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1、Advaned Optimization Topics in Robust Beamforming Lecture TopicsIntroduction Optimization Problem and ReviewTake examples for Engineering Background involving in Optimization Problem Outage-based Robust Beamforming(QCQP SDP)NP-hard problem involving 3-SAT,Partition,Nonconvex QP combined with Multice
2、ll Coordinated BeamformingProximity,Bregman Distance,Proximity Gradient Method combined with Bayesian Statistics InferenceIntroduction Optimization Problem and Reviewconvex optimization problem in standard formRemark:P and D are a convex on Rn and are affine on Rn Fakas lemma&Duality theoryKKT Theor
3、emPower Allocation in Parallel AWGN ChannelCapacity of Channels known to TransmittersSubspaces Affine setsthe affine set C can be expressed as i.e as a subspace plus an offset Convex setsconvex set expansion and convex combination of points in CHyperplanes and halfspacesIntersectionexampleEllipsoids
4、 Worst-Case Robust Beamforming ProblemSecond order cone ConesLinear-fractional transformationin addition epigraph and sublevel Convex Optimization ProblemConvex optimization problem in standard formconvex and a few of PDF affine equality constraintsfeasible set is convex and dual problemExamples on
5、optimization problemApproximate the FIR via chebyshev Error Weighted value desirable filter Standard Form for Filter ScenariosExamples for different issues involving in Optimization algorithm Information Theory on Space-Time Channel Capacity the mutual information is maxmum if H(Y/H)is maxmum.the ca
6、pacity C(H)for a linear flat fading MIMOPartition Problem MAXCUT Sat Boolean least-square problem basic problem in digital communicationscould check all 2n possible value of xan NP-Hard problem,and very hard in practisemany heuristics for approximate solutionwherethe cost of having-witha feasible x
7、corresponds to the partitionthe objective is to find the patition with least total costMuti-Group Multicast Transmit BeamformingDownlink transmission:based station has K antennas,m receiversn multicast groups beamforming vector for Gksignal sent to group Gktransmitted signal can be expressed as foll
8、ows:optmization problem and its applicationQCQP Problem assume each receiver has one antenna,withchannel vector hiFor ULA,Los.far field,Vandermondesignal at receiver SINR for Receiver ith UserSINRtransmit beamforming problem:mimimize transmit power,subject to QoS constraint for each receiver in each
9、 groupFurther extend to NP-Complete problem,scopthere exist NP-Hard problem such that NP-Complete problem,Given integers，there exist binary variable A plolynomial reduction to the single group multicast transmit beamforming over the reals a case in point Cognitive RadioComplexity of QCQPlet n=2m+1 a
10、nd let the complex-valued decision vector beConsider the representation defined as follows:claim:Yes or no for Sat problem NP-Hardiff the following QCQP over Cn has a minimum value of n this gives rise to a set of linear equationssince we can let Distributed muticell coordinated Beamforming tx signa
11、l of ith cell:rx signal of user j in the ith cellwhere hm,i,j is the channel from mth cell to user j in the ith cell,define CSIT Rm,i,j in the same way as before SINRQCQP ProblemADMM the idea is to reformulate of()in a consesus optSchur complementConvex Relaxation of QCQPNonlinear Optimization Probl
12、em based on Bayesian Statistics Inference ProblemProximity Bregman Distance Proximity Gradient MethodEM Algorithm for Wavelet-Based on Image Restoration EM method is exploited to make addrssing the problemdenotes linear observation operator(i.e a matrix)discrete inverse wavelet transform,wavelet coe
13、fficients,and AGWN respectively.Bayesian Statistics Inference soft-threshold function so-called shrinkage methodlet f be convex(possibly nonsmooth),for every x,proximity operator of f is defined asProximity Bregman Distance Proximity Gradient Methodthe Morean envelop of f is defined asit has been widely used in CS ADMM Image proccesing minmize over compact set at point x*ADMM AlgorithmADMM algorithm is attached importance to 5G.so does in big data application Thanks a lot!
