SI 251-Convex Optimization,Spring 2017
Homework 4
Due on ,April 6,2017,before class
Note:Please compress your codes into one file and sent it to TAs,and print your figures or results and answer the questions on A4paper.
Finish your simulation with CVX package (MATLAB/Python/···).And initialize your program with com-mands to fix your randomized results and make sure that your results are repeatable.For example,if you are using MATLAB,you may add rng(’default’);rng(1);in the preamble.And you may need to reprogram the given MATLAB code segments to other programming languages that you’d like to choose.1.Feasibility
1)(Multiuser transmit beamforming.)Power minimization problem in wireless communication
P :minimize
w 1,···,w K
K  k =1
w k  2
subject to SINR k ≥γk ,k =1,···,K,
(1)
where w 1,···,w K ∈C n are the beamforming vectors for receiver k =1,···,K .Signal-to-interference-plus-noise-ratio for the k -th user SINR k is given by
SINR k =|h H k w k |2 i =k |h H k
w i |2+σ2,(2)
where h k ∈C n is the channel coeffcient vector between the transmitter and the k -th receiver and
σ2is noise power.In the simulation,consider
the complex Gaussian h k ∼CN (0,s 2I )in which s =1/√K .And the noise power σ2can be set as 1without loss of generality.Each target SINR γk ≥0and it’s often represented with dB,which is de
fined as 10log γk .
(a)Consider
the relationship between target SINR and the feasibility of P .Please draw the phase
transition 1figure where X-axis is target SINR in dB (γ1=···=γK =γ),and Y-axis is the ratio when the problem is feasible over multiple realizations of
R =
#{P is feasible }
#of tests(channel realizations)
.
(3)
Assume K =50,n =3.You need to run 20times and take average.(5points)
(b)Please draw the phase transition figure about the relationship between the number of users K
and the feasibility of P .Assume n =3,γ=−15dB.You need to run 20times and take average.(5points)
(c)Please draw the phase transition figure about the relationship between the number of antennas
n and the feasibility of P .Assume K =100,γ=−10dB.You need to run 20times and take average.(5points)
2)(Second-order cone optimization problem.)Randomly generate standard SOCP
P SOCP :minimize x ∈R
n
f T x
subject to
A i x +b i  ≤c T i x +d i ,i =1,···,K
(4)
where each entry of A i ∈R m ×n ,b i ∈R m ,c i ∈R n ,d i ∈R is all draw of i.i.d.standard Gaussian
distribution N (0,1).Please draw the phase transition figure about the relationship between the number of constriants K and the feasibility of P SOCP .Assume m =20,n =100.You need to run 20times and take average.(10points)
1For
more about phase transition,refer to Dennis Amelunxen et al.:Living on the edge:Phase transitions in convex programs
with random data,in:Information and Inference 2014,iau005
凸优化2017作业及答案
2.Optimization problems.
(a)(LASSO.)We wish to recover a sparse vector x∈R n from measurements y∈R m.Our measurement
model tells us that
y=Ax+v,
where A∈R m×n is a known matrix and v∈R m is unknown measurement error.The entries of v are drawn IID from the distribution N(0,σ2).We canfirst try to recover x by solving the optimization problem
min
Ax−y 22+γ||x||22.(5)
x
This problem is called ridge regression.A more successful approach is to solve the LASSO problem
min
Ax−y 22+γ||x||1.(6)
x
Please use the code below to define n,m,A,x,and y.
1
2
3
4
5
6
7
(a)Use CVX to estimate x from y using ridge regression and LASSO problem,respectively.(15
points)
(b)Plot your result to compare the estimated x with the true x.(5points)
(c)How many measurements m are needed tofind an accurate x with ridge regression?How about
with the LASSO?(5points)
(b)(Portfolio Optimization.)Find minimum-risk portfolios with the same expected return as the uniform
portfolio(w=(1/n)1),with risk measured by portfolio return variance,and the following portfolio constraints(in addition to1T w=1):
•No(additional)constraints.
•Long-only:w 0.
•Limit on total short position:1T w−≤0.5,where(w−)i=max{w i,0}.
(a)Use CVX to compare the optimal risk in these portfolios with each other and the uniform
portfolio.(10points)
(b)Plot the optimal risk-return trade-offcurves for the long-only portfolio,and for total short
position limited to0.5,in the samefigure.Comment on the relationship between the two trade-
offcurves.(10points)
(c)(Energy Storage Trade-offs.)We consider the use of a storage device(say,a battery)to reduce the
total cost of electricity consumed over one day.We divide the day into T time periods,and let p t denote the(positive,time-varying)electricity price,and u t denote the(nonnegative)usage or consumption,in period t,for t=1,...,T.Without the use of a battery,the total cost is p T u.Let q t denote the(nonnegative)energy stored in the battery in period t.For simplicity,we neglect energy loss(although this is easily handled as well),so we have q t+1=q t+c t,t=1,...,T1,where c t is the charging of the battery in period t;c t<0means the battery is discharged.We will require that q1=q T+c ,wefinish with the same battery charge that we start with.With the battery operating,the net consumption in period t is u t+c t;we require this to be ,we do not pump power back into the grid).The t
otal cost is then p T(u+c).The battery is characterized by three parameters:The capacity Q,where q t≤Q;the maximum charge rate C,where c t≤C;
and the maximum discharge rate D,where c t≥D.(The parameters Q,C,and D are nonnegative.)
wispring是什么意思(a)Explain how tofind the charging profile c∈R T(and associated stored energy profile q∈R T)
that minimizes the total cost,subject to the constraints.(5points)
p T(u+c)
min
q,c
s.t q t+1=q t+c t,t=1,...,T−1
q1=q T+c T
0≤q t≤Q,t=1,...,T
−D≤c t≤C,t=1,...,T
0≤u t+c t,t=1,...,T
(b)Use CVX to solve the problem above with Q=35,C=D=3as well as p and u defined by the
following code:
1
2
3
4
5
Plot u t,p t,c t,and q t versus t.(15points)
(c)Storage Trade-offs Plot the minimum total cost versus the storage capacity Q,using p and u
below,and charge/discharge limits C=D=3.Repeat for charge/discharge limits C=D=1.
(Put these two trade-offcurves on the same plot.)Give an interpretation of the endpoints of the trade-offcurves.(10points)
SI 251-Convex Optimization,Spring 2017
Homework 4
Due on ,April 6,2017,before class
Note:Please compress your codes into one file and sent it to TAs,and print your figures or results and answer the questions on A4paper.
Finish your simulation with CVX package (MATLAB/Python/···).And initialize your program with com-mands to fix your randomized results and make sure that your results are repeatable.For example,if you are using MATLAB,you may add rng(’default’);rng(1);in the preamble.And you may need to reprogram the given MATLAB code segments to other programming languages that you’d like to choose.1.Feasibility
1)(Multiuser transmit beamforming.)Power minimization problem in wireless communication
P :minimize
w 1,···,w K
K ∑k =1
∥w k ∥2
subject to
SINR k ≥γk ,k =1,···,K,
(1)
where w 1,···,w K ∈C n are the beamforming vectors for receiver k =1,···,K .Signal-to-interference-plus-noise-ratio for the k -th user SINR k is given by
SINR k =|h H k w k |2∑i =k |h H k
w i |2+σ2,(2)
where h k ∈C n is the channel coeffcient vector between the transmitter and the k -th receiver and
σ2is noise power.In the simulation,consider
the complex Gaussian h k ∼CN (0,s 2I )in which s =1/√K .And the noise power σ2can be set as 1without loss of generality.Each target SINR γk ≥0and it’s often represented with dB,which is defined as 10log γk .
(a)Consider the relationship between target SINR and the feasibility of P .Please draw the phase
transition 1figure where X-axis is target SINR in dB (γ1=···=γK =γ),and Y-axis is the ratio when the problem is feasible over multiple realizations of
R =
#{P is feasible }
#of tests(channel realizations)
.
(3)
Assume K =50,n =3.You need to run 20times and take average.(5points)
(b)Please draw the phase transition figure about the relationship between the number of users K
and the feasibility of P .Assume n =3,γ=−15dB.You need to run 20times and take average.(5points)
(c)Please draw the phase transition figure about the relationship between the number of antennas
n and the feasibility of P .Assume K =100,γ=−10dB.You need to run 20times and take average.(5points)Solution:
1For
more about phase transition,refer to Dennis Amelunxen et al.:Living on the edge:Phase transitions in convex programs
with random data,in:Information and Inference 2014,iau005
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