Conjugate Calculus Rules
g ∗ (y)
g(x)
Pm
i=1
Pm
fi (xi )
i=1
fi∗ (yi )
αf ∗ (y/α)
αf (x) (α > 0)
αf (x/α) (α > 0)
f∗
f (A(x − a)) + hb, xi + c
αf ∗ (y)
(AT )−1 (y − b) + ha, yi − c − ha, bi
Conjugate Functions
f∗
f
dom(f )
ex
R
− log x
R++
−1 − log(−y) (dom(f ∗ ) = R−− )
max{1 − x, 0}
R
y + δ[−1,0] (y)
1
|x|p
p
R
1
|y|q
q
p
− xp
R+
1 T
x Ax
2
+ bT x + c
Rn
1 T
x Ax
2
+ bT x + c
Rn
assumptions
y log y − y (dom(f ∗ ) = R+ )
−
(−y)q
q
p > 1, p1 +
0 < p < 1,
− b)T A−1 (y − b) − c
A ∈ Sn++ , b ∈ Rn , c ∈
R
1
(y
2
Rn
maxi {xi }
Rn
δ∆n (y)
δC (x)
C
σC (y)
C⊆E
σC (x)
E
δcl(conv(C)) (y)
C⊆E
kxk
p
− α2 − kxk2
p
α2 + kxk2
E
B[0, α]
δBk·k∗ [0,1] (y)
p
α kyk2∗ + 1
p
−α 1 − kyk2∗
(dom f ∗ = Bk·k∗ [0, 1])
1
kxk2
2
E
i=1 xi log xi
Pn
i=1 xi log xi
Pn
− i=1 log xi
P
log ( ni=1 exi )
1
kxk2
2
1
kxk2
2
Rn+
∆n
Rn++
E
+ δC (x)
C
− 21 d2C (x)
E
=1
(dom(f ∗ ) = R−− )
− b)T A† (y − b) − c
(dom(f ∗ ) = b + Range(A))
Pn yi −1
i=1 e
P
log ( ni=1 eyi )
P
−n − ni=1 log(−yi )
Pn
∗
i=1 yi log yi (dom(f ) = ∆n )
Pn
1
q
1
(y
2
1
p
+
1
q
=1
A ∈ Sn+ , b ∈ Rn , c ∈
R
α>0
α>0
1
kyk2∗
2
1
kyk2
2
− 12 d2C (y)
1
kyk2
2
+ δC (y)
∅ =
6 C ⊆ E, E Euclidean
∅=
6 C ⊆ E closed convex
Conjugates of Symmetric Spectral Functions over Sn
g(X)
dom(g)
g ∗ (Y)
dom(g ∗ )
λmax (X)
Sn
δΥn (Y)
Υn
αkXkF (α > 0)
Sn
δBk·kF [0,α] (Y)
Bk·kF [0, α]
αkXk2F (α > 0)
Sn
1
kYk2F
4α
Sn
αkXk2,2 (α > 0)
Sn
δBk·kS
[0,α] (Y)
Bk·kS1 [0, α]
αkXkS1 (α > 0)
Sn
δBk·k2,2 [0,α] (Y)
Bk·k2,2 [0, α]
− log det(X)
Sn++
Sn−−
λi (X) log(λi (X))
Sn+
−n − log det(−Y)
n
X
eλi (Y)−1
n
X
1
i=1
n
X
Sn
i=1
λi (X) log(λi (X))
Υn
Pn
log
i=1
eλi (Y)
Sn
i=1
Conjugates of Symmetric Spectral Functions over Rm×n
g ∗ (Y)
dom(g ∗ )
g(X)
dom(g)
ασ1 (X) (α > 0)
Rm×n
δBk·kS
[0,α] (Y)
Bk·kS1 [0, α]
αkXkF (α > 0)
Rm×n
δBk·kF [0,α] (Y)
Bk·kF [0, α]
αkXk2F (α > 0)
Rm×n
1
kYk2F
4α
Rm×n
αkXkS1 (α > 0)
Rm×n
δBk·kS
1
∞
[0,α] (Y)
Bk·kS∞ [0, α]