RL 2015
1/ 13
SMR
Notation and
definitions
The reduction
lemma
The Sticky
Face Lemma
Proof
Review
Projections and the Reduction Lemma
Stephen M. Robinson
University of Wisconsin-Madison
Work supported by AFOSR Grant FA9550-10-1-0101
Terryfest, Limoges, FR 18.05.2015
Notation
RL 2015
2/ 13
SMR
Notation and
definitions
The reduction
lemma
The Sticky
Face Lemma
Proof
Review
If P is a nonempty polyhedral convex subset of Rn , then
ΠP denotes the Euclidean projector on P
TP and NP are respectively the tangent-cone and
normal-cone operators of P.
For subsets X and Y of Rn the excess of X over Y is
e(X , Y ) = inf{α ≥ 0 | X ⊂ Y + αB n },
where B n is the unit ball.
Critical face and critical cone
RL 2015
3/ 13
SMR
Notation and
definitions
The reduction
lemma
The Sticky
Face Lemma
Proof
Definition
Let P be a nonempty polyhedral convex subset of Rn . For any
point z ∈ Rn let x = ΠP (z) and n∗ = z − x. Then
φP (z) := P ∩ {x + [pos(−n∗ )]◦ }
is the critical face of P for z. Also,
Review
κP (z) := TP (x) ∩ [pos(−n∗ )]◦
is the critical cone of P for z.
The critical face φP (z) is a nonempty face of P that is also
expressible as NP−1 (n∗ ). The critical cone κP (z) is a nonempty
face of TP (x).
Illustration
RL 2015
4/ 13
SMR
Notation and
definitions
The reduction
lemma
The Sticky
Face Lemma
Proof
Review
The reduction lemma
RL 2015
5/ 13
SMR
Notation and
definitions
The reduction
lemma
The Sticky
Face Lemma
Proof
Review
Lemma
Let P be a nonempty polyhedral convex subset of Rn and let
z0 ∈ Rn . Let x0 = ΠP (z0 ) and n0∗ = z0 − x0 . There is a
neighborhood Q of the origin in R2n such that
[(x0 , n0∗ ) + Q] ∩ NP = (x0 , n0∗ ) + [Q ∩ NκP (z0 ) ].
(1)
This lemma is extremely useful in the local analysis of
variational problems, because it lets us change the underlying
set from P to κP (z0 ), which often has a simpler structure.
Example
RL 2015
6/ 13
SMR
Notation and
definitions
The reduction
lemma
The Sticky
Face Lemma
Proof
Review
Variational inequality problem: find x ∈ P such that
hf (q, x), x 0 − xi ≥ 0 for each x 0 ∈ P, where q is a
parameter and f is continuous.
Rewrite: 0 ∈ f (q, x) + NP (x), equivalently
[x, −f (q, x)] ∈ NP .
Given a solution x0 for parameter value q0 , suppose we
want to investigate parameter values q 0 near q0 . Write
this as [x, −f (q 0 , x)] ∈ NP .
The reduction lemma says that for (x, q) close to (x0 , q0 )
we can work instead with [u, −g (q 0 , u)] ∈ NV , where
V = κP (x0 ) and g (q 0 , u) = f (q 0 , x0 + u) − f (q0 , x0 ).
Rewrite the new problem as 0 ∈ g (q 0 , u) + NV (x); now we
have replaced P by the critical cone V = κP (x0 ).
Structure of the proof
RL 2015
7/ 13
SMR
Notation and
definitions
The reduction
lemma
The Sticky
Face Lemma
Proof
Review
Proof of reduction lemma given in Math. Oper. Res. 1991
is very short and simple (13 lines), but it requires a fact
about polyhedral convex sets: the sticky face lemma.
Direct proof in the 2d edition of Dontchev and
Rockafellar’s book Implicit Functions and Solution
Mappings: A View from Variational Analysis, does not
require the sticky face lemma, but it is longer.
Unfortunately, the 1984 proof of the sticky face lemma is
long, complicated, and not intuitive. We will give a shorter
and more accessible proof.
Having that, we also have a short and accessible proof of
the reduction lemma.
The sticky face lemma
RL 2015
8/ 13
SMR
Notation and
definitions
The reduction
lemma
The Sticky
Face Lemma
Lemma
Let P be a nonempty polyhedral convex subset of Rn . Let
x0∗ ∈ Rn and define F = NP−1 (x0∗ ). Then there is a
neighborhood U of x0∗ such that whenever x ∗ ∈ U one has
Proof
Review
NP−1 (x ∗ ) = NF−1 (x ∗ ).
(2)
The reason for the lemma’s name is that it says the faces
NP−1 (x ∗ ) cannot escape from the face F as long as x ∗ remains
near x0 .
This is intuitively clear
RL 2015
9/ 13
SMR
Notation and
definitions
The reduction
lemma
The Sticky
Face Lemma
Proof
Review
Preparing the proof
RL 2015
10/ 13
Notation and
definitions
We say two subsets X and Y of Rn are locally identical at
z ∈ Rn if there is a neighborhood N of z such that
X ∩ N = Y ∩ N.
A fact about barrier cones:
The reduction
lemma
Lemma
SMR
The Sticky
Face Lemma
Proof
Review
Let P be a convex subset of Rn such that bc P is polyhedral,
and let (x0 , x0∗ ) ∈ NP . Let F = NP−1 (x0∗ ). Then:
a. bc F is polyhedral: specifically,
bc F = (bc P) + pos{−x0∗ };
b. bc F and bc P are locally identical at x0∗ .
The proof of this lemma is straightforward convex analysis,
using the properties of barrier cones.
(3)
Key arguments
RL 2015
11/ 13
SMR
Notation and
definitions
The reduction
lemma
The Sticky
Face Lemma
Proof
Review
Dispose of the case F = ∅
Use the lemma on the preceding slide to produce a
neighborhood Q of x0∗ such that Q ∩ bc F = Q ∩ bc P.
Observe that NP−1 is a polyhedral multifunction (i.e., its
graph is a finite union of polyhedral convex sets) and
therefore there are a nonnegative λ and a neighborhood
W of x0∗ such that whenever x ∗ ∈ W one has
e[NP−1 (x ∗ ), NP−1 (x0∗ )] ≤ λkx ∗ − x0∗ k
Observe that among the faces G of P that are not
contained in F there is one that minimizes e(G , F ) and
this minimum ρ is positive
Combining the arguments
RL 2015
12/ 13
SMR
Notation and
definitions
The reduction
lemma
The Sticky
Face Lemma
Proof
Review
Choose a positive α such that λα < ρ and let
U := Q ∩ W ∩ B(x0∗ , α)
If x ∗ ∈ U then e[NP−1 (x ∗ ), NP−1 (x0∗ )] < ρ and therefore
NP−1 (x ∗ ) ⊂ F
If NP−1 (x ∗ ) is empty then so is NP−1 (x ∗ ) (x ∗ ∈ Q)
If NP−1 (x ∗ ) is nonempty then hx ∗ , ·i has at least one
maximizer on P, and all such lie in F and so are also
maximizers on F . Then NP−1 (x ∗ ) = NF−1 (x ∗ ) and we are
finished
What we’ve covered
RL 2015
13/ 13
SMR
Notation and
definitions
The reduction
lemma
The Sticky
Face Lemma
Proof
Review
The reduction lemma is an important tool for local analysis
of variational problems posed over polyhedral convex sets.
It has a very short proof provided that one has the sticky
face lemma.
We now have a fairly short and geometrically clear proof
of the sticky face lemma.