscomplementarity condition parameterized by x .

07 27 2020 24 43 MPEC Problem Definition A mathematical optimization with part of theconstraints being in the form of another.

optimization bi level optim Optimization w r t yas constraint parametrized by x 07 27 2020 25 43.

MPEC Problem Features Bi level multi level structure Suit for the Stackelberg game analysisFollower .

07 27 2020 26 43 Two Tier Power Control System Model Two tier heterogeneous networks Overlapping frequency reuse.

Multi channel transmissions Rate maximization07 27 2020 27 43 Two Tier Power Control Problem Formulation.

Stackelberg gameLeader Macro cell users Follower Small cell Hierarchical power control07 27 2020 28 43 Two Tier Power Control.

Analysis Follower s power allocationWater filling 07 27 2020 29 43 Two Tier Power Control.

Analysis Leader s power allocationMPECreformulationformulation as an optimization Backward induction.

Follower soptimal asconstraint07 27 2020 30 43 Two Tier Power Control.

Analysis Leader s power allocationMPEC reformulation as an optimization Water fillingBased on follower s.

p is an implicit function07 27 2020of q 31 43 Two Tier Power Control Analysis.

Water filling based part Implicit function part Denote the implicit function as and applyLeader s powerallocation as a fixed .

point iteration07 27 2020 32 43 Two Tier Power Control Analysis Structural results on lower water filling.

07 27 2020 33 43 Two Tier Power Control Analysis Solution to the MPEC leader s problem The structural results of the follower s problem is.

the crucial to solve the MPEC problem 07 27 2020 34 43 Jamming Aided Eavesdropping System Model Multi channel communications.

Full duplex eavesdropper Improve eavesdropping by jamming attacks07 27 2020 35 43 Jamming Aided Eavesdropping Problem Formulation.

Stackelberg gameLeader s problem Follower s problem 07 27 2020 36 43 Jamming Aided Eavesdropping Analysis.

Follower s problemConvex problem and solved with LagrangeGeneralized water Implicit function of07 27 2020 37 43.

Jamming Aided Eavesdropping Analysis Leader s problembackward induction intractable optimizationIntractable optimization due to.

the lack of explicit analyticalexpression07 27 2020 38 43 Jamming Aided Eavesdropping Analysis.

MPEC problemoptimizatioparametersThe lower optimalcondition as a constraint.

rather than part of the07 27 2020 39 43 Jamming Aided Eavesdropping Analysis Equivalent optimization problem.

optimizatioparametersviolating the KKT condition inconstraint replace of theA well defined optimization .

qualification CQ optimality conditionSolved with regular method07 27 2020 40 43 Jamming Aided Eavesdropping Replacing the optimality constraints with its KKT.

equivalent is very useful in solving MPEC The violation of CQ is a common problem whichshould be tackled carefully 07 27 2020 41 43 Smoothing Method for General.

General Problem07 27 2020 42 43 Smoothing Method for General Equivalent Reformulationviolating the.

constraintqualification CQ Well definedoptimizationwith slack.

variable z07 27 2020 43 43 Smoothing Method for General Smoothing Function Definition .

Properties 07 27 2020 44 43 Smoothing Method for General Smoothing Approximation Parameterized optimization.

Well defined optimization07 27 2020 45 43 Introduction Stackelberg Game.

Variational Inequalities Definition Power Control with Aggregated InterferenceConstraints Definition.

Examples Definition Examples 46 43 Department of ElectricalIntroduction.

and Computer EngineeringComput Reduce latencyand transmissionData Service OperatorsData Fog Nodes.

Center FNs s Authorized Data Service Subscribers Department of ElectricalMotivationand Computer Engineering.

Large managenumber of computingdeployed at How tovarious Equilibrium performlocations Problem with large scale.

optimizationEquilibrium ConstraintsCompetitio How to EPEC n among multiple.

conflictingmultiple entities AlternatingMethod of method Multipliers.

Real Time Data Service ScenarioDepartment of Electricaland Computer EngineeringK DSOs N ADSSs Computing resources from data centers or FNs .

Computing Resource Blocks CRBs DSOs manage CRBs for serving the ADSSs ADSSs request for CRBs from the DSOs Utility Functions of DSOs and ADSSsDepartment of Electrical.

and Computer Engineering Price set for one CRB by DSO i for ADSS j Number of CRBs purchased from DSO i byADSSfunctionUtility j xi jfor DSO i Utility function for ADSS j.

Revenue Operational Revenuefrom the and fromCRBs measureme workloadprovided nt costs data Maximization of Profits for DSOs and ADSSs.

Department of Electricaland Computer EngineeringOptimization problem for DSO i Optimization problem for ADSS jConstraintConstraint.

Constraint available Constraion service CRBs nt ondelay service DSOs and ADSSs aim to maximize their own profits optimal i j and xi j .

DSOs provide incentives to ADSSs to choose xi j toobtain optimal i j Equilibrium Problem withDepartment of Electricaland Computer Engineering.

Equilibrium Constraints EPEC OptimizatioOptimizatio Optimization of the utilities of DSOs while consideringthe utilities of ADSSs conflicting objectives .

Two level hierarchical optimization problem Equilibria and constraints exist at both upper and lower Alternating Direction Method of Multipliers ADMM Department of Electricaland Computer Engineering.

Constraint Method for large scale optimization Fast convergence when objective function is convexEPEC ADMMConflicting objectives Large.

network ADMM based EPEC in Fog ComputingDepartment of Electricaland Computer EngineeringOptimizatio.

Optimizatio The outer loop terminates when the total profit of the DSOsconverges to an optimal value i e Value at current iteration Value at previous iteration Error Total Profit of ADSSs vs Number of ADSSs.

Department of Electricaland Computer Engineering Optimization using ADMM increases the total profit ofthe ADSSs in Fog Computing compared to Cloud Computing Use GNEP to tackle the common constraints in.

VI assisted convergence analysis in distributedalgorithms Use Quasi NE to tackle the non convex games The VI theory is a very powerful mathematical toolwhich has applications in many areas .

By building up the equivalence between VI andgame the VI theory provides us alternativemanner in the investigation on the properties ofNash equilibrium particularly on the uniquenessand convergence .

MPEC and EPEC07 27 2020 56 43 References 1 S Guruacharya D Niyato D I Kim and E Hossain Hierarchicalcompetition for downlink power allocation in OFDMA femtocell networks .

IEEE Trans Wireless Commun vol 12 no 4 pp 1543 1553 Apr 2013 2 K Zhu E Hossain and A Anpalagan Downlink power control in two tiercellular ofdma networks under uncertainties a robust Stackelberg game IEEE Trans Commun vol 63 no 2 pp 520 543 Feb 2015 3 J Wang M Peng S Jin and C Zhao A generalized nash equilibrium.

approach for robust cognitive radio networks via generalized variationalinequalities IEEE Trans Wireless Commun vol 13 no 7 pp 3701 3714 Jul 2014 4 Q Han B Yang X Wang K Ma C Chen and X Guan Hierarchical game based uplink power control in femtocell networks IEEE Trans Veh .

Technol vol 63 no 6 pp 2819 2843 Jul 2014 5 Z Luo J Pang and D Ralph Mathematical Programs with EquilibriumConstraints Cambridge UK Cambridge University Press 1996 Xi an Jiaotong University University of07 27 2020 57 43.

Internet of Things. Big Data. Cloud Computing. Fog Computing. Data Service Operators (DSOs) Computing and Storage. Authorized Data Service Subscribers (ADSSs) Fog Nodes (FNs) Data Centers. Reduce latency and transmission costs. Department of Electrical and Computer Engineering

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