26.8 Bibliographic remarks 298
andclosedif its complementAc=Rd\Ais open. TheboundaryofAis
denoted by∂Aand is the set of points inx∈Rdsuch that for allε >0 the
setBε(x) contains points fromAandAc. Note that points in the boundary
need not be inA. Some examples:∂[0,∞) ={ 0 }=∂(0,∞) and∂Rn=∅.
26.8 Bibliographic remarks
The main source for these notes is the excellent book by Rockafellar [2015]. The
basic definitions are in Part I. The Fenchel dual is analyzed in Part III while
Legendre functions are found in Part V. Convex optimization is a huge topic.
The standard text is by Boyd and Vandenberghe [2004].
26.9 Exercises
26.1 Prove Jensen’s inequality (Theorem 26.2). Precisely, letX∈Rdbe a
random variable for whichE[X] exists andf:Rd→R ̄a convex function. Prove
thatE[f(x)]≥f(E[X]).
Hint Letx 0 =E[X]∈Rdand define a linear functiong:Rd→Rsuch that
g(x 0 ) =f(x 0 ) andg(x)≤f(x) for allx∈Rd. To guarantee the existence ofgyou
may find thesupporting hyperplane theorem, which states that ifS⊂Rm
is a convex set ands∈∂Sthen there exists a supporting hyperplane containing
s.
26.2Letf:Rd→R∪{−∞,∞}.
(a) Prove thatf∗∗(x)≤f(x).
(b)Assume thatf is convex and differentiable on int(dom(f)). Show that
f∗∗(x) =f(x) forx∈int(dom(f)).
As mentioned in the text, the assumption thatfbe differentiable can be
relaxed to an assumption that the epigraph offis closed in which case the
result holds over the whole domain. The proof is not hard, but you will need
to use the subdifferential rather than the gradient and the boundary must
be treated with care.
26.3For each of the real-valued functions below decide whether or not it is
Legendre on the given domain.
(a)f(x) =x^2 on [− 1 ,1].
(b)f(x) =−
√
xon [0,∞).
(c)f(x) = log(1/x) on [0,∞) withf(0) =∞.
(d)f(x) =xlog(x) on [0,∞) withf(0) = 0.
