Homework Set #1 - due on April 12 in class

EE 150 - Applications of Convex Optimization in Signal Processing and CommunicationsDr. Andre Tkacenko, JPLThird Term 2011-2012Due on Thursday, April 12 in class.<strong>Homework</strong> <strong>Set</strong> <strong>#1</strong>1. (10 points) Send a brief message with your name in full to the course e-mail address(ee150.acospc@gmail.com) in order to get on the mailing list.2. (10 points) (Young’s inequality for products, Hölder’s inequality, and Minkowski’sinequality:) The goal of this problem is to prove several useful inequalities for the l p -norm.Each of the inequalities referenced above can be proved using the one previously mentioned(except, of course, for the first one).(a) (Young’s inequality for products:) Suppose p, q > 0 are such that 1 p + 1 qprove thatab ≤ app + bqq ,with equality if and only if a p = b q .= 1. If a, b ≥ 0,Hint: Use the fact that the exponential function is convex, which implies that for0 < θ < 1, we have e θx+(1−θ)y ≤ θe x + (1 − θ)e y , with equality if and only if x = y.(b) (Hölder’s inequality:) Let x, y ∈ C n with x k = [x] kand y k = [y] kfor 1 ≤ k ≤ n. Showthatn∑|x k y k | ≤ ||x|| p||y|| q,with equality if and only if |x k| p||x|| p pk=1= |y k| q||y|| q qfor all 1 ≤ k ≤ n.(c) (Minkowski’s inequality:) Suppose that x and y are as in part (b). Prove that||x + y|| p≤ ||x|| p+ ||y|| p,with equality if and only if x k = λ k y k or y k = λ k x k for some λ k ≥ 0 for all 1 ≤ k ≤ nfor p = 1 and x = λy or y = λx for some λ ≥ 0 for p > 1. This is the triangleinequality for the l p -norm.Hint: First show that |c + d| ≤ |c| + |d| for c, d ∈ C with equality if and only if c = Kdor d = Kc for some K ≥ 0 (this is the triangle inequality for complex scalars). Thenapply Hölder’s inequality (twice) and trace back all of the conditions required forequality.3. (10 points) (Dual of the l p -norm:) If ||·|| is a norm on C n , recall that, for the standard innerproduct, the dual norm ||·|| is defined as follows:{ [ ] }||z|| = sup Re z † x : ||x|| ≤ 1 .Suppose that p, q > 0 and satisfy 1 p + 1 q = 1. Prove that the dual of the l p-norm ||·|| pis thel q -norm ||·|| q.

Hint: Use the equivalent formulation of the dual norm given by{∣ }∣∣z||z|| = sup † x∣ : ||x|| = 1 ,along with Hölder’s inequality from the previous problem to show that ∣ ∣z † x ∣ ∣ ≤ ||z|| q. Thenfind a clever choice of x to achieve equality.4. (10 points) (Block matrix inversion and the matrix inversion lemma:) Let A, B, C, and Dare matrices of dimensions m × m, m × n, n × m, and n × n, respectively. Suppose that Aand D are invertible. This problem deals with the (m + n) × (m + n) block matrix M givenby[ ]A BM =.C D(a) Show the following two identities:[] [ ] [ ]I m 0 m×n A B Im −A −1 B−CA −1 I n C D 0 n×m I n[ ] [ ] []Im −BD −1 A B I m 0 m×n0 n×m I n C D −D −1 C I n==[A0 m×n0 n×m D − CA −1 B[A − BD −1 C 0 m×n0 n×m D]] .(b) Using part (a), show that[] [I m 0 m×n A 0 m×nM =CA −1 I n 0 n×m D − CA −1 B[ ] [Im BD −1 A − BD −1 C 0 m×n=0 n×m I n 0 n×m D] [ ]Im A −1 B0 n×m I n] [] .I m 0 m×nD −1 C I nFrom this result, show that we havedet(M) = det(A) det ( D − CA −1 B ) = det ( A − BD −1 C ) det(D) .Hint: For the first part, use the fact that[ ] −1 [ ]Im FIm −F=0 n×m I n 0 n×m I nand[ ] −1 [ ]Im 0 m×n Im 0 m×n=,G I n −G I nfor any m × n matrix F and n × m matrix G. To help with the second part, use thefact that([])R 0 m×ndet(PQ) = det(P) det(Q) and det= det(R) det(S) ,0 n×m Sfor any k × k matrices P and Q, as well as any m × m matrix R and n × n matrix S.

Notice that Σ v is the Schur complement of Σ y in Σ z .Hint: From our knowledge of conditional probability, the pdf of v, denoted f v (v), is given by( [f v (v) = f (x,y)(x, y 0 ) f z vTy T ] ) T0=,f y (y 0 )f y (y 0 )where f y (y 0 ) is the pdf of y evaluated at y 0 , which can be obtained using the multivariateGaussian distribution expression above as it can be shown that y ∼ N ( µ y , Σ y). Also, besure to use the results from the previous problem.Reading assignments:1. Look over parts of The Matrix Cookbook as needed and start reading the cvx Users’ Guide.Reminders:Late homework policy for EE 150: Late homeworks will not be accepted. There will be noexceptions to this other than institute established emergency reasons, in which case a signedletter is required from an authorized official.NCT Problems: Remember that problems with an asterisk, such as *7 are no collaboration type(NCT) problems.Texts: The abbreviation CO-BV corresponds to the textbook “Convex Optimization” by StephenBoyd and Lieven Vandenberghe. In addition, CO-AE refers to the Additional Exercises forConvex Optimization, also by Boyd and Vandenberghe. Finally, CVX corresponds to the cvxUsers’ Guide by Michael Grant and Stephen Boyd.