Strong Subdifferential Results
∂f (x)
f (x)
assumptions
Bk·k∗ [0, 1]
kxk

 X
kxk1

x
kxk2
o
max(x)
i∈I(x)
x 6= 0,
,
E = Rn , I(x) = {i :
max(x) = xi }
E = Rn , x = αe for some
α∈R
∅=
6 S⊆E
NS (x)
(
{y ∈ E∗ : kyk∗ ≤ hy, xi} , kxk ≤ 1,
δB[0,1] (x)
∅,
kxk > 1.
X
kAx + bk1
sgn (aTi x
(
kAx + bk2
[−ai , ai ]
i∈I0 (x)
AT (Ax+b)
,
kAx+bk2
AT Bk·k2 [0, 1],
Ax + b 6= 0,
λi sgn (aTi x + bi )ai :
i∈Ix
P
i∈Ix λi = 1
λi ≥ 0
AT Bk·k1 [0, 1]
kAx + bk∞

X

i∈I(x)
i∈I(x)
{x − PC (x)}
dC (x)
)
E = Rn , A ∈ Rm×n , b ∈
Rm , Ix = {i : kAx + bk∞ =
|aTi x + bi |}, Ax + b 6= 0
same as above but with
Ax + b = 0


X
λi ai :
λi = 1, λi ≥ 0

 n
o
 x−PC (x) ,
E = Rn , A ∈ Rm×n , b ∈
Rm , I6= (x) = {i : aTi x + bi 6=
0}, I0 (x) = {i : aTi x + bi =
0}
E = Rn , A ∈ Rm×n , b ∈
Rm
Ax + b = 0.
(
X
X
+ bi )ai +
i∈I6= (x)
dC (x)
E = Rn , I(x) = {i :
kxk∞ = |xi |}, x 6= 0
∆n
δS (x)
1
d (x)2
2 C
E = Rn
i∈I(x)
max(x)
maxi {aTi x +
b}
E = Rn , I6= (x) = {i : xi 6=
0}, I0 (x) = {i : xi = 0}.

 Bk·k [0, 1], x = 0.
2


X

λi = 1 
X

λi sgn (xi )ei : i∈I(x)


i∈I(x)

λi ≥ 0


X

X
λi e i :
λi = 1, λi ≥ 0


kxk∞
kAx + bk∞
[−ei , ei ]
i∈I0 (x)
 n

kxk2
X
sgn (xi )ei +
i∈I6= (x)
x=0


E = Rn ,ai ∈ Rn , bi ∈ R,
I(x) = {i : f (x) = aTi x+bi }
C - nonempty closed and
convex, E - Euclidean
x∈
/ C,
 NC (x) ∩ B[0, 1] x ∈ C.
C - nonempty closed and
convex, E - Euclidean