Convex sets and convex functions taking the infinity value

If C is convex, then its interior int C and its closureC are convex Convexity is preserved by the following operations : Let I be an arbitrary set... Convex combination and convex hull o

Chapter 1.

Convex sets and convex functions taking the infinity value

Convex set

Definition A subset C of IRn is convexif

∀x, y ∈ C ∀t ∈ [0, 1] tx + (1 − t)y ∈ CProposition If C is convex, then its interior int C and its closureC

are convex

Convexity is preserved by the following operations :

Let I be an arbitrary set If Ci ⊆ IRn, i ∈ I , are convex, then

C = ∩i ∈ICi is convex

Let C and D be two convex sets in IRn and let a and b be two realnumbers Then the following set is convex:

aC + bD := {ac + bd | c ∈ C , d ∈ D}

Examples of convex sets

The following are some examples of convex sets :

(1) Hyperplane: S = {x |pTx = α}, where p is a nonzero vector in IRn,called the normal to the hyperplane, and α is a scalar

(2) Half-space: S = {x |pTx ≤ α}, where p is a nonzero vector in IRn,and α is a scalar

(3) Open half-space: S = {x |pTx < α}, where p is a nonzero vector in

IRn and α is a scalar

(4) Polyhedral set : S = {x |Ax ≤ b}, where A is an m × n matrix, and b

is an m vector (Here the inequality should be interpreted

elementwise.)

Examples of convex sets

(5) Polyhedral cone: S = {x |Ax ≤ 0}, where A is an m × n matrix

(6) Cone spanned by a finite number of vectors :

S = {x |x =Pm

j =1λjaj|λj ≥ 0, j = 1, , m}, where a1, , am aregiven vectors in IRn

(7) Neighborhood: Nε(¯x ) = {x ∈ IRn|kx − ¯x k < ε}, where ¯x is a fixedvector in IRn and ε > 0

Convex cone

Some of the geometric optimality conditions that we will study use convexcones

Definition A nonempty set C in IRn is called a cone with vertex zero

if x ∈ C implies that αx ∈ C for all α ≥ 0 If, in addition, C is convex,then C is called a convex cone

0

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Convex combination and convex hull of a set

Definition x is said to bea convex combination of x1, , xm if thereexist α1 ≥ 0, , αm ≥ 0 such that

x = α1x1+ · · · + αmxm, and α1+ · · · + αm= 1

The convex hull of C (denoted conv C ) is the intersection of all convexsubsets containing C

Proposition (Carath´eodory’s lemma) Let C ⊆ IRn Then each

element of conv C is a convex combination of at most n + 1 points of C

Illustration

Closed convex hull

Remark The convex hull of an open subset is open But the convex hull

of a closed set is not necessarily closed

Example Let C = {(0, 0)} ∪ {(x , y ) | x ≥ 0, xy = 1} Then

conv C = {(0, 0)} ∪ {(x , y ) | x > 0, y > 0} is not closed

See the figure below

So we have to use the closed convex hull

Closed convex hull

Definition The closed convex hull of a subset C of IRn is the

smallest closed convex subset containing C It is denoted conv C

Proposition The closed convex hull of a subset C is equal to theclosure of its convex hull, i.e.,

conv C = conv C

Proposition The convex hull of a bounded set is bounded The

convex hull of a compact set is compact

Domain and Epigraph

Definition Let f : IRn→ IR ∪ {+∞} The domain of f is the set

dom f = { x ∈ IRn| f (x) < +∞ }

The function f is proper if dom f is nonempty

If f is proper, then the epigraph of f is the nonempty set defined byepi f = {(x , r )|f (x ) ≤ r }

R

x z y

Example The indicator function

Let S be a nonempty subset of IRn The indicator function of S is definedby

δS : IRn→ IR ∪ {+∞} δS(x ) =



0 if x ∈ S+∞ otherwise

The indicator function δS is a proper function whose domain is S

Since epi δS = S × IR+, we have that δS is convex if and only if S isconvex Moreover,

f is proper and convex

S ⊆ IRn nonempty and convex

dom f ∩ S 6= ∅







=⇒ f + δS is proper and convex

We have a correspondencebetween convex sets andconvex functions:

Operations preserving convexity

Proposition Let C be a convex subset of IRn and let f1, , fm beconvex functions Let also and w1, , wm ≥ 0 Then

w1f1+ · · · + wmfm andmax1≤i ≤mfi(x )are convex functions

More generally, let {fi}i ∈I be a family of convex functions Then

f = supi ∈Ifi is a convex function

Passing from sets to functions

A common device to construct convex functions on IRn is to construct aconvex set F in IRn+1 and then take the function whose graph is the lowerboundary of F in the following sense

Proposition Let F be any convex set in IRn+1 and let

f (x ) = inf{µ | (x , µ) ∈ F }

Then f is a convex function

Example Let fi, i ∈ I be proper convex functions on IRn Then

F = ∩i ∈Iepi fi is convex and F is the epigraph of the convex function

f = supi ∈Ifi

Let us introduce another operation called the infimal convolution

Infimal convolution

We introduce the functional operation which corresponds to the addition

of epigraphs as sets in IRn+1 Let f1, f2 be proper convex functions on IRn.Let Fi = epi fi and let F = F1+ F2 Then F is convex

(f1⊕ f2)(x ) := inf{f1(x1) + f2(x2) : x1+ x2= x }

Infimal Convolution

Proposition Let f1 and f2 be two proper convex functions If theyhave a common affine lower bound : for some (s, b) ∈ IRn× IR,

fj(x ) ≥ hs, x i − b for j = 1, 2 and all x ∈ IRn,

then their infimal convolution is also proper and convex Furthermore

epis(f1⊕ f2) = epis(f1) + epis(f2)

where epis(f ) = {(x , r ) ∈ IRn× IR | f (x) < r }

Convex hull of a function

Let f : IRn→ IR ∪ {+∞} be proper and minorized by an affine function,i.e., there exists (s, b) ∈ IRn× IR such that

f (x ) ≥ < s, x > −b for all x ∈ IRn.Let F = conv epi f andg (x ) = inf{r : (x , r ) ∈ conv epi f } Then g isconvex It is denoted conv f and is called the convex hull of f

Proposition The convex hull of f coincides with the following twofunctions g1 and g2 on IRn :

g1(x ) = sup{h(x ) : h is proper and convex, h ≤ f }

g2(x ) = inf{Pk

j =1αjf (xj) : k = 1, 2, ;

α ∈ ∆k, xj ∈ dom f ,Pk

j =1αjxj = x }

Other concepts of convexity

Definition A function f : IRn→ IR ∪ {+∞} isstrictly convex if it isconvex and

A strongly convex function is strictly convex but the converse is not true(example : f (x ) = x4)

Sublevel sets

Definition Let f : IRn→ IR ∪ {+∞} and r ∈ IR ∪ {+∞} The

sublevel set of f at level r is the set (possibly empty)

Sr(f ) = {x ∈ IRn| f (x) ≤ r }