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Slide1
Discrete Optimization Lecture 2 – Part I
M. Pawan Kumarpawan.kumar@ecp.fr
Slides available online http://
cvn.ecp.fr
/personnel/
pawan
Slide2Recap
V
a
V
b
V
c
d
a
d
b
d
c
Label
l
0
Label
l
1
D
: Observed data (image)
V
: Unobserved variables
L
: Discrete, finite label set
Labeling
f
:
V
L
Slide3Recap
V
a
V
b
V
c
d
a
d
b
d
c
Label
l
0
Label
l
1
D
: Observed data (image)
V
: Unobserved variables
L
: Discrete, finite label set
Labeling
f
:
V
L
V
a
is assigned l
f(a)
Slide4Recap
V
a
V
b
V
c
2
5
4
2
6
3
0
1
1
0
0
2
1
3
Q(f;
) = ∑
a
a;f(a)
+ ∑
(a,b)
ab;f(a)f(b)
Label
l
0
Label
l
1
bc;
f
(b)
f
(c)
a;f(a)
d
a
d
b
d
c
Slide5Recap
V
a
V
b
V
c
2
5
4
2
6
3
0
1
1
0
0
2
1
3
Q(f;
) = ∑
a
a;f(a)
+ ∑
(a,b)
ab;f(a)f(b)
Label
l
0
Label
l
1
f* = argmin
f
Q(f;
)
d
a
d
b
d
c
Slide6Recap
V
a
V
b
V
c
2
5
4
2
6
3
0
1
1
0
0
2
1
3
Q(f;
) = ∑
a
a;f(a)
+ ∑
(a,b)
ab;f(a)f(b)
Label
l
0
Label
l
1
f* = argmin
f
Q(f;
)
Slide7Outline
Convex Optimization
Integer Programming Formulation
Convex
Relaxations
Comparison
Generalization of Results
Slide8Mathematical Optimization
min g
0
(
x
)
s.t. g
i
(
x
) ≤ 0
h
i
(x) = 0
Objective function
Inequality constraints
Equality constraints
x
is a
feasible point
g
i
(
x
) ≤ 0, h
i
(x) = 0
x
is a
strictly
feasible point gi(x) < 0, hi
(x) = 0
Feasible region
- set of all feasible points
Slide9Convex Optimization
min g
0
(
x
)
s.t. g
i
(
x
) ≤ 0
h
i
(x) = 0
Objective function
Inequality constraints
Equality constraints
Objective function is
convex
Feasible region is
convex
Convex set
?
Convex function
?
Slide10Convex Set
x
1
x
2
c
x
1
+ (1 - c)
x
2
c
[0,1]
Line Segment
Endpoints
Slide11Convex Set
x
1
x
2
All points on the line segment lie within the set
For all line segments with endpoints in the set
Slide12Non-Convex Set
x
1
x
2
Slide13Examples of Convex Sets
x
1
x
2
Line Segment
Slide14Examples of Convex Sets
x
1
x
2
Line
Slide15Examples of Convex Sets
Hyperplane
a
T
x
- b = 0
Slide16Examples of Convex Sets
Halfspace
a
T
x
- b ≤ 0
Slide17Examples of Convex Sets
Second-order Cone
||
x
|| ≤ t
t
x
2
x
1
Slide18Examples of Convex Sets
Semidefinite Cone
{
X
|
X
0}
a
T
Xa
≥ 0, for all a R
n
All eigenvalues of
X are non-negative
a
T
X
1a ≥ 0
a
T
X2a ≥ 0
a
T
(c
X1 + (1-c)
X2)a ≥ 0
Slide19Operations that Preserve Convexity
Intersection
Polyhedron / Polytope
Slide20Operations that Preserve Convexity
Intersection
Slide21Operations that Preserve Convexity
Affine Transformation x
Ax + b
Slide22Convex Function
x
g(x)
Blue
point always lies above
red
point
x
1
x
2
Slide23Convex Function
x
g(x)
g( c
x
1
+ (1 - c)
x
2
) ≤ c g(
x
1
) + (1 - c) g(
x
2
)
x
1
x
2
Domain of g(.) has to be convex
Slide24Convex Function
x
g(x)
x
1
x
2
-g(.) is concave
g( c
x
1
+ (1 - c)
x
2
) ≤ c g(
x
1
) + (1 - c) g(
x
2
)
Slide25Convex Function
Once-differentiable functions
g(
y
) +
g(
y
)
T
(
x - y) ≤ g(x)
x
g(x)
(
y
,g(
y
))
g(
y
) +
g(
y
)
T
(
x
-
y
)
Twice-differentiable functions
2
g(
x
) 0
Slide26Convex Function and Convex Sets
x
g(x)
Epigraph of a convex function is a convex set
Slide27Examples of Convex Functions
Linear function
a
T
x
p-Norm functions
(x
1
p
+ x
2p + xnp)1/p, p ≥ 1
Quadratic functions
xT Q x
Q
0
Slide28Operations that Preserve Convexity
Non-negative weighted sum
x
g
1
(x)
w
1
x
g
2
(x)
+ w
2
+ ….
x
T
Q
x
+
a
T
x
+ b
Q
0
Slide29Operations that Preserve Convexity
Pointwise maximum
x
g
1
(x)
max
x
g
2
(x)
,
Pointwise minimum of concave
functions is concave
Slide30Convex Optimization
min g
0
(
x
)
s.t. g
i
(
x
) ≤ 0
h
i
(x) = 0
Objective function
Inequality constraints
Equality constraints
Objective function is convex
Feasible region is convex
Slide31Linear Programming
min g
0
(
x
)
s.t. g
i
(
x
) ≤ 0
h
i
(x) = 0
Objective function
Inequality constraints
Equality constraints
min g
0
T
x
s.t. g
i
T
x
≤ 0
h
i
T
x
= 0
Linear function
Linear constraints
Linear constraints
Slide32Quadratic Programming
min g
0
(
x
)
s.t. g
i
(
x
) ≤ 0
h
i
(x) = 0
Objective function
Inequality constraints
Equality constraints
min
x
T
Qx
+
a
T
x
+ b
s.t. g
i
T
x
≤ 0
h
i
T
x
= 0
Quadratic function
Linear constraints
Linear constraints
Slide33Second-Order Cone Programming
min g
0
(
x
)
s.t. g
i
(
x
) ≤ 0
h
i
(x) = 0
Objective function
Inequality constraints
Equality constraints
min g
0
T
x
s.t.
x
T
Q
i
x
+
a
i
T
x
+ bi ≤ 0
h
i
Tx = 0
Linear function
Quadratic
constraints
Linear constraints
Slide34Semidefinite Programming
min g
0
(
x
)
s.t. g
i
(
x
) ≤ 0
h
i
(x) = 0
Objective function
Inequality constraints
Equality constraints
min
Q
X
s.t.
X
0
A
i
X
= 0
Linear function
Semidefinite constraints
Linear constraints
Slide35Outline
Convex Optimization
Integer Programming Formulation
Convex
Relaxations
Comparison
Generalization of Results
Slide36Integer Programming Formulation
2
5
4
2
0
1
3
0
V
1
V
2
Label
‘
0
’
Label
‘
1
’
Unary Cost
Unary Cost Vector
u
= [ 5
Cost of V
1
= 0
2
Cost of V
1
= 1
; 2 4 ]
Labeling =
{1 , 0}
Slide372
5
4
2
0
1
3
0
V
1
V
2
Label
‘
0
’
Label
‘
1
’
Unary Cost
Unary Cost Vector
u
= [ 5
2
; 2 4 ]
T
Label vector
x
= [ -1
V
1
0
1
V
1
= 1
; 1 -1 ]
T
Recall that the aim is to find the optimal
x
Integer Programming Formulation
Labeling =
{1 , 0}
Slide382
5
4
2
0
1
3
0
V
1
V
2
Label
‘
0
’
Label
‘
1
’
Unary Cost
Unary Cost Vector
u
= [ 5
2
; 2 4 ]
T
Label vector
x
= [ -1
1
; 1 -1 ]
T
Sum of Unary Costs =
1
2
∑
i
u
i
(1 +
x
i
)
Integer Programming Formulation
Labeling =
{1 , 0}
Slide392
5
4
2
0
1
3
0
V
1
V
2
Label
‘
0
’
Label
‘
1
’
Pairwise Cost
0
Cost of V
1
= 0 and V
1
= 0
0
0
0
0
Cost of V
1
= 0 and V
2
= 0
3
Cost of V
1
= 0 and V
2
= 1
1
0
0
0
0
0
1
0
3
0
Pairwise Cost Matrix
P
Integer Programming Formulation
Labeling =
{1 , 0}
Slide402
5
4
2
0
1
3
0
V
1
V
2
Label
‘
0
’
Label
‘
1
’
Pairwise Cost
Pairwise Cost Matrix
P
0
0
0
0
0
3
1
0
0
0
0
0
1
0
3
0
Sum of Pairwise Costs
1
4
∑
ij
P
ij
(1 +
x
i
)(1+
x
j
)
Integer Programming Formulation
Labeling =
{1 , 0}
Slide412
5
4
2
0
1
3
0
V
1
V
2
Label
‘
0
’
Label
‘
1
’
Pairwise Cost
Pairwise Cost Matrix
P
0
0
0
0
0
3
1
0
0
0
0
0
1
0
3
0
Sum of Pairwise Costs
1
4
∑
ij
P
ij
(1 +
x
i
+
x
j
+
x
i
x
j
)
1
4
∑
ij
P
ij
(1 +
x
i
+
x
j
+
X
ij
)
=
X
=
x x
T
X
ij
=
x
i
x
j
Integer Programming Formulation
Labeling =
{1 , 0}
Slide42Constraints
Uniqueness Constraint
∑
x
i
= 2 - |L|
i
V
a
Integer Constraints
x
i
{-1,1}X =
x xTInteger Programming Formulation
Slide43x
* = argmin
1
2
∑
u
i
(1 +
x
i
)
+
1
4
∑
Pij (1 + xi + xj + X
ij)
∑ xi
= 2 - |L|
i Va
x
i {-1,1}
X
=
x xT
Convex
Non-Convex
Integer Programming Formulation
Slide44Outline
Convex Optimization
Integer Programming Formulation
Convex
Relaxations
Linear Programming (LP-S)
Semidefinite Programming (SDP-L)
Second Order Cone Programming (SOCP-MS)Comparison
Generalization of Results
Slide45LP-S
x
* = argmin
1
2
∑
u
i
(1 +
x
i)
+
1
4
∑
Pij (1 + xi
+ xj + Xij)
∑
xi = 2 - |L|
i Va
x
i
{-1,1}
X
=
x xT
Retain Convex Part
Schlesinger, 1976
Relax Non-Convex
Constraint
Slide46LP-S
x
* = argmin
1
2
∑
u
i
(1 +
x
i)
+
1
4
∑
Pij (1 + xi
+ xj + Xij)
∑
xi = 2 - |L|
i Va
x
i
[-1,1]
X
=
x xT
Retain Convex Part
Schlesinger, 1976
Relax Non-Convex
Constraint
Slide47LP-S
X
=
x x
T
Schlesinger, 1976
X
ij
[-1,1]
1 + xi + xj + Xij ≥ 0
∑
Xij = (2 - |L|) xi
j Vb
Slide48LP-S
x
* = argmin
1
2
∑
u
i
(1 +
x
i)
+
1
4
∑
Pij (1 + xi
+ xj + Xij)
∑
xi = 2 - |L|
i Va
x
i
[-1,1]
X
=
x xT
Retain Convex Part
Schlesinger, 1976
Relax Non-Convex
Constraint
Slide49LP-S
x
* = argmin
1
2
∑
u
i
(1 +
x
i)
+
1
4
∑
Pij (1 + xi
+ xj + Xij)
∑
xi = 2 - |L|
i Va
x
i
[-1,1],
Retain Convex Part
Schlesinger, 1976
X
ij
[-1,1]
1 +
xi + xj
+ Xij ≥ 0
∑
Xij = (2 - |L|) xi
j V
b
LP-S
Slide50Outline
Convex Optimization
Integer Programming Formulation
Convex
Relaxations
Linear Programming (LP-S)
Semidefinite Programming (SDP-L)
Second Order Cone Programming (SOCP-MS)Comparison
Generalization of Results
Slide51SDP-L
x
* = argmin
1
2
∑
u
i
(1 +
x
i)
+
1
4
∑
Pij (1 + xi
+ xj + Xij)
∑
xi = 2 - |L|
i Va
x
i
{-1,1}
X
=
x xT
Retain Convex Part
Lasserre, 2000
Relax Non-Convex
Constraint
Slide52SDP-L
x
* = argmin
1
2
∑
u
i
(1 +
x
i)
+
1
4
∑
Pij (1 + xi
+ xj + Xij)
∑
xi = 2 - |L|
i Va
x
i
[-1,1]
X
=
x xT
Retain Convex Part
Relax Non-Convex
Constraint
Lasserre, 2000
Slide53x
1
x
2
x
n
1
.
.
.
1
x
1
x
2
.
.
.
x
n
1
x
T
x
X
=
Rank = 1
X
ii
= 1
Positive Semidefinite
Convex
Non-Convex
SDP-L
Slide54x
1
x
2
x
n
1
.
.
.
1
x
1
x
2
.
.
.
x
n
X
ii
= 1
Positive Semidefinite
Convex
SDP-L
1
x
T
x
X
=
Slide55Schur
’s Complement
A
B
B
T
C
=
I
0
B
T
A
-1
I
A
0
0
C -
B
T
A
-1
B
I
A
-1
B
0
I
0
A 0
C -B
T
A
-1
B 0
Slide56X
-
xx
T
0
1
x
T
x
X
=
1
0
x
I
1
0
0
X - xx
T
I
x
T
0
1
Schur
’
s Complement
SDP-L
Slide57SDP-L
x
* = argmin
1
2
∑
u
i
(1 +
x
i)
+
1
4
∑
Pij (1 + xi
+ xj + Xij)
∑
xi = 2 - |L|
i Va
x
i
[-1,1]
X
=
x xT
Retain Convex Part
Relax Non-Convex
Constraint
Lasserre, 2000
Slide58SDP-L
x
* = argmin
1
2
∑
u
i
(1 +
x
i)
+
1
4
∑
Pij (1 + xi
+ xj + Xij)
∑
xi = 2 - |L|
i Va
x
i
[-1,1]
Retain Convex Part
X
ii
= 1
X
-
xx
T
0
Accurate
SDP-L
Inefficient
Lasserre, 2000
Slide59Outline
Convex Optimization
Integer Programming Formulation
Convex
Relaxations
Linear Programming (LP-S)
Semidefinite Programming (SDP-L)
Second Order Cone Programming (SOCP-MS)Comparison
Generalization of Results
Slide60SOCP Relaxation
x
* = argmin
1
2
∑
u
i
(1 +
x
i)
+
1
4
∑
Pij (1 + xi
+ xj + Xij)
∑
xi = 2 - |L|
i Va
x
i
[-1,1]
X
ii
= 1
X
-
xx
T
0
Derive SOCP relaxation from the SDP relaxation
Further Relaxation
Slide611-D Example
X
-
xx
T
0
X
-
x2 ≥ 0
For two semidefinite matrices,
Frobenius inner product is non-negative
A
A
0
x2 X
SOC of the form ||
v ||2 st
= 1
Slide622-D Example
X
11
X
12
X
21
X
22
1
X
12
X
12
1
=
X
=
x
1
x
1
x
1
x2
x
2
x
1
x
2x2
xxT
=
x12
x
1x2
x1
x2
=
x
2
2
Slide632-D Example
(
X
-
xx
T)
1 -
x12
X
12-x1x2
0
1
0
0
0
X
12
-
x
1
x2
1 - x
22
x
12
1
-1
x1 1
C
1
0
C1 0
Slide642-D Example
(
X
-
xx
T)
1 -
x12
X
12-x1x2
0
0
0
0
1
X
12
-
x
1
x2
1 - x
22
C
2
0
C2 0
x
2
2
1
-1 x
2 1
Slide652-D Example
(
X
-
xx
T)
1 -
x12
X
12-x1x2
0
1
1
1
1
X
12
-
x
1
x2
1 -
x22
C
3
0
C3
0
(x
1
+ x2)2
2 + 2X12
SOC of the form ||
v ||2 st
Slide662-D Example
(
X
-
xx
T)
1 -
x12
X
12-x1x2
0
1
-1
-1
1
X
12
-
x
1
x2
1 -
x22
C
4
0
C4
0
(x
1
- x2)2
2 - 2X12
SOC of the form ||
v ||2 st
Slide67SOCP Relaxation
Consider a matrix
C
1
=
UU
T 0
(
X -
xxT)
||U
Tx ||2 X
C1
C1 0
Continue for C
2, C3, … , C
n
SOC of the form || v ||2
st
Kim and Kojima, 2000
Slide68SOCP Relaxation
How many constraints for SOCP = SDP ?
Infinite. For all C 0
Specify constraints similar to the 2-D example
x
i
x
j
X
ij
(x
i
+
x
j
)
2
2 + 2
X
ij
(x
i
+
x
j
)
2
2 - 2
X
ij
Slide69SOCP-MS
x
* = argmin
1
2
∑
u
i
(1 +
x
i)
+
1
4
∑
Pij (1 + xi
+ xj + Xij)
∑
xi = 2 - |L|
i Va
x
i
[-1,1]
X
ii
= 1
X
-
xx
T
0
Muramatsu and Suzuki, 2003
Slide70SOCP-MS
x
* = argmin
1
2
∑
u
i
(1 +
x
i)
+
1
4
∑
Pij (1 + xi
+ xj + Xij)
∑
xi = 2 - |L|
i Va
x
i
[-1,1]
(x
i
+
x
j)2 2 + 2
Xij
(x
i - xj)
2 2 - 2Xij
Specified only when
P
ij 0
Muramatsu and Suzuki, 2003
Slide71Outline
Convex Optimization
Integer Programming Formulation
Convex
Relaxations
Comparison
Generalization of Results
Kumar, Kolmogorov and
Torr, JMLR 2010
Slide72Dominating Relaxation
For all MAP Estimation problem (
u
,
P
)
A dominates B
A
B
≥
Dominating relaxations are better
Slide73Equivalent Relaxations
A dominates B
A
B
=
B dominates A
For all MAP Estimation problem (
u
,
P
)
Slide74Strictly Dominating Relaxation
A dominates B
A
B
>
B does not dominate A
For at least one MAP Estimation problem (
u
,
P
)
Slide75SOCP-MS
(x
i
+
x
j
)
2 2 + 2Xij
(x
i - xj)2 2 - 2X
ij
Muramatsu and Suzuki, 2003
Pij ≥ 0
(x
i + xj)
2
2
- 1
X
ij
=
Pij < 0
(x
i
- xj
)2
2
1 -
X
ij
=
SOCP-MS is a QP
Same as QP by Ravikumar and Lafferty, 2005
SOCP-MS
≡
QP-RL
Slide76LP-S vs. SOCP-MS
Differ in the way they relax
X
=
xx
T
X
ij
[-1,1]
1 + xi +
xj + Xij ≥ 0
∑ Xij = (2 - |L|) xi
j Vb
LP-S
(x
i
+
xj)2 2 + 2X
ij
(xi - xj)2 2 - 2X
ij
SOCP-MS
F
(LP-S)
F
(SOCP-MS)
Slide77LP-S vs. SOCP-MS
LP-S strictly dominates SOCP-MS
LP-S strictly dominates QP-RL
Where have we gone wrong?
A Quick Recap !
Slide78Recap of SOCP-MS
x
i
x
j
X
ij
1
1
1
1
C =
(x
i
+
x
j
)
2
2 + 2
X
ij
Slide79Recap of SOCP-MS
x
i
x
j
X
ij
1
-1
-1
1
C =
(x
i
-
x
j
)
2
2 - 2
X
ij
Can we use different C matrices ??
Can we use a different subgraph ??
Slide80Outline
Convex Optimization
Integer Programming Formulation
Convex
Relaxations
Comparison
Generalization of Results
SOCP Relaxations on TreesSOCP Relaxations on Cycles
Kumar, Kolmogorov and Torr, JMLR 2010
Slide81SOCP Relaxations on Trees
Choose any arbitrary tree
Slide82SOCP Relaxations on Trees
Choose any arbitrary C 0
Repeat over trees to get relaxation SOCP-T
LP-S strictly dominates SOCP-T
LP-S strictly dominates QP-T
Slide83Outline
Convex Optimization
Integer Programming Formulation
Convex
Relaxations
Comparison
Generalization of Results
SOCP Relaxations on Trees
SOCP Relaxations on CyclesKumar, Kolmogorov and Torr, JMLR 2010
Slide84SOCP Relaxations on Cycles
Choose an arbitrary even cycle
P
ij
≥ 0
P
ij
≤ 0
OR
Slide85SOCP Relaxations on Cycles
Choose any arbitrary C 0
Repeat over even cycles to get relaxation SOCP-E
LP-S strictly dominates SOCP-E
LP-S strictly dominates QP-E
Slide86SOCP Relaxations on Cycles
True for odd cycles with
P
ij
≤ 0
True for odd cycles with
Pij ≤ 0 for only one edge
True for odd cycles with
Pij ≥ 0 for only one edge
True for all combinations of above cases
Slide87The SOCP-C Relaxation
Include all LP-S constraints
True SOCP
a
b
0
0
1
1
0
0
0
b
c
1
0
0
0
0
0
0
c
a
0
0
1
1
0
0
0
0
1
0
Submodular
Non-submodular
Submodular
Slide88The SOCP-C Relaxation
a
b
Include all LP-S constraints
True SOCP
0
0
1
1
0
0
0
b
c
1
0
0
0
0
0
0
c
a
0
0
1
1
0
0
0
0
1
0
Frustrated Cycle
Slide89The SOCP-C Relaxation
a
b
Include all LP-S constraints
True SOCP
0
0
1
1
0
0
0
b
c
1
0
0
0
0
0
0
c
a
0
0
1
1
0
0
0
0
1
0
LP-S Solution
a
b
1
0
-1
-1
0
0
0
b
c
-1
0
1
1
0
0
0
c
a
0
-1
-1
0
0
0
1
-1
1
1
Objective Function = 0
Slide90The SOCP-C Relaxation
a
b
Include all LP-S constraints
True SOCP
0
0
1
1
0
0
0
b
c
1
0
0
0
0
0
0
c
a
0
0
1
1
0
0
0
0
1
0
LP-S Solution
a
b
1
0
-1
-1
0
0
0
b
c
-1
0
1
1
0
0
0
c
a
0
-1
-1
0
0
0
1
-1
1
1
Define an SOC Constraint using C =
1
Slide91The SOCP-C Relaxation
a
b
Include all LP-S constraints
True SOCP
0
0
1
1
0
0
0
b
c
1
0
0
0
0
0
0
c
a
0
0
1
1
0
0
0
0
1
0
LP-S Solution
a
b
1
0
-1
-1
0
0
0
b
c
-1
0
1
1
0
0
0
c
a
0
-1
-1
0
0
0
1
-1
1
1
(x
i
+
x
j
+
x
k
)
2
3 + 2 (
X
ij
+
X
jk
+
X
ki
)
Slide92The SOCP-C Relaxation
a
b
Include all LP-S constraints
True SOCP
0
0
1
1
0
0
0
b
c
1
0
0
0
0
0
0
c
a
0
0
1
1
0
0
0
0
1
0
SOCP-C Solution
Objective Function = 0.75
SOCP-C strictly dominates LP-S
a
b
0.8
0.6
-0.8
-0.8
-0.6
0.6
-0.6
b
c
0.6
-0.6
0
0
c
a
0
0
0.8
0.6
-0.6
0.4
-0.4
-0.4
0.4
-0.3
0.3
0.3
-0.3
Slide93The SOCP-Q Relaxation
Include all cycle inequalities
True SOCP
a
b
c
d
Clique of size n
Define an SOCP Constraint using C =
1
(
Σ
x
i
)
2
≤ n +
(
Σ
Xij)
SOCP-Q strictly dominates LP-S
SOCP-Q strictly dominates SOCP-C
Slide944-Neighbourhood MRF
Test SOCP-C
50 binary MRFs of size 30x30
u
≈
N
(0,1)
P
≈
N
(0,
σ
2
)
Slide954-Neighbourhood MRF
σ
= 2.5
Slide968-Neighbourhood MRF
Test SOCP-Q
50 binary MRFs of size 30x30
u
≈
N
(0,1)
P
≈
N (0,σ2)
Slide978-Neighbourhood MRF
σ
= 1.125
Slide98Conclusions
Large class of SOCP/QP dominated by LP-S
New SOCP relaxations dominate LP-
S
But better LP relaxations exist