对于任意
α
∈
M
+
?
(
X
)
,
ε
>
0
\alpha \in \mathcal{M}^{+*} \left( \mathcal{X} \right), \varepsilon > 0
α∈M+?(X),ε>0,Softmin operator定义为
对于任意
f
∈
C
(
X
)
f \in \mathcal{C} \left( \mathcal{X} \right)
f∈C(X)
Smin
?
α
ε
(
f
)
?
?
ε
log
?
?
α
,
exp
?
(
?
f
ε
)
?
\operatorname{Smin}_{\alpha}^{\varepsilon} \left( f \right) \triangleq - \varepsilon \log \left\langle \alpha, \exp \left( -\frac{f}{\varepsilon} \right) \right\rangle
Sminαε?(f)??εlog?α,exp(?εf?)?
( α n ? α ?and? f n ? ∥ ? ∥ ∞ f ) ? Smin ? α n ε ( f n ) → Smin ? α ε ( f ) \left(\alpha_n \rightharpoonup \alpha \text { and } f_n \stackrel{\|\cdot\|_{\infty}}{\longrightarrow} f\right) \Longrightarrow \operatorname{Smin}_{\alpha_n}^{\varepsilon}\left(f_n\right) \rightarrow \operatorname{Smin}_\alpha^{\varepsilon}(f) (αn??α?and?fn??∥?∥∞??f)?Sminαn?ε?(fn?)→Sminαε?(f)
?
α
∈
M
1
+
(
X
)
,
?
α
,
f
?
?
ε
→
+
∞
Smin
?
α
ε
(
f
)
?
ε
→
0
min
?
x
∈
Supp
?
(
α
)
f
(
x
)
\forall \alpha \in \mathcal{M}_1^{+}(\mathcal{X}),\langle\alpha, f\rangle \stackrel{\varepsilon \rightarrow+\infty}{\longleftarrow } \operatorname{Smin}_\alpha^{\varepsilon}(f) \stackrel{\varepsilon \rightarrow 0}{\longrightarrow} \min _{x \in \operatorname{Supp}(\alpha)} f(x)
?α∈M1+?(X),?α,f??ε→+∞?Sminαε?(f)?ε→0?x∈Supp(α)min?f(x)
?
(
f
,
g
)
∈
C
(
X
)
,
f
?
g
?
Smin
?
α
ε
(
f
)
?
Smin
?
α
ε
(
g
)
\forall(f, g) \in \mathcal{C}(\mathcal{X}), f \leqslant g \Longrightarrow \operatorname{Smin}_\alpha^{\varepsilon}(f) \leqslant \operatorname{Smin}_\alpha^{\varepsilon}(g)
?(f,g)∈C(X),f?g?Sminαε?(f)?Sminαε?(g)
?
K
∈
R
,
Smin
?
α
ε
(
f
+
K
)
=
Smin
?
α
ε
(
f
)
+
K
\forall K \in \mathbb{R}, \operatorname{Smin}_\alpha^{\varepsilon}(f+K)=\operatorname{Smin}_\alpha^{\varepsilon}(f)+K
?K∈R,Sminαε?(f+K)=Sminαε?(f)+K
对于任意的
α
∈
M
+
?
(
X
)
\alpha \in \mathcal{M}^{+*} \left( \mathcal{X} \right)
α∈M+?(X),Softmin是1-Lipschitz，即
?
(
f
,
g
)
∈
C
(
X
)
,
∣
Smin
?
α
ε
(
f
)
?
Smin
?
α
ε
(
g
)
∣
≤
∥
f
?
g
∥
∞
\forall \left( f, g \right) \in \mathcal{C} \left( \mathcal{X} \right), \quad \left| \operatorname{Smin}_{\alpha}^{\varepsilon} \left( f \right) - \operatorname{Smin}_{\alpha}^{\varepsilon} \left( g \right) \right| \le \| f - g \|_{\infty}
?(f,g)∈C(X),∣Sminαε?(f)?Sminαε?(g)∣≤∥f?g∥∞?
证明：
令
u
t
=
t
(
g
?
f
)
+
f
u_t = t \left( g - f \right) + f
ut?=t(g?f)+f,其中
t
∈
[
0
,
1
]
t \in \left[ 0, 1 \right]
t∈[0,1]
∣
Smin
?
α
ε
(
f
)
?
Smin
?
α
ε
(
g
)
∣
=
∣
∫
0
1
d
d
t
Smin
?
α
ε
(
u
t
)
d
t
∣
=
∣
∫
0
1
?
α
,
(
g
?
f
)
exp
?
(
u
t
ε
)
?
α
,
exp
?
(
u
t
ε
)
?
?
d
t
∣
≤
∫
0
1
∣
?
α
,
(
g
?
f
)
exp
?
(
u
t
ε
)
?
α
,
exp
?
(
u
t
ε
)
?
?
∣
d
t
≤
∫
0
1
∣
?
α
,
∥
f
?
g
∥
∞
exp
?
(
u
t
ε
)
?
α
,
exp
?
(
u
t
ε
)
?
?
∣
d
t
=
∥
f
?
g
∥
∞
∫
0
1
∣
?
α
,
exp
?
(
u
t
ε
)
?
α
,
exp
?
(
u
t
ε
)
?
?
∣
d
t
=
∥
f
?
g
∥
∞
\begin{aligned} \quad \left| \operatorname{Smin}_{\alpha}^{\varepsilon} \left( f \right) - \operatorname{Smin}_{\alpha}^{\varepsilon} \left( g \right) \right| &= \quad \left| \int_{0}^{1} \frac{d}{dt}\operatorname{Smin}_{\alpha}^{\varepsilon} \left( u_{t} \right) \mathrm{d}t \right|= \left| \int_{0}^{1} \left\langle \alpha, \left( g - f \right) \frac{\exp \left( \frac{u_{t}}{\varepsilon} \right) }{\left\langle \alpha, \exp \left( \frac{u_{t}}{\varepsilon} \right)\right\rangle } \right\rangle \mathrm{d}t \right| \\ &\le \int_{0}^{1} \left| \left\langle \alpha, \left( g - f \right) \frac{\exp \left( \frac{u_{t}}{\varepsilon} \right) }{\left\langle \alpha, \exp \left( \frac{u_{t}}{\varepsilon} \right)\right\rangle } \right\rangle \right| \mathrm{d}t \\ &\le \int_{0}^{1} \left| \left\langle \alpha, \| f - g \|_{\infty} \frac{\exp \left( \frac{u_{t}}{\varepsilon} \right) }{\left\langle \alpha, \exp \left( \frac{u_{t}}{\varepsilon} \right)\right\rangle } \right\rangle \right| \mathrm{d}t\\ &= \| f - g \|_{\infty}\int_{0}^{1} \left| \left\langle \alpha, \frac{\exp \left( \frac{u_{t}}{\varepsilon} \right) }{\left\langle \alpha, \exp \left( \frac{u_{t}}{\varepsilon} \right)\right\rangle } \right\rangle \right| \mathrm{d}t\\ &=\| f - g \|_{\infty} \end{aligned}
∣Sminαε?(f)?Sminαε?(g)∣?=
?∫01?dtd?Sminαε?(ut?)dt
?=
?∫01??α,(g?f)?α,exp(εut??)?exp(εut??)??dt
?≤∫01?
??α,(g?f)?α,exp(εut??)?exp(εut??)??
?dt≤∫01?
??α,∥f?g∥∞??α,exp(εut??)?exp(εut??)??
?dt=∥f?g∥∞?∫01?
??α,?α,exp(εut??)?exp(εut??)??
?dt=∥f?g∥∞??
定义
S
α
(
f
)
(
y
)
=
?def.?
Smin
?
α
ε
(
C
(
?
,
y
)
?
f
)
,
S
β
(
g
)
(
x
)
=
?def.?
Smin
?
β
ε
(
C
(
x
,
?
)
?
g
)
\mathcal{S}_\alpha(f)(y) \stackrel{\text { def. }}{=} \operatorname{Smin}_\alpha^{\varepsilon}(\mathrm{C}(\cdot, y)-f), \quad \quad \mathcal{S}_\beta(g)(x) \stackrel{\text { def. }}{=} \operatorname{Smin}_\beta^{\varepsilon}(\mathrm{C}(x, \cdot)-g)
Sα?(f)(y)=?def.?Sminαε?(C(?,y)?f),Sβ?(g)(x)=?def.?Sminβε?(C(x,?)?g)
假设
C
C
C在
X
2
\mathcal{X}^2
X2上连续，对于任意
α
\alpha
α-integrable function
f
f
f，
S
α
(
f
)
\mathcal{S}_\alpha(f)
Sα?(f)是连续函数
如果
C
C
C对于每个输入都是是
γ
?
Lipschitz
\gamma-\text{Lipschitz}
γ?Lipschitz，则
S
α
(
f
)
\mathcal{S}_\alpha(f)
Sα?(f)是
γ
?
Lipschitz
\gamma-\text{Lipschitz}
γ?Lipschitz
证明：
∣
S
α
(
f
)
(
y
)
?
S
β
(
g
)
(
x
)
∣
≤
∥
C
(
?
,
y
)
?
C
(
x
,
?
)
∥
≤
γ
d
X
(
x
,
y
)
\left| \mathcal{S}_\alpha(f)(y) - \mathcal{S}_\beta(g)(x) \right| \le \| \mathrm{C}(\cdot, y) - \mathrm{C}(x, \cdot)\| \le \gamma \mathrm{d}_{\mathcal{X}} \left( x, y \right)
∣Sα?(f)(y)?Sβ?(g)(x)∣≤∥C(?,y)?C(x,?)∥≤γdX?(x,y)