Iterative Optimization in Inverse Problems

Fenchel Duality 243

As for the fourth claim, if a ∈ S,then

y, a−max

s∈S

y, s

achieves its maximum value of zero at y =0,sof

∗

(a)=0.

Finally, we have

min

q∈Q

max

p∈P

p, Aq =min

q∈Q

f(Aq)=max

a∈S

g

∗

(a)=max

a∈S

min

q∈Q

p, Aq.

Therefore,

min

q∈Q

max

p∈P

p, Aq =max

p∈P

min

q∈Q

p, Aq.

17.4 Exercises

Ex. 17.1 Show that the exponential function f(x)=exp(x)=e

x

has con-

jugate

exp

∗

(a)=a log a − a, (17.12)

if a>0, 0 if a =0,and+∞ if a<0.

Ex. 17.2 Show that the function f(x)=−log x,forx>0, has the conju-

gate function f

∗

(a)=−1 − log(−a),fora<0.

Ex. 17.3 Show that the function f(x)=

|x|

p

p

has conjugate f

∗

(a)=

|a|

q

q

,

where p>0, q>0,and

1

p

+

1

q

=1. Therefore, the function f(x)=

1

2

x

2

2

is its ...