Discrete Optimization Lecture 2 – Part I M. Pawan Kumar [email protected] Slides available online

Discrete Optimization Lecture 2 – Part I

M. Pawan Kumar

[email protected]

Slides available online http://cvn.ecp.fr/personnel/pawan

Recap

Va Vb Vc

da db dc

Label l0

Label l1

D : Observed data (image)

V : Unobserved variables

L : Discrete, finite label set

Labeling f : V L

Recap

Va Vb Vc

da db dc

Label l0

Label l1

D : Observed data (image)

V : Unobserved variables

L : Discrete, finite label set

Labeling f : V L

Va is assigned lf(a)

Recap

Va Vb Vc

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 l0

Label l1

bc;f(b)f(c)

a;f(a)

da db dc

Recap

Va Vb Vc

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 l0

Label l1

f* = argminf Q(f; )

da db dc

Recap

Va Vb Vc

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 l0

Label l1

f* = argminf Q(f; )

Outline

• Convex Optimization

• Integer Programming Formulation

• Convex Relaxations

• Comparison

• Generalization of Results

Mathematical Optimization

min g0(x)

s.t. gi(x) ≤ 0

hi(x) = 0

Objective function

Inequality constraints

Equality constraints

x is a feasible point gi(x) ≤ 0, hi(x) = 0

x is a strictly feasible point gi(x) < 0, hi(x) = 0

Feasible region - set of all feasible points

Convex Optimization

min g0(x)

s.t. gi(x) ≤ 0

hi(x) = 0

Objective function



Objective function is convex

Feasible region is convex

Convex set? Convex function?

Convex Set

x1 x2

c x1 + (1 - c) x2

c [0,1]

Line Segment

Endpoints

Convex Set

x1 x2

All points on the line segment lie within the set

For all line segments with endpoints in the set

Non-Convex Set

x1 x2

Examples of Convex Sets

x1 x2

Line Segment


x1 x2

Line


Hyperplane aTx - b = 0


Halfspace aTx - b ≤ 0


Second-order Cone ||x|| ≤ t

t

x2

x1


Semidefinite Cone {X | X 0}

aTXa ≥ 0, for all a Rn

All eigenvalues of X are non-negative

aTX1a ≥ 0 aTX2a ≥ 0

aT(cX1 + (1-c)X2)a ≥ 0

Operations that Preserve ConvexityIntersection

Polyhedron / Polytope

Operations that Preserve ConvexityIntersection

Operations that Preserve ConvexityAffine Transformation x Ax + b

Convex Function

x

g(x)

Blue point always lies above red point

x1

x2

Convex Function

x

g(x)

g( c x1 + (1 - c) x2 ) ≤ c g(x1) + (1 - c) g(x2)

x1

x2

Domain of g(.) has to be convex

Convex Function

x

g(x)

x1

x2

-g(.) is concave

g( c x1 + (1 - c) x2 ) ≤ c g(x1) + (1 - c) g(x2)

Convex 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

2g(x) 0

Convex Function and Convex Sets

x

g(x)

Epigraph of a convex function is a convex set

Examples of Convex Functions

Linear function aTx

p-Norm functions (x1p + x2

p + xnp)1/p, p ≥ 1

Quadratic functions xT Q x

Q 0

Operations that Preserve ConvexityNon-negative weighted sum

x

g1(x)

w1

x

g2(x)

+ w2 + ….

xT Q x + aTx + b

Q 0

Operations that Preserve ConvexityPointwise maximum

x

g1(x)

max

x

g2(x)

,

Pointwise minimum of concave functions is concave

Convex Optimization

min g0(x)

s.t. gi(x) ≤ 0

hi(x) = 0

Objective function



Objective function is convex

Feasible region is convex

Linear Programming

min g0(x)

s.t. gi(x) ≤ 0

hi(x) = 0

Objective function



min g0Tx

s.t. giTx ≤ 0

hiTx = 0

Linear function

Linear constraints

Linear constraints

Quadratic Programming

min g0(x)

s.t. gi(x) ≤ 0

hi(x) = 0

Objective function



min xTQx + aTx + b

s.t. giTx ≤ 0

hiTx = 0

Quadratic function

Linear constraints

Linear constraints

Second-Order Cone Programming

min g0(x)

s.t. gi(x) ≤ 0

hi(x) = 0

Objective function



min g0Tx

s.t. xTQix + aiTx + bi ≤ 0

hiTx = 0

Linear function

Quadraticconstraints

Linear constraints

Semidefinite Programming

min g0(x)

s.t. gi(x) ≤ 0

hi(x) = 0

Objective function



min Q X

s.t. X 0

Ai X = 0

Linear function

Semidefinite constraints

Linear constraints

Outline




• Comparison


Integer Programming Formulation

2

5

4

2

0

1 3

0

V1 V2

Label ‘0’

Label ‘1’Unary Cost

Unary Cost Vector u = [ 5

Cost of V1 = 0

2

Cost of V1 = 1

; 2 4 ]

Labeling = {1 , 0}

2

5

4

2

0

1 3

0

V1 V2

Label ‘0’


Unary Cost Vector u = [ 5 2 ; 2 4 ]T

Label vector x = [ -1

V1 0

1

V1 = 1

; 1 -1 ]T

Recall that the aim is to find the optimal x


Labeling = {1 , 0}

2

5

4

2

0

1 3

0

V1 V2

Label ‘0’


Unary Cost Vector u = [ 5 2 ; 2 4 ]T

Label vector x = [ -1 1 ; 1 -1 ]T

Sum of Unary Costs = 12

∑i ui (1 + xi)


Labeling = {1 , 0}

2

5

4

2

0

1 3

0

V1 V2

Label ‘0’

Label ‘1’Pairwise Cost

0Cost of V1 = 0 and V1 = 0

0

00

0Cost of V1 = 0 and V2 = 0

3

Cost of V1 = 0 and V2 = 11 0

00

0 0

10

3 0

Pairwise Cost Matrix P


Labeling = {1 , 0}

2

5

4

2

0

1 3

0

V1 V2

Label ‘0’



0 0

00

0 3

1 0

00

0 0

10

3 0

Sum of Pairwise Costs14

∑ij Pij (1 + xi)(1+xj)


Labeling = {1 , 0}

2

5

4

2

0

1 3

0

V1 V2

Label ‘0’



0 0

00

0 3

1 0

00

0 0

10

3 0

Sum of Pairwise Costs14

∑ij Pij (1 + xi +xj + xixj)

14

∑ij Pij (1 + xi + xj + Xij)=

X = x xT Xij = xi xj


Labeling = {1 , 0}

Constraints

• Uniqueness Constraint

∑ xi = 2 - |L|i Va

• Integer Constraints

xi {-1,1}

X = x xT


x* = argmin 12

∑ ui (1 + xi) + 14

∑ Pij (1 + xi + xj + Xij)

∑ xi = 2 - |L|i Va

xi {-1,1}

X = x xT

ConvexNon-Convex


Outline



• Convex Relaxations– Linear Programming (LP-S)– Semidefinite Programming (SDP-L)– Second Order Cone Programming (SOCP-MS)

• Comparison


LP-S

x* = argmin 12

∑ ui (1 + xi) + 14


∑ xi = 2 - |L|i Va

xi {-1,1}

X = x xT

Retain Convex PartSchlesinger, 1976

Relax Non-ConvexConstraint

LP-S

x* = argmin 12

∑ ui (1 + xi) + 14


∑ xi = 2 - |L|i Va

xi [-1,1]

X = x xT



LP-S

X = x xT

Schlesinger, 1976

Xij [-1,1]

1 + xi + xj + Xij ≥ 0

∑ Xij = (2 - |L|) xij Vb

LP-S

x* = argmin 12

∑ ui (1 + xi) + 14


∑ xi = 2 - |L|i Va

xi [-1,1]

X = x xT



LP-S

x* = argmin 12

∑ ui (1 + xi) + 14


∑ xi = 2 - |L|i Va

xi [-1,1],


Xij [-1,1]

1 + xi + xj + Xij ≥ 0

∑ Xij = (2 - |L|) xij Vb

LP-S

Outline• Convex Optimization



• Comparison


SDP-L

x* = argmin 12

∑ ui (1 + xi) + 14


∑ xi = 2 - |L|i Va

xi {-1,1}

X = x xT

Retain Convex PartLasserre, 2000


SDP-L

x* = argmin 12

∑ ui (1 + xi) + 14


∑ xi = 2 - |L|i Va

xi [-1,1]

X = x xT

Retain Convex Part


Lasserre, 2000

x1

x2

xn

1

...

1 x1 x2... xn

1 xT

x X

=

Rank = 1

Xii = 1

Positive SemidefiniteConvex

Non-Convex

SDP-L

x1

x2

xn

1

...

1 x1 x2... xn

Xii = 1

Positive SemidefiniteConvex

SDP-L

1 xT

x X

=

Schur’s Complement

A B

BT C

=I 0

BTA-1 I

A 0

0 C - BTA-1B

I A-1B

0 I

0

A 0 C -BTA-1B 0

X - xxT 0

1 xT

x X

=1 0

x I

1 0

0 X - xxT

I xT

0 1

Schur’s Complement

SDP-L

SDP-L

x* = argmin 12

∑ ui (1 + xi) + 14


∑ xi = 2 - |L|i Va

xi [-1,1]

X = x xT

Retain Convex Part


Lasserre, 2000

SDP-L

x* = argmin 12

∑ ui (1 + xi) + 14


∑ xi = 2 - |L|i Va

xi [-1,1]

Retain Convex Part

Xii = 1 X - xxT 0

Accurate

SDP-L

Inefficient

Lasserre, 2000

Outline




• Comparison


SOCP Relaxation

x* = argmin 12

∑ ui (1 + xi) + 14


∑ xi = 2 - |L|i Va

xi [-1,1]

Xii = 1 X - xxT 0

Derive SOCP relaxation from the SDP relaxation

Further Relaxation

1-D ExampleX - xxT 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

2-D Example

X11 X12

X21 X22

1 X12

X12 1

=X =

x1x1 x1x2

x2x1 x2x2

xxT =x1

2 x1x2

x1x2

=x2

2

2-D Example(X - xxT)

1 - x12 X12-x1x2

01 0

0 0 X12-x1x2 1 - x22

x12 1

-1 x1 1

C1 0 C1 0


1 - x12 X12-x1x2

00 0

0 1 X12-x1x2 1 - x22

C2 0 C2 0

x22 1

-1 x2 1


1 - x12 X12-x1x2

01 1

1 1 X12-x1x2 1 - x22

C3 0 C3 0

(x1 + x2)2 2 + 2X12



1 - x12 X12-x1x2

01 -1

-1 1 X12-x1x2 1 - x22

C4 0 C4 0

(x1 - x2)2 2 - 2X12


SOCP Relaxation

Consider a matrix C1 = UUT 0

(X - xxT)

||UTx ||2 X C1

C1 0

Continue for C2, C3, … , Cn


Kim and Kojima, 2000

SOCP Relaxation

How many constraints for SOCP = SDP ?

Infinite. For all C 0

Specify constraints similar to the 2-D example

xi xj

Xij

(xi + xj)2 2 + 2Xij

(xi + xj)2 2 - 2Xij

SOCP-MS

x* = argmin 12

∑ ui (1 + xi) + 14


∑ xi = 2 - |L|i Va

xi [-1,1]

Xii = 1 X - xxT 0

Muramatsu and Suzuki, 2003

SOCP-MS

x* = argmin 12

∑ ui (1 + xi) + 14


∑ xi = 2 - |L|i Va

xi [-1,1]

(xi + xj)2 2 + 2Xij (xi - xj)2 2 - 2Xij

Specified only when Pij 0


Outline




• Comparison


Kumar, Kolmogorov and Torr, JMLR 2010

Dominating Relaxation

For all MAP Estimation problem (u, P)

A dominates B

A B

≥

Dominating relaxations are better

Equivalent Relaxations

A dominates B

A B

=

B dominates A

For all MAP Estimation problem (u, P)

Strictly Dominating Relaxation

A dominates B

A B

>

B does not dominate A

For at least one MAP Estimation problem (u, P)

SOCP-MS

(xi + xj)2 2 + 2Xij (xi - xj)2 2 - 2Xij


• Pij ≥ 0(xi + xj)2

2- 1Xij =

• Pij < 0(xi - xj)2

21 -Xij =

SOCP-MS is a QP

Same as QP by Ravikumar and Lafferty, 2005

SOCP-MS ≡ QP-RL

LP-S vs. SOCP-MSDiffer in the way they relax X = xxT

Xij [-1,1]

1 + xi + xj + Xij ≥ 0

∑ Xij = (2 - |L|) xij Vb

LP-S

(xi + xj)2 2 + 2Xij

(xi - xj)2 2 - 2Xij

SOCP-MS

F(LP-S)

F(SOCP-MS)

LP-S vs. SOCP-MS

• LP-S strictly dominates SOCP-MS

• LP-S strictly dominates QP-RL

• Where have we gone wrong?

• A Quick Recap !

Recap of SOCP-MS

xi xj

Xij

1 1

1 1

C =

(xi + xj)2 2 + 2Xij

Recap of SOCP-MS

xi xj

Xij

1 -1

-1 1

C =

(xi - xj)2 2 - 2Xij

Can we use different C matrices ??

Can we use a different subgraph ??

Outline




• Comparison

• Generalization of Results– SOCP Relaxations on Trees– SOCP Relaxations on Cycles


SOCP Relaxations on Trees

Choose any arbitrary tree

SOCP 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

Outline




• Comparison

• Generalization of Results– SOCP Relaxations on Trees– SOCP Relaxations on Cycles


SOCP Relaxations on Cycles

Choose an arbitrary even cycle

Pij ≥ 0 Pij ≤ 0OR


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


• True for odd cycles with Pij ≤ 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

The SOCP-C Relaxation Include all LP-S constraints True SOCP

a b

0

01 1

0

0

0

b c

1

00 0

0

0

0

c a

0

01 1

0

0

0

0 1 0

Submodular Non-submodular Submodular

The SOCP-C Relaxation

a b

Include all LP-S constraints True SOCP

0

01 1

0

0

0

b c

1

00 0

0

0

0

c a

0

01 1

0

0

0

0 1 0

Frustrated Cycle


a b


0

01 1

0

0

0

b c

1

00 0

0

0

0

c a

0

01 1

0

0

0

0 1 0

LP-S Solution

a b

1

0-1 -1

0

0

0

b c

-1

01 1

0

0

0

c a

0-1 -1

0

0

0

1 -1 1

1

Objective Function = 0


a b


0

01 1

0

0

0

b c

1

00 0

0

0

0

c a

0

01 1

0

0

0

0 1 0

LP-S Solution

a b

1

0-1 -1

0

0

0

b c

-1

01 1

0

0

0

c a

0-1 -1

0

0

0

1 -1 1

1

Define an SOC Constraint using C = 1


a b


0

01 1

0

0

0

b c

1

00 0

0

0

0

c a

0

01 1

0

0

0

0 1 0

LP-S Solution

a b

1

0-1 -1

0

0

0

b c

-1

01 1

0

0

0

c a

0-1 -1

0

0

0

1 -1 1

1

(xi + xj + xk)2 3 + 2 (Xij + Xjk + Xki)


a b


0

01 1

0

0

0

b c

1

00 0

0

0

0

c a

0

01 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

0.80.6

-0.60.4

-0.4 -0.4

0.4

-0.3

0.3 0.3

-0.3

The 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 xi)2 ≤ n + (Σ Xij)

SOCP-Q strictly dominates LP-S

SOCP-Q strictly dominates SOCP-C

4-Neighbourhood MRF

Test SOCP-C

50 binary MRFs of size 30x30

u ≈ N (0,1)

P ≈ N (0,σ2)

4-Neighbourhood MRF

σ = 2.5

8-Neighbourhood MRF

Test SOCP-Q

50 binary MRFs of size 30x30

u ≈ N (0,1)

P ≈ N (0,σ2)

8-Neighbourhood MRF

σ = 1.125

Conclusions

• Large class of SOCP/QP dominated by LP-S

• New SOCP relaxations dominate LP-S

• But better LP relaxations exist

Documents

Discrete Optimization Lecture 2 – Part I M. Pawan Kumar [email protected] Slides available online