Convex Functions
This chapter is devoted to a class of functions from ℝ n into ℝ∪ +∞ called convex functions and to give a first important property of such functions. Any convex function is continuous on the interior of its domain if this one is nonempty. If the domain of
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This chapt er is devot ed to a class of fun cti ons from jRn into jR U { + oo} called convex fun ctions and to give a first important prop erty of such functions. Any convex fun cti on is cont inuous on the interior of it s dom ain if this on e is non empty. If the domain of a convex fun ction f has an empty interior, then the restriction of f t o t he affine set spa nned by it s domain is cont inuous on the relative interior of its dom ain (this expression makes sense b ecause the dom ain of a convex fun cti on is a convex set) . Another nice propert y of convex functions is that for any pointwise converging sequence of real-valu ed convex fun ctions defined on a relatively open set U , the limi t fun cti on of this sequence is the uniform limit on any compact subse t of U of this sequence . Our definition of convex fun cti ons is less genera l than t he one given by Rockafellar [51]. Rockafellar defines convexity for functi ons from jRn -> iR, by the convexity of their epigraph. But he shows and we will verify (Exercise 5.19) that for a fun cti on jRn -> jR U {+ oo} , his definition of convexity coincides wit h t he definit ion we will give below.
5.1 Basic definitions and properties Let f : A -> jR U {+ oo} be a fun ction defined from A, a nonempty set of R" , into jR U {+oo}. We require that A is convex. We say that f is convex on A if:
If f is real-valued then f is convex on A if:
The fun cti on
f is said to be strictly convex if
Va l E A , Va2 E A , V>.. E ]0, 1[, f (>.. al The doma in of f is dom(J ) := {x E is convex.
+ (1 jRn
>.. )a2) < >"f(at}
+ (1 -
>")f(a2)'
I f (x) < +oo }. Ob serve that dom(J)
M. Florenzano et al., Finite Dimensional Convexity and Optimization © Springer-Verlag Berlin Heidelberg 2001
74
5. Convex functions
The function f : A -+ JR U {-oo} is concave on A if - f is convex. Explicitly, f is concave on A if
If f is real-valued then f is concave on A if:
It is strictly concave on A if: Val E A ,Va2 E A, V>. E ]0, 1[, f(>.al + (1 - >.)a2) > >'f(al)
+ (1 - >.)f(a2).
Its domain will be: dom(J) = {x E JRn I f(x) > -oo} . We define the epigraph of a convex function f : A -+ JR U {+oo}: epi(J) := {(x, /L) E A x JR I /L 2: f(x)}. One can show that f : A -+ JR U {+oo} is convex if, and only if, epi(J) is convex. We have also the property that if f : A -+ JR U {+oo} is convex then, Va E JR, the level set lev'I) when>. = 0, we adopt the following rule : (i) if>' > then the function (>'I) is defined by (>'I)(x) = >'f(x), Vx E dom(J) and (>.f)(x) = +00 if x¢: dom(J), (ii) if>' = then (>'I) = 0, i.e. (>.f)(x) = 0, Vx E JRn. With this rule , on can easily check that a function f is convex on JRn if, and only if
° °
°.
Vx E JRn, Vy E JR n, V>. E [0,1], f(>.x + (1 - >.)y)
~
(>.f)(x)
+ ((1 -
>.)f)(y) .
Observe that we do not impose in this inequality>. to be in the interior of the interval [0,1], but we use the notation (>'I)(x) instead of >.j(x). In this chapter and the next chapters 6 and 7, we will adopt this rule.
5.1 Basic definitions and properties
75
Proposition 5.1.1
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