Lec21 Hw Probs

7/26/2019 Lec21 Hw Probs

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AdditionalHomeworkProblems

RobertM.Freund

April,2004

1

2004 Massachusetts Institute of Technology.


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1 Exercises

1. LetIRn ++

denotethenonnegativeorthant,namelyIRn ={xIRn |xj 0, j = 1, . . . , n}. ConsideringIRn asacone,provethat IRn = IRn+ + +,thusshowingthatIRn isself-dual.+

n n 22. LetQ = xIRn |x1 j=2xj . Qn iscalled thesecond-order

cone, the Lorentz cone, or the ice-cream cone (I am not making thisup). ConsideringQn asacone,provethat(Qn) =Qn,thusshowingthatQn isself-dual.

3. ProveCorollary3ofthenotesondualitytheory,whichassertsthatthe

existence

of

Slater

point

for

the

conic

dual

problem

guarantees

strongdualityandthattheprimalattains itsoptimum.

4. Considerthefollowingminimaxproblems:

min max (x, y) and max min (x, y)xX yY yY xX

where X and Y are nonempty compact convex sets in IRn and IRm,respectively,and(x, y)isconvexinxforfixedy,andisconcaveinyforfixedx.

(a) ShowthatminxX maxyY (x, y) maxyY minxX (x, y)intheabsenceofanyconvexity/concavityassumptionsonX,Y,

and/or

(, ).

(b) Showthatf(x) := maxyY (x, y) isaconvex function inxandthatg(y) := minxX(x, y) isaconcavefunction iny.

(c) Useaseparatinghyperplanetheoremtoprove:

min max (x, y) = max min (x, y) .xX yY yY xX

5. Let X and Y be nonempty sets in IRn, and let f(), g() : IRn IR. Consider the followingconjugate functions f () and g () defined as

follows:f(u) := inf{f(x)utx} ,

xX

and

tg (u) := sup {g(x)u x}.

xX


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(a) Constructageometric interpretationoff () and g (). (b) Show that f () is a concave function on X := {u | f(u) >

}, and g ()isaconvexfunctiononY :={u|g(u)


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(c) Starting from u = (1,2), perform one iteration of the steepestascent

method

for

the

dual

problem.

In

particular,

solve

the

followingproblemwhered=L (u):

max L ( u

+d)s.t. u+d0 ,

0 .

7. Prove Remark 1 of the notes on conic duality, that the dual of thedual is the primal for the conic dual problems of Section 13 of thedualitynotes.

8. Considerthefollowingverygeneralconicproblem:

TGCP: z =minimumx c x

s.t. AxbK1

xK2 ,

whereK1

IRm andK2

IRn areeachaclosedconvexcone. Derive

thefollowingconicdualforthisproblem:

GCD: v = maximumy bTy

s.t.

c

ATy

K2

y

K ,1

and show that the dual of GCD is GCP. How is the conic format ofSection13ofthedualitynotesaspecialcaseofGCP?

n

9. Fora(square)matrixM IRnn,definetrace(M) = Mjj ,andforj=1

twomatricesA,BIRkl define

k l

A

B

:=

Aij

Bij

.

i=1j=1

Provethat:


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(a) A B=trace(ATB).

(b) trace(M N)=trace(N M).

10. LetSkk denotetheconeofpositivesemi-definitesymmetricmatrices,+nnamelySkk ={X Skk |vTXv0 forallv }. Considering+

SkkSkk asacone,provethat =Snn,thusshowingthatSkk+ + + +isself-dual.

11. Considertheproblem:

P : z =minimumx1,x2,x3 x1

s.t.

x2 +x3 = 0

x1

10

(x1, x2) x3 ,

where denotestheEuclideannorm. Thennotethatthisproblemisfeasible(setx1

=x2

=x3

=0),andthatthefirstandthirdconstraintscombinetoforcex1 =0 inanyfeasible,solution,wherebyz

= 0.

Wewilldualizeonthethefirsttwoconstraints,setting

X :={(x1, x2, x3) | (x1, x2) x3} .

Using multipliers u1, u2

for the first two constraints, our Lagrangianis:

L(x1, x2, x3, u1, u2) = x1+u1(x2+x3)+u2(x110)=10u2+(1u2, u1, u1)T(x1, x2, x3)

Then

L (u1, u2) = min 10u2

+ (1 u2, u1, u1)T(x1, x2, x3).

(x1,x2)x3

(i) Showthat:

10u2 if

(1

u2, u1) u1L (u1, u2) =

if (1 u2, u1)> u1

,


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andhencethedualproblemcanbewrittenas:

D : v = maximumu1,u2 10u2

s.t. (1 u2, u1) u1

u1 IR, u2 0 .

(ii) Showthatv =10,andthatthesetofoptimalsolutionsofDiscomprisedofthosevectors(u1, u2) = (, 1)forall0. HencebothPandDattaintheiroptimawithafinitedualitygap.

(iii) In this example the primal problem is a convex problem. Whyis

there

nevertheless

a

duality

gap?

What

hypotheses

are

absent

that

otherwisewouldguaranteestrongduality?

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