covex_assignment

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Problem Set 1

January 7, 2014

1. (a) Prove that for rectangular matricesB,D, and invertible matricesA,C

A(A + BCD)1 =I B(C1 + DA1B)1DA1.

(b) Using the above result, prove that

(A1 + BTC1B)1BTC1 =ABT(BABT + C)1.

(c) Find an expression for(A+ xyT)1 in terms ofA1, x and y for invertible

matrixA and (column) vectorsx andy.

2. Leti(A)denote an eigenvalue ofA

(a) i(AB) =i(BA)whereA,Bare not necessarily square.

(b) IfA= AT, find the following in terms ofi(A)

i. Tr(Ap).

ii. i(I +cA).

iii. i(A cI).iv. i(A

1).

3. The notationA 0denotes the fact that A is positive definite. Prove the followingresults forA 0:

(a) A1 0(b) [A]ii 0for alli, where[A]iidenotes thei-th diagonal entry ofA.(c) For anyB, rank(BABT) =rank(B).

(d) IfBis full row-rank, thenBABT 0.(e) IfA 0, then for any matrix X,XAXT =0 AX= 0.


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4. Prove the following for the operator norm (with a = b, and all vector norms are.

a).

(a)Ia,a = 1(b)Aa,a = supx=1 Axa(c)Aa,a = supx=0 Axx .(d) LetATA 0, then A2

2,2= maxii(A

TA)

(e)A1,1

= maxj

i|[A]i,j|(f)Ax Aa,a x(g)ABa,a Aa,a Ba,a

5. Consider X ={x Rn|Ax = b} forA Rmn andb Rm with b R(A).Prove that the following two statements are equivalent (i.e., (a) (b) and (b)(a))

(a) cTx= d for allx X(b) There existsv Rm such thatc= ATvandd = bTv.

6. Prove that

(a)u + v22

= u22+ v2

2if and only ifuTv= 0.

(b) 2< a,b> +2 < x,y>= + < a x,b y>(c)x

1 n x

2

(d)x1 x

2 x

7. ConsiderA Rmn wherem nand rank ofA isn. Suppose there exists B suchthatBA= I and thatAB= BTAT.

(a) LetX=AB. Show that

i. X2 =X

ii. Tr(X) =rank(X)

(b) Find an expression forBin terms ofA.

(c) For any two vectors b Rn and c Rm, prove thatAb c2

ABc c2

.


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EE609 Assignment 2 January 8, 2015

Weight: 1% if submitted by Jan 15, 2014.

Assignment will not be graded. No need to submit practice problems.

1 Assignment Problems

1. Show that the set{x Rn|x x12 x x22}is actually a half space. and express itin canonical form.

2. Show that the set{x Rn|x1 2}is a polyhedra and express it as an intersection of halfspaces (and hyperplanes, if required).

3. Show that the set{x R2++|x1x2 1}is convex and express it as an intersection of infinitenumber of half-spaces.

4. Is the following set convex:{x Rn|x 0,xTy 1 for ally with y2 = 1}? Show thatthe set {x Rn|xTy 1 for ally withy C} is always convex, even ifCis not convex.

5. Consider the two-dimensional positive semidefinite coneS2+defined as

{X= [ x yy z] |x,y ,z R,X 0} (1)

Show that it can equivalently be expressed as {x,y ,z R|x 0, z 0, xz y2}.

2 Practice Problems

6. Show that a set is affine if and only if its intersection with any line is affine.

7. What is the minimum distance between two parallel halfspaces{x Rn|aTx b1} and{x Rn|aTx b2} (Hint: it depends on where the origin is)?

8. Under what conditions ona,a,b, andb, is the following true?

{x Rn|aTx b} {x Rn|aTx b} (2)

9. For m < n, let b Rm and A Rmn be full row rank. Show that any affine set{x R

n|Ax = b} can be expressed in the form {Cu+ v|u Rm}. For example, the set{x R2|x1+x2= 1} can be expressed as {[u1 u]

T|u R}

10. What is the affine hull of the set {x R3|x21+x22= 1, x3= 1}.

11. Is the following set affine:{x Rn|x x11 x x21}?

12. Given two vectorsy, z Rn, consider the setS{a,b} = {x Rn|x = ay+bz}. Show that

the following set is a polyhedra, and find its boundaries.

a[1,1]

b[1,1]

S{a,b} (3)

13. Show that the set {x Rn|x 1} is a polyhedra and express it as an intersection of halfspaces (and hyperplanes, if required).


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EE609 Assignment 2, Page 2 of 2 January 8, 2015

14. Show that the set of all doubly stochastic matrices is convex polyhedral in Rnn. A doubly

stochastic matrix is a square matrix with nonnegative entries with the property that the sum of

entries in every row and column is exactly 1.

15. What kind of set is C = {x R2|p(0) = 1, |p(t)| 1, for1 t 2} where p(t) =x1+x2t?

16. Given, consider the setS ={x Rn|x a2 x b2}for a = b. Show thatS isa halfspace for = 1, convex for 1.

17. Find the separating hyperplane between the two setsC = {x R2

|x2 0} and D = {x R2+|x1x2 1}.

18. Express the following norm balls as intersection half-spaces (a) {x Rn|x2 1}, (b){x Rn|x1 1}, and (c) {x R

n|x 1}.

19. Which of the following sets are convex (provide proof or counterexample)

(a) {x R2|x21+ 2ix1x2+i2x22 1 i= 1, 2, . . . , 10}

(b) {x R2|x21+ix1x2+i2x22 1 i= 1, 2, . . . , 10}

(c) {x Rn| minixi= 1}

(d)

20. Show that the perspective function transformation, P(C) := {x/t|[xTt]T C} preservesconvexity.

21. Show that the set {x Rn|(Ax + b)T(Ax + b) (cTx +d)2, cTx +d >0} is a convexcone.


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Weight: 1% if submitted by Jan 22, 2014.

Assignment will not be graded. No need to submit practice problems.


1. Starting from Jensens inequality, show thatxy1 x + (1 )y.2. Consider a differentiable functionf : R R with R+ domf, and its running average,

defined as

F(x) = 1

x x0

f(t)dt (1)

with domF = R++. Show thatF(x)is convex iff(x)is convex.

3. Show that the harmonic meanf(x) = (n

i=11/xi)1

is concave.

4. Definex[j] as thej -th largest component ofx; e.g.,x[1] = maxi{xi}, andx[n] = mini{xi}.Given any non-negative numbers 1 2 . . . r 0, prove thatf(x) =

ri=1ix[i]

is convex.

5. Prove the reverse Jensens inequality for a convexf with dom f = Rn, i > 0 and 1ni=2i = 1

f(1x1 2x2 . . . nxn)1f(x1) 2f(x2) . . . nf(xn) (2)

2 Practice Problems

6. Consider an increasing and convex functionf : R R with domf={a, b}. Let the functiong : R Rdenote its inverse, i.e., g(f(x)) = xfor a x band dom g ={f(a), f(b)}.Show thatg is a concave function.

7. Show that the running average of the non-differentiable functionf(x)is also convex. Sincef

is not differentiable, you must use the zeroth order condition to prove the convexity ofF.8. Give an example of a functionf(x)whose epigraph is (a) half-space, (b) norm cone, and (c)

polyhedron.

9. Give an example of a concave positive function (f(x)> 0) with domain Rn.

10. Show that the functionf(x)is convex if and only if the functionf(a+tb)is convex for alla +tb domf andt.

11. Show that the harmonic function:f(x) = (n

i=1xai )

1/afora


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(d) f(x1, x2) =x1 x

12 for01.

(e) f(x1, x2=

x1x2

14. Show that the harmonic meanf(x) = (n

i=11/xi)1

is concave.

15. Show that the functionf(x, t) = xTx/ton domf= Rn R++is convex in(x, t).16. Show that the function f(x, s , t) = log(st xTx) is convex on dom f = {(x, s , t)

Rn+2|st > xTx, s, t >0}.

17. Show that the functionf(x, t) =x33/t2 is convex on {(x, t)|t >0}.

18. Show that the functionf(x) = Ax+bcTx+d

is convex on {x|cTx +d >0}.19. Prove the following for non-differentiable functionsfandg.

(a) Iff andg are convex, both nondecreasing (or nonincreasing), and positive functions onan interval, thenf g is convex.

(b) Iff,g are concave, positive, with one nondecreasing and the other nonincreasing, thenf g is concave.

(c) Iffis convex, nondecreasing, and positive, andgis concave, nonincreasing, and positive,thenf /gis convex.

(d) Iffis nonnegative and convex, andg is positive and concave, thenf2/gis convex.

20. A functionf(x) is log-concave iflog(f(x)) is a concave function on{x|f(x) > 0}. Showthat the Gaussian cdfF(x)is log-concave for allx.

21. Show that the maximum of a convex functionfover the box B:={xl xxu} is achievedat one of its2n vertices.

22. Using the following intermediate steps, show that the functionf(x, t) = log(t2 xTx)overdomf={(x, t)Rn+1|t >x2} is convex.(a) Show thatt xTx/tis convex over domf.(b) Show that log(t xTx/t)is convex over domf.

23. Show that the following function is convex on {x|x21}

f(x) =Ax b22

1 x22(3)

24. Show that the weighted geometric mean

f(x) =n

i=1

xkk (4)

with domf= Rn++ is concave fork0 and ni=1k = 1.

25. Show that the Huber function

f(x) =

x22/2 x21x22 1/2 x2> 1

(5)

is convex on Rn.


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No submission required.


1. What is the solution of the following linear program

min cTx (1)

s. t.0 xi 1 i= 1, . . . , n . (2)

2. Consider the following linear program

min cTx (3)

s. t. Ax b (4)

whereA is square and full rank.

(a) When is the problem infeasible?

(b) When is the problem unbounded below?

(c) When does the problem have a finite solution, and what is it?

3. Show that any linear programming problem can be expressed as

min cTx (5)

s. t. Ax = b (6)

xi 0 i= 1, . . . , n (7)

4. Solve the following quadratic optimization problem

min cTx (8)

s. t. x2 1 (9)

5. Consider the sinusoidal measurement model in Gaussian noise described as,

y(n) =d+ccos(2f0n) +ssin(2f0n) +w(n), 0 n N 1, (10)

wherew(n) is additive white Gaussian noise and E|w(n)|2

= 2n. Answer the questions

that follow

(a) Formulate the LS estimation problem for parametersd, c, s.

(b) Derive the LS estimator ofd, c, swith a suitable approximation for large N.


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2 Practice Problems


min cTx (11)

s. t. Ax = b (12)





min cTx (13)

s. t. aTx b (14)

wherea = 0.




8. Given thatci 0 for i = 1, 2, . . . , n, solve the following problem

min cTx (15)

s. t. i xi bi i= 1, . . . , n (16)

wherei bifor alli = 1, 2, . . . , n.


min cTx (17)

s. t.

n

i=1

xi = 2 (18)

0 xi 1 i= 1, . . . , n (19)


min cTx (20)

s. t.

ni=1

xi = 1 (21)

xi 0 i= 1, . . . , n (22)

11. Consider the two sets:

S1= {v1,v2, . . . ,vp} (23)

S2= {u1,u2, . . . ,uq} (24)


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Formulate the linear optimization problem to determine the separating hyperplane, i.e., find

a Rn andb R such that

aTvi b i= 1, . . . , p (25)

aTui b i= 1, . . . , q (26)

Ensure that your problem excludes the trivial solution a = b = 0.

12. Consider the following quadratic optimization problem

min cTx (27)

s. t. xTAx 1 (28)

Letifori= 1, 2, . . . nbe the real eigenvalues of a symmetric matrix A. What is the solutionof this problem for the following cases.

(a) Ifi >0 for i = 1, 2, . . . n.

(b) If1 0 for i = 2, . . . n.

(c) If1= 0, whilei >0 for i = 2, . . . n.

13. Solve the following optimization problem forA 0,

min cT

x (29)s. t. (x xc)

TA(x xc) 1 (30)

14. Solve the following optimization problem forA 0,

min xTAx (31)

s. t. x22 1 (32)

Hint: Given that the eigenvalue decomposition A = UUT, use the change of variabley= UTx.

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