Outer limits of subdifferentials
for min-max type functions
Workshop on Variational Analysis with Applications
13-14 December 2016, PolyU
Vera Roshchina (School of Science, RMIT University, Australia)
Joint work with Andrew Eberhard and Tian Sang, with some help from
Kaiwen Meng (Southwest Jiaotong University, Chendu) and Minghua Li
(Chongqing university of Arts and Sciences)
Fréchet subdifferential
We will only consider continuous functions f : Rn → R.
Fréchet subdifferential ∂f is a generalisation of the MoreauRockafellar subdifferential to nonconvex functions:






f (x0) − f (x) − hv, x0 − xi
≥0 .
∂f (x) = v | lim inf
0
0


x
→x
kx
−
x
k


0
x 6=x
Compare with the subdifferential of a convex function:
∂f (x) = {v | f (x0 ) ≥ f (x) + hv, x0 − xi
y
y
f(x)
y
f(x)
nonsmooth
nonconvex function
nonsmooth
convex function
∀x0 ∈ Rn }.
empty Fréchet
subdifferential
f(x)
p
p
x
x
x 1/16
Limiting subdifferential
Limiting (Mordukhovich) subdifferential
∂¯f (x̄) = Lim sup ∂f (x).
x→x̄
For example, consider the following function at x = 0.
f (x) = 2kxk∞ − kxk2
2/16
An old result
Let ϕ1, ϕ2, . . . , ϕr : Rn → R be locally Lipschitz and approximate
convex at x̄. Then




[
\




∂ϕi(x̄) ∪ 
Cx̄(g) ,
∂¯ min ϕi (x̄) ⊂ 
i∈1:r
i∈I(x̄)
g∈S n−1
Cx̄(g) =



v̄ | v̄ ∈ Arg maxhv, gi, hv̄, gi = min
∂ϕi0 (x̄)
max hv, gi
i∈I(x̄) v∈∂ϕi (x̄)



Approximate convexity is a technical assumption. Note that Daniilidis and
Georgiev showed that a.c. locally Lipschitz functions coincide with the class
of lower-C1 functions in Rn .
[Ngai, Luc, Thera (2000)], [Daniilidis, Georgiev (2004)], [R. (2010)]
3/16
Equality in the polyhedral case
x2
3
-2
2
-3
x1
∂f(0)
[R. (2007)]




\
\




∂¯ min ϕi (x̄) = 
∂ϕi(x̄) ∪ 
Cx̄(g) ,
i∈1:r
i∈I(x̄)
g∈S n−1
Cx̄(g) =



v̄ | v̄ ∈ Arg maxhv, gi, hv̄, gi = min
∂ϕi0 (x̄)
max hv, gi
i∈I(x̄) v∈∂ϕi (x̄)



4/16
Motivation
Error bound bound [Fabian, Henrion, Kruger and Outrata, 2010]. For
our purposes we assume that g : Rn → R is continuous. Then


Er g(x̄) ≥ lim inf dist (0n, ∂g(x)) = dist 0n, Lim sup ∂g(x).


x→x̄
g(x)>g(x̄)
x→x̄
g(x)>g(x̄)
∂ >g(x̄) = Lim sup ∂g(x)
x→x̄
g(x)>g(x̄)
1. Regular locally Lipschitz function [Li, Meng, Yang, 2016]
dist(0, ∂ >g(x̄)) ≤ Erg(x̄) ≤ dist(0, ∂σ>∂g(x̄) (0))
Open question:
∂σ>∂g(x̄) (0) ⊂ ∂ >g(x̄)?
The left hand side can be expressed via the end set,
∂σ>∂g(x̄) (0) = cl end (∂σ∂g(x̄) (0)).
5/16
Min-max type functions
2. Can we obtain some sort of characterisation for min-max
functions, like the one for convex polyhedral functions [Cánovas,
Henrion, López and Parra (2016)]?
Let g(x) = max gj (x), where gj (x) = haj , xi − bj , then
j∈J
[
Lim sup ∂g(x) =
x→x̄
g(x)>g(x̄)
co{aj , j ∈ D},
D∈D(x̄)
where D(x̄) ⊂ 2J(x̄) consists of
all subsets D ⊂ J(x̄) s.t. ∃d:

ha , di = 1,
j
ha , di < 1,
j
j ∈ D,
j ∈ J(x̄) \ D
6/16
Pointwise max of C 1
3. Let g(x) = max gj (x), where gj ∈ C 1. Then
j∈J
[
D∈DAI (x̄)
co{∇gj (x̄), j ∈ D} ⊂ Lim sup ∂g(x),
x→x̄
g(x)>g(x̄)
where DAI (x̄) contains all D from D(x̄) such that {∇gj (x̄), j ∈ D}
are affinely independent.
Here D(x̄) ⊂ 2J(x̄), where J(x̄) is the active index set, contains
all subsets D ⊂ J(x̄) such that the following system is consistent:

h∇g (x̄), di = 1, j ∈ D,
i
h∇g (x̄), di < 1,
j ∈ J(x̄) \ D
i
in the variable d ∈ Rn. [Cánovas, Henrion, López and Parra (2016)]
Open question: can we drop affine dependence?
7/16
Main result
Theorem 1. Let f : Rn → R, f (x) = mini∈I fi(x), where fi(x) is
(Hadamard) directionally differentiable with sublinear directional
derivative for all i ∈ I (think locally Lipschitz regular). Then
[
p∈S
0
f (x̄;p)>0
\
Arg maxhv, pi ⊂ Lim sup ∂f (x),
i∈I(x,p) v∈∂fi (x̄)
(1)
x→x̄
f (x)>f (x̄)
where




I(x, p) = i0 ∈ I(x) max hv, pi = min max hv, pi .


v∈∂fi0 (x)
i∈I(x) v∈∂fi (x)
8/16
Geometric interpretation
9/16
Question 1
Corollary 2. Let f : X → R be Hadamard directionally differentiable at every point x ∈ X, where X is an open subset of
Rn; moreover assume that the directional derivative f 0(x̄; ·) is a
sublinear function for every fixed x̄ ∈ X, then
[
p∈S
0
f (x̄;p)>0
Arg maxhv, pi ⊂ Lim sup ∂f (x).
v∈∂f (x̄)
(2)
x→x̄
f (x)>f (x̄)
10/16
Question 2
Theorem 3. Let f : X → R be as before (i.e. finite min), and in
addition assume that for every i ∈ I the function fi is piecewise
affine, i.e.
fi(x) = max(haij , xi + bij )
j∈Ji
∀i ∈ I,
where Ji’s are finite index sets for each i ∈ I. Then
[
p∈S
0
f (x̄;p)>0
\
Arg maxhv, pi = Lim sup ∂f (x),
i∈I(x,p) v∈∂fi (x̄)
x→x̄
f (x)>f (x̄)
where as before
I(x, p) =




i ∈ I(x) | max hv, pi = min max hv, pi .
 0

v∈∂fi0 (x)
i∈I(x) v∈∂fi (x)
11/16
Question 3
Corollary 4. Let g(x) := max gj (x), with gj : X → R continuously
j∈J
differentiable for all j ∈ J, where |J| < ∞, and let x̄ ∈ Rn. Then
[
n
o
conv ∇gj (x̄), j ∈ D =
D∈D(x̄)
[
Arg maxhv, pi
p∈S,g 0 (x̄,p)>0 v∈∂g(x̄)
⊆ Lim sup ∂g(x),
x→x̄
g(x)>g(x̄)
in other words, the subsets DAI (x̄) can be replaced by D(x̄).
Moreover when all {gj }j∈J are affine we have an identity.
12/16
Main technical result
Lemma 5. Let f : X → R be a pointwise minimum of finitely
many functions with sublinear Hadamard directional derivatives.
Then for every x̄ ∈ X, p ∈ S and
y∈
\
Arg maxhv, pi
(3)
i∈I(x̄,p) v∈∂fi (x̄)
there exist sequences {xk }and {yk } such that
xk → x̄,
xk − x̄
−→ p,
kxk − x̄k k→∞
yk ∈ ∂f (xk ),
yk −→ y.
k→∞
13/16
References
A.Y. Kruger, On Fréchet subdifferentials, Journal of Mathematical Sciences (2003).
H.V. Ngai, D.T. Luc, M. Thera, Approximate convex functions,
Nonlinear Convex Anal. (2000).
V. Roshchina, Mordukhovich subdifferential of pointwise minimum of approximate convex functions, Optim. Meth. Softw.
(2010).
A. Daniilidis, P. Georgiev, Approximate convexity and submonotonicity, J. Math. Anal. Appl. (2004).
V. Roshchina, Relationships between upper exhausters and the
basic subdifferential in variational analysis, J. Math. Anal. Appl.
(2007)
14/16
References
M. J. Cánovas, R. Henrion, M. A. López, J. Parra, Outer Limit
of Subdifferentials and Calmness Moduli in Linear and Nonlinear
Programming, JOTA (2016).
M.J. Fabian, R. Henrion, A. Kruger and J. Outrata, Error Bounds:
Necessary and Sufficient Conditions, Set-Valued Analysis (2010).
A. Kruger, Error Bounds and Holder Metric Subregularity, SetValued Var. Anal (2015).
A. Kruger, Error bounds and metric subregularity, Optimization
(2015).
M. Li, K, Meng, X. Yang, On Error Bound Moduli for Locally
Lipschitz and Regular Functions (arxiv:1608.03360, 2016)
A. Eberhard, V. Roshchina, T. Sang: Preprint in preparation.
15/16
Thank you