lecture 9: basic convex analysismath.xmu.edu.cn/group/nona/damc/lecture09.pdfbasic convex analysis...

Lecture 9 Basic Convex Analysis

April 22 - 24 2020

Basic Convex Analysis Lecture 9 April 22 - 24 2020 1 30

1 Notation

R = (minusinfin+infin) field of real numbers

Rn n-dimensional Euclidean space with inner product 〈middot middot〉Givens sets A and B A sube B denotes that A is a subset (possiblyequal to) B and A sub B means that A is a strict subset of B intAand clA denote the interior and the closure of A respectively

Given a norm 983042 middot 983042 its dual norm 983042 middot 983042lowast is defined as

983042z983042lowast = sup〈zx〉 | 983042x983042 le 1

We have983042x983042 = sup〈zx〉 | 983042z983042lowast le 1

ℓp norm (1 le p le infin)

983042x983042p =

983091

983107n983131

j=1

|xj |p983092

9831081p

p isin [1infin) 983042x983042infin = maxj

|xj |


Holderrsquos inequality

for p q isin [1infin] satisfying1

p+

1

q= 1

〈xy〉 le 983042x983042p983042y983042q

and moreover that 983042 middot 983042p and 983042 middot 983042q are dual norms

Generalized Cauchy-Schwarz inequality

for any pair of dual norms 983042 middot 983042 and 983042 middot 983042lowast

〈xy〉 le 983042x983042983042y983042lowast

Fenchel-Young inequality

for any pair of dual norms 983042 middot 983042 983042 middot 983042lowast and any η gt 0

〈xy〉 le η

2983042x9830422 + 1

2η983042y9830422lowast


2 Convex sets

The term ldquoconvexrdquo can be applied both to sets and to functions

A set C isin Rn is a convex set if the straight line segment connectingany two points in C lies entirely inside C Formally

forall xy isin C α isin [0 1] αx+ (1minus α)y isin C

Example A convex set (left) and a non-convex set (right)


21 Basic properties of convex sets

If α isin R and C is convex then

αC = αx x isin C

is convex

The Minkowski sum of two convex sets C1 and C2 defined by

C1 + C2 = x1 + x2 x1 isin C1x2 isin C2

is convex

If αi isin R and all Ci are convex then

C =

m983131

i=1

αiCi

is convex


If Cα are convex sets for each α isin A where A is an arbitrary indexset then the intersection

C =983135

αisinACα

is convex

The convex hull of a set of points x1 middot middot middot xm isin Rn defined by

convx1 middot middot middot xm =

983083m983131

i=1

λixi λi ge 0

m983131

i=1

λi = 1

983084

is convex


Theorem 1 (Projection onto closed convex sets)

Let C be a closed convex set and x isin Rn Then there is a unique pointπC(x) called the projection of x onto C such that

983042xminus πC(x)9830422 = infyisinC

983042y minus x9830422

that isπC(x) = argmin

yisinC983042y minus x98304222

A point z = πC(x) is the projection of x onto C if and only if

〈xminus zy minus z〉 le 0

for all y isin C


Projection of the point x onto the set C (with projection πC(x))exhibiting 〈xminus πC(x)y minus πC(x)〉 le 0


Corollary 2 (Nonexpansiveness)

Projections onto convex sets are nonexpansive in particular

983042πC(x)minus y9830422 le 983042xminus y9830422

for any x isin Rn and y isin C

Theorem 3 (Strict separation of points)

Let C be a closed convex set Given any point x isin C there is a vector vsuch that

〈vx〉 gt supyisinC

〈vy〉

Moreover we can take the vector v = xminus πC(x) and

〈vx〉 ge supyisinC

〈vy〉+ 983042v98304222


Separation of the point x from C by the vector v = xminus πC(x)


For nonempty sets S1 and S2 satisfying S1 cap S2 = empty if there existvector v ∕= 0 and scalar b such that

〈vx〉 ge b for all x isin S1

and〈vx〉 le b for all x isin S2

thenx isin Rn | 〈vx〉 = b

is called a separating hyperplane for nonempty sets S1 and S2

Theorem 4 (Strict separation of convex sets)

Let C1 C2 be closed convex sets with C2 compact and C1 cap C2 = empty Thenthere is a vector v such that

infxisinC1

〈vx〉 gt supxisinC2

〈vx〉


Strict separation of convex sets


For a set S and x isin bdS = clS intS if vector v ∕= 0 satisfies

〈vx〉 ge 〈vy〉 for all y isin S

thenz isin Rn | vT(zminus x) = 0

is called a supporting hyperplane supporting S at x

Theorem 5 (Supporting hyperplane theorem)

For convex set C and any x isin bdC theres exists a supportinghyperplane supporting C at x ie exist v ∕= 0 satisfying

〈vx〉 ge 〈vy〉 for all y isin C


Supporting hyperplanes to a convex set


Is a supporting hyperplane supporting C at x unique

Theorem 6 (Halfspace intersections)

Let C sub Rn be a closed convex set Then C is the intersection of all thehalfspaces containing it


3 Convex functions

A function f Rn 983041rarr R cup +infin is a convex function if its domaindom(f) is convex and for all xy isin dom(f) α isin [0 1]

f(αx+ (1minus α)y) le αf(x) + (1minus α)f(y)

(strictly convex means lt)The epigraph of a function f is defined as

epif = (x t) f(x) le tA function is convex if and only if its epigraph is a convex setExample Convex function f(x) = maxx2minus2xminus 02


Theorem 7 (First-order convexity condition)

Differentiable f is convex if and only if dom(f) is convex and

f(y) ge f(x) +nablaf(x)T(y minus x) forall xy isin dom(f)

Theorem 8 (Second-order convexity conditions)

Assume f is twice continuously differentiable Then f is convex if andonly if dom(f) is convex and

nabla2f(x) ≽ 0 forall x isin dom(f)

that is nabla2f(x) is positive semidefinite

A function f is called closed if its epigraph is a closed set

Theorem 9 (Continuous over closed domain rArr closedness)

Suppose f Rn 983041rarr R cup +infin is continuous over its domain anddom(f) is closed Then f is closed


Lemma 10 (Convexity + compactness rArr boundedness)

Let f be convex and defined on the ℓ1 ball in n dimensions

B1 = x isin Rn 983042x9830421 le 1

Then there exist minusinfin lt m le M lt infin such that

m le f(x) le M forall x isin B1

More general convex f on a compact domain is bounded

Theorem 11 (Convexity + compactness rArr L-continuity)

Let f be convex and defined on a convex set C with non-empty interiorLet B sube intC be compact Then there is a constant L such that

|f(x)minus f(y)| le L983042xminus y983042

on B that is f is L-Lipschitz continuous on B


Definition The directional derivative of a function f at a point xin the direction d is

f prime(xd) = limαrarr0+

f(x+ αd)minus f(x)

α

Theorem 12 (Convexity rArr existence of directional derivative)

For convex f at any point x isin intdom(f) and for any d thedirectional derivative f prime(xd) exists and is

f prime(xd) = infαgt0

f(x+ αd)minus f(x)

α

Moreover g(d) = f prime(xd) is convex and there exists a constant L lt infinsuch that

|g(d)| = |f prime(xd)| le L983042d983042

for any d isin Rn If f is Lipschitz continuous with respect to the norm983042 middot 983042 we can take L to be the Lipschitz constant of f


Theorem 13 (Any local minimizer of a convex function is global)

Let f Rn 983041rarr R be convex and x be a local minimizer of f (resp a localminimizer of f over a convex set C) Then x is a global minimizer of f(resp a global minimizer of f over C)

Proof If x is a local minimizer of f over a convex set C then for anyy isin C we have for small enough t gt 0 that

f(x) le f(x+ t(y minus x)) or 0 le f(x+ t(y minus x))minus f(x)

t

We now use the criterion of increasing slopes that is for any convexfunction f and any u isin Rn the function φ(t)

φ(t) =f(x+ tu)minus f(x)

t

is increasing in t gt 0 Therefore forallt gt 0 we have

0 le f(x+ t(y minus x))minus f(x)

t

Setting t = 1 yields f(x) le f(y) for any y isin CBasic Convex Analysis Lecture 9 April 22 - 24 2020 20 30

The slopesf(x+ t)minus f(x)

tincrease with t1 lt t2 lt t3


4 Subgradient and subdifferential

Definition A vector g isin Rn is a subgradient of f at a point x if

f(y) ge f(x) + 〈gy minus x〉 for all y isin Rn

The subdifferential denoted partf(x) is the set of all subgradients off at x

Example g1 = nablaf(x1) g2g3 isin partf(x2)


Examples

part983042x983042 =

983083g isin Rn 983042g983042lowast = 1 〈gx〉 = 983042x983042 if x ∕= 0

g isin Rn 983042g983042lowast le 1 if x = 0

For 983042x9830422 we have part983042x9830422 =983083x983042x9830422 if x ∕= 0

g isin Rn 983042g9830422 le 1 if x = 0

For case n = 1 we have part|x| =

983099983105983103

983105983101

minus1 if x lt 0

[minus1 1] if x = 0

1 if x gt 0

Theorem 14 (Nonemptiness closedness convexity boundedness ofthe subdifferential set at interior points of the domain)

Suppose f is convex Let x isin intdom(f) Then partf(x) is nonemptyclosed convex and bounded


Theorem 15 (Nonemptiness of subdifferential sets rArr convexity)

Let f Rn 983041rarr R cup +infin be proper and assume that dom(f) is convexSuppose that for any x isin dom(f) the set partf(x) is nonempty Then fis convex

Theorem 16 (First-order characterizations of strong convexity)

Let f Rn 983041rarr R cup +infin be proper closed and convex Then for a givenγ gt 0 the following three claims are equivalent

(i) f is γ-strongly convex

(ii) For any x satisfying partf(x) ∕= empty y isin dom(f) and g isin partf(x)

f(y) ge f(x) + 〈gy minus x〉+ γ

2983042y minus x9830422

(iii) For any x and y satisfying partf(x) ∕= empty partf(y) ∕= empty and gx isin partf(x)gy isin partf(y)

〈gx minus gyxminus y〉 ge γ983042xminus y9830422


Theorem 17 (Equivalent characterization of subdifferential)

An equivalent characterization of the subdifferential partf(x) of convex fat x is

partf(x) = g 〈gd〉 le f prime(xd) forall d isin Rn

Theorem 18 (Max formula of directional derivative)

Suppose f is closed convex and partf(x) ∕= empty Then

f prime(xd) = supgisinpartf(x)

〈gd〉

Theorem 19 (Subgradient bounded by Lipschitz constant)

Suppose that convex function f is L-Lipschitz continuous with respectto the norm 983042 middot 983042 over a set C where C sub intdom(f) Then

sup983042g983042lowast g isin partf(x)x isin C le L


Theorem 20 (Minimizer of convex function over convex set)

Let f be convex The point x983183 isin intdom(f) minimizes f over a closedconvex set C if and only if there exists a subgradient g isin partf(x983183) suchthat 〈gy minus x983183〉 ge 0 for all y isin C

The point x983183 minimizes f

over C(the shown level curves)

Active case x983183 isin bdCminusg supporting hyperplane

Inactive case x983183 isin intCg = 0 rArr 0 isin partf(x983183)


5 Calculus rules with subgradients

Scaling

If h(x) = αf(x) for some α ge 0 then parth(x) = αf(x)

Finite sums

Suppose that fi i = 1 m are convex functions and let f =

m983131

i=1

fi

If x isin intdom(fi) i = 1 m then partf(x) =

m983131

i=1

partfi(x)

Exercise x isin Rm 983042x9830421 =983123m

i=1 fi(x) fi(x) = |xi| part983042x9830421 =

Affine transformations

Let f Rm 983041rarr R be convex and A isin Rmtimesn and b isin Rm Thenh Rn 983041rarr R defined by h(x) = f(Ax+ b) is convex and hassubdifferential

parth(x) = ATpartf(Ax+ b)

Exercises (1) proof (2) part983042Ax+ b9830421= (3) part983042Ax+ b9830422 =


Maximum of a finite collection of convex functions

Let fi i = 1 m be convex functions and f(x) = max1leilem

fi(x)

Then we haveepif =

983135

1leilem

epifi

which is convex and therefore f is convex

If x isin intdom(fi) i = 1 m then the subdifferential partf(x) is theconvex hull of the subgradients of active functions (those attainingthe maximum) at x that is

partf(x) = convpartfi(x) fi(x) = f(x)

If there is only a single unique active function fi then

partf(x) = partfi(x)


Exercise x isin Rm 983042x983042infin = max1leilem

fi(x) fi(x) = |xi| part983042x983042infin =

Exercise

f(x) = maxf1(x) f2(x) f1(x) = x2 f2(x) = minus2xminus 15

x0 = minus1 +983155

45 partf(x0) =


Supremum of an infinite collection of convex functions

Considerf(x) = sup

αisinAfα(x)

where A is an arbitrary index set and fα is convex for each α

If the supremum is attained then

partf(x) supe convpartfα(x) fα(x) = f(x)

If the supremum is not attained the function f may not besubdifferentiable at x


1 Notation

R = (minusinfin+infin) field of real numbers

Rn n-dimensional Euclidean space with inner product 〈middot middot〉Givens sets A and B A sube B denotes that A is a subset (possiblyequal to) B and A sub B means that A is a strict subset of B intAand clA denote the interior and the closure of A respectively

Given a norm 983042 middot 983042 its dual norm 983042 middot 983042lowast is defined as

983042z983042lowast = sup〈zx〉 | 983042x983042 le 1

We have983042x983042 = sup〈zx〉 | 983042z983042lowast le 1

ℓp norm (1 le p le infin)

983042x983042p =

983091

983107n983131

j=1

|xj |p983092

9831081p

p isin [1infin) 983042x983042infin = maxj

|xj |




p+

1

q= 1

〈xy〉 le 983042x983042p983042y983042q




〈xy〉 le 983042x983042983042y983042lowast



〈xy〉 le η

2983042x9830422 + 1

2η983042y9830422lowast


2 Convex sets








αC = αx x isin C

is convex



is convex


C =

m983131

i=1

αiCi

is convex



C =983135

αisinACα

is convex



983083m983131

i=1

λixi λi ge 0

m983131

i=1

λi = 1

983084

is convex





983042y minus x9830422





for all y isin C











〈vy〉



〈vy〉+ 983042v98304222











infxisinC1


〈vx〉


















3 Convex functions



















B1 = x isin Rn 983042x9830421 le 1











f(x+ αd)minus f(x)

α




f(x+ αd)minus f(x)

α


|g(d)| = |f prime(xd)| le L983042d983042







t



t



t











Examples

part983042x983042 =




g isin Rn 983042g9830422 le 1 if x = 0


983099983105983103

983105983101

minus1 if x lt 0

[minus1 1] if x = 0

1 if x gt 0











2983042y minus x9830422










〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)








p+

1

q= 1

〈xy〉 le 983042x983042p983042y983042q




〈xy〉 le 983042x983042983042y983042lowast



〈xy〉 le η

2983042x9830422 + 1

2η983042y9830422lowast


2 Convex sets








αC = αx x isin C

is convex



is convex


C =

m983131

i=1

αiCi

is convex



C =983135

αisinACα

is convex



983083m983131

i=1

λixi λi ge 0

m983131

i=1

λi = 1

983084

is convex





983042y minus x9830422





for all y isin C











〈vy〉



〈vy〉+ 983042v98304222











infxisinC1


〈vx〉


















3 Convex functions



















B1 = x isin Rn 983042x9830421 le 1











f(x+ αd)minus f(x)

α




f(x+ αd)minus f(x)

α


|g(d)| = |f prime(xd)| le L983042d983042







t



t



t











Examples

part983042x983042 =




g isin Rn 983042g9830422 le 1 if x = 0


983099983105983103

983105983101

minus1 if x lt 0

[minus1 1] if x = 0

1 if x gt 0











2983042y minus x9830422










〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)






2 Convex sets








αC = αx x isin C

is convex



is convex


C =

m983131

i=1

αiCi

is convex



C =983135

αisinACα

is convex



983083m983131

i=1

λixi λi ge 0

m983131

i=1

λi = 1

983084

is convex





983042y minus x9830422





for all y isin C











〈vy〉



〈vy〉+ 983042v98304222











infxisinC1


〈vx〉


















3 Convex functions



















B1 = x isin Rn 983042x9830421 le 1











f(x+ αd)minus f(x)

α




f(x+ αd)minus f(x)

α


|g(d)| = |f prime(xd)| le L983042d983042







t



t



t











Examples

part983042x983042 =




g isin Rn 983042g9830422 le 1 if x = 0


983099983105983103

983105983101

minus1 if x lt 0

[minus1 1] if x = 0

1 if x gt 0











2983042y minus x9830422










〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)








αC = αx x isin C

is convex



is convex


C =

m983131

i=1

αiCi

is convex



C =983135

αisinACα

is convex



983083m983131

i=1

λixi λi ge 0

m983131

i=1

λi = 1

983084

is convex





983042y minus x9830422





for all y isin C











〈vy〉



〈vy〉+ 983042v98304222











infxisinC1


〈vx〉


















3 Convex functions



















B1 = x isin Rn 983042x9830421 le 1











f(x+ αd)minus f(x)

α




f(x+ αd)minus f(x)

α


|g(d)| = |f prime(xd)| le L983042d983042







t



t



t











Examples

part983042x983042 =




g isin Rn 983042g9830422 le 1 if x = 0


983099983105983103

983105983101

minus1 if x lt 0

[minus1 1] if x = 0

1 if x gt 0











2983042y minus x9830422










〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)







C =983135

αisinACα

is convex



983083m983131

i=1

λixi λi ge 0

m983131

i=1

λi = 1

983084

is convex





983042y minus x9830422





for all y isin C











〈vy〉



〈vy〉+ 983042v98304222











infxisinC1


〈vx〉


















3 Convex functions



















B1 = x isin Rn 983042x9830421 le 1











f(x+ αd)minus f(x)

α




f(x+ αd)minus f(x)

α


|g(d)| = |f prime(xd)| le L983042d983042







t



t



t











Examples

part983042x983042 =




g isin Rn 983042g9830422 le 1 if x = 0


983099983105983103

983105983101

minus1 if x lt 0

[minus1 1] if x = 0

1 if x gt 0











2983042y minus x9830422










〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)









983042y minus x9830422





for all y isin C











〈vy〉



〈vy〉+ 983042v98304222











infxisinC1


〈vx〉


















3 Convex functions



















B1 = x isin Rn 983042x9830421 le 1











f(x+ αd)minus f(x)

α




f(x+ αd)minus f(x)

α


|g(d)| = |f prime(xd)| le L983042d983042







t



t



t











Examples

part983042x983042 =




g isin Rn 983042g9830422 le 1 if x = 0


983099983105983103

983105983101

minus1 if x lt 0

[minus1 1] if x = 0

1 if x gt 0











2983042y minus x9830422










〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)















〈vy〉



〈vy〉+ 983042v98304222











infxisinC1


〈vx〉


















3 Convex functions



















B1 = x isin Rn 983042x9830421 le 1











f(x+ αd)minus f(x)

α




f(x+ αd)minus f(x)

α


|g(d)| = |f prime(xd)| le L983042d983042







t



t



t











Examples

part983042x983042 =




g isin Rn 983042g9830422 le 1 if x = 0


983099983105983103

983105983101

minus1 if x lt 0

[minus1 1] if x = 0

1 if x gt 0











2983042y minus x9830422










〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)















infxisinC1


〈vx〉


















3 Convex functions



















B1 = x isin Rn 983042x9830421 le 1











f(x+ αd)minus f(x)

α




f(x+ αd)minus f(x)

α


|g(d)| = |f prime(xd)| le L983042d983042







t



t



t











Examples

part983042x983042 =




g isin Rn 983042g9830422 le 1 if x = 0


983099983105983103

983105983101

minus1 if x lt 0

[minus1 1] if x = 0

1 if x gt 0











2983042y minus x9830422










〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)






















3 Convex functions



















B1 = x isin Rn 983042x9830421 le 1











f(x+ αd)minus f(x)

α




f(x+ αd)minus f(x)

α


|g(d)| = |f prime(xd)| le L983042d983042







t



t



t











Examples

part983042x983042 =




g isin Rn 983042g9830422 le 1 if x = 0


983099983105983103

983105983101

minus1 if x lt 0

[minus1 1] if x = 0

1 if x gt 0











2983042y minus x9830422










〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)



















B1 = x isin Rn 983042x9830421 le 1











f(x+ αd)minus f(x)

α




f(x+ αd)minus f(x)

α


|g(d)| = |f prime(xd)| le L983042d983042







t



t



t











Examples

part983042x983042 =




g isin Rn 983042g9830422 le 1 if x = 0


983099983105983103

983105983101

minus1 if x lt 0

[minus1 1] if x = 0

1 if x gt 0











2983042y minus x9830422










〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)








f(x+ αd)minus f(x)

α




f(x+ αd)minus f(x)

α


|g(d)| = |f prime(xd)| le L983042d983042







t



t



t











Examples

part983042x983042 =




g isin Rn 983042g9830422 le 1 if x = 0


983099983105983103

983105983101

minus1 if x lt 0

[minus1 1] if x = 0

1 if x gt 0











2983042y minus x9830422










〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)










t



t



t











Examples

part983042x983042 =




g isin Rn 983042g9830422 le 1 if x = 0


983099983105983103

983105983101

minus1 if x lt 0

[minus1 1] if x = 0

1 if x gt 0











2983042y minus x9830422










〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)















Examples

part983042x983042 =




g isin Rn 983042g9830422 le 1 if x = 0


983099983105983103

983105983101

minus1 if x lt 0

[minus1 1] if x = 0

1 if x gt 0











2983042y minus x9830422










〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)













2983042y minus x9830422










〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)












〈gd〉













Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)














Scaling


Finite sums


m983131

i=1

fi


m983131

i=1

partfi(x)


i=1 fi(x) fi(x) = |xi| part983042x9830421 =








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)








fi(x)

Then we haveepif =

983135

1leilem

epifi









Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)








Exercise


x0 = minus1 +983155

45 partf(x0) =



Considerf(x) = sup

αisinAfα(x)







Considerf(x) = sup

αisinAfα(x)






lecture 9: basic convex analysismath.xmu.edu.cn/group/nona/damc/lecture09.pdfbasic convex analysis...

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