Convex-Analysis-and-Nonlinear-Optimization-第三章习题解答.pdf

STATEKEYLABORATORYOFCADCG,ZHEJIANGUNIVERSITY

ExerciseSolutionsforConvexAnalysis

WeizhongZhang,zhangweizhongzju@

1CHAPTER3FENCHELDUALITY

3.1SubgradientsandConvexFunctions

1.Proof

(1)

ifthefunctionfissublinear,theitisobviousthatfissubaditive.

f(x)f(x)(1)

alsof(x)f(1x)1f(x),

i.e.f(x)f(x)(2)

from(1)and(2)wehave

f(x)f(x)

sofispositivelyhomogenepus.

f(xy)f(x)f(y)f(x)f(y)

theproofcompleted.

(2)

Theonlythingweshouldproveisthat:

1

x,ylinfxylinf

f(xy)f(x)f(y)f(x)f(y)f(xy);

f(xy)f((xy))f(xy)

sof(xy)f(xy),i.e.xylinf

2.Proof

weshoulddeducethat0core(D)and0int(D)

itisveryeasytoverifyit.

3.Itisveryobvious.

4.Proof

¯¯¯

dC(x),wehaved,xxC(x)C(x)C(x)0,xE

¯

ifxC,thend,xx0

otherwisexcanbeanyvectorinE

¯¯

SoC(x)NC(x).

5.Proof

(a)

x(1)yx(1)y

sof(x)xisconvex.

{1d1},x0

f(x){1},x0

{1},x0

(b)

Todeduceconvex,weshouldconsiderthefollowingcases:

(1)xandyarebothnonnegitive,

(2)otherwise

{d0},x0

f(x){0},x0

,x0

(c)

2

f(x),x0

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