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Convex Programming. Brookes Vision Reading Group. Huh?. What is convex ??? What is programming ??? What is convex programming ???. Huh?. What is convex ??? What is programming ??? What is convex programming ???. Convex Function. f(t x + (1-t) y) <= t f(x) + (1-t) f(y). Convex Function.

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Convex programming

Convex Programming

Brookes Vision Reading Group


Convex programming
Huh?

  • What is convex ???

  • What is programming ???

  • What is convex programming ???


Convex programming
Huh?

  • What is convex ???

  • What is programming ???

  • What is convex programming ???


Convex function
Convex Function

f(t x + (1-t) y) <= t f(x) + (1-t) f(y)


Convex function1
Convex Function

Is a linear function convex ???


Convex set
Convex Set

Region above a convex function is a convex set.


Convex set1
Convex Set

Is the set of all positive semidefinite matrices convex??


Convex programming
Huh?

  • What is convex ???

  • What is programming ???

  • What is convex programming ???


Programming

Objective function

Example

Constraints

Programming

  • Objective function to be minimized/maximized.

  • Constraints to be satisfied.


Example
Example

Optimal solution

Vertices

Objective function

Feasible region


Convex programming
Huh?

  • What is convex ???

  • What is programming ???

  • What is convex programming ???


Convex programming1
Convex Programming

  • Convex optimization function

  • Convex feasible region

  • Why is it so important ???

  • Global optimum can be found in polynomial time.

  • Many practical problems are convex

  • Non-convex problems can be relaxed to convex ones.


Convex programming2
Convex Programming

  • Convex optimization function

  • Convex feasible region

  • Examples ???

  • Linear Programming

    • Refer to Vladimir/Pushmeet’s reading group

  • Second Order Cone Programming

    • What ???

  • Semidefinite Programming

    • All this sounds Greek and Latin !!!!


Outline
Outline

  • Convex Optimization

    • Second Order Cone Programming (SOCP)

    • Semidefinite Programming (SDP)

  • Non-convex optimization

    • SDP relaxations

    • SOCP relaxations

  • Optimization Algorithms

    • Interior Point Method for SOCP

    • Interior Point Method for SDP

2 out of 3 is not bad !!!


Outline1
Outline

  • Convex Optimization

    • Second Order Cone Programming (SOCP)

    • Semidefinite Programming (SDP)

  • Non-convex optimization

    • SDP relaxations

    • SOCP relaxations

  • Optimization Algorithms

    • Interior Point Method for SOCP

    • Interior Point Method for SDP


Second order cone
Second Order Cone

  • || u || < t

    • u - vector of dimension ‘d-1’

    • t - scalar

    • Cone lies in ‘d’ dimensions

  • Second Order Cone defines a convex set

  • Example: Second Order Cone in 3D

x2 + y2 <= z2


X 2 y 2 z 2

Hmmm

ICE CREAM !!

x2 + y2 <= z2


Second order cone programming

Linear Objective Function

Affine mapping of SOC

Second Order Cone Programming

Minimize fTx

Subject to || Ai x+ bi || <= ciT x + di

i = 1, … , L

Constraints are SOC of ni dimensions

Feasible regions are intersections of conic regions



Why socp
Why SOCP ??

  • A more general convex problem than LP

    • LP  SOCP

  • Fast algorithms for finding global optimum

    • LP - O(n3)

    • SOCP - O(L1/2) iterations of O(n2∑ni)

  • Many standard problems are SOCP-able


Socp able problems
SOCP-able Problems

  • Convex quadratically constrained quadratic programming

  • Sum of norms

  • Maximum of norms

  • Problems with hyperbolic constraints


Socp able problems1
SOCP-able Problems

  • Convex quadratically constrained quadratic programming

  • Sum of norms

  • Maximum of norms

  • Problems with hyperbolic constraints


Convex programming
QCQP

Minimize xT P0 x + 2 q0T x + r0

Subject to xT Pi x + 2 qiT x + ri

Pi >= 0

|| P01/2 x + P0-1/2 x ||2 + r0 -q0TP0-1p0


Convex programming
QCQP

Minimize xT P0 x + 2 q0T x + r0

Subject to xT Pi x + 2 qiT x + ri

Minimize t

Subject to || P01/2 x + P0-1/2 x || < = t

|| P01/2 x + P0-1/2 x || < = (r0 -q0TP0-1p0)1/2


Socp able problems2
SOCP-able Problems

  • Convex quadratically constrained quadratic programming

  • Sum of norms

  • Maximum of norms

  • Problems with hyperbolic constraints


Sum of norms
Sum of Norms

Minimize  || Fi x + gi ||

Minimize  ti

Subject to || Fi x + gi || <= ti

Special Case: L-1 norm minimization


Socp able problems3
SOCP-able Problems

  • Convex quadratically constrained quadratic programming

  • Sum of norms

  • Maximum of norms

  • Problems with hyperbolic constraints


Maximum of norms
Maximum of Norms

Minimize max || Fi x + gi ||

Minimize t

Subject to || Fi x + gi || <= t

Special Case: L-inf norm minimization



Socp able problems4
SOCP-able Problems

  • Convex quadratically constrained quadratic programming

  • Sum of norms

  • Maximum of norms

  • Problems with hyperbolic constraints


Hyperbolic constraints
Hyperbolic Constraints

x >= 0 , y >= 0

w2 <= xy

|| [2w; x-y] || <= x+y



Outline2
Outline

  • Convex Optimization

    • Second Order Cone Programming (SOCP)

    • Semidefinite Programming (SDP)

  • Non-convex optimization

    • SDP relaxations

    • SOCP relaxations

  • Optimization Algorithms

    • Interior Point Method for SOCP

    • Interior Point Method for SDP


Semidefinite programming

Linear Objective Function

Linear Constraints

Semidefinite Programming

Minimize C  X

Subject to Ai X = bi

X >= 0

Linear Programming on Semidefinite Matrices


Why sdp
Why SDP ??

  • A more general convex problem than SOCP

    • LP  SOCP  SDP

  • Generality comes at a cost though

    • SOCP - O(L1/2) iterations of O(n2∑ni)

    • SDP - O((∑ni)1/2) iterations of O(n2∑ni2)

  • Many standard problems are SDP-able


Sdp able problems
SDP-able Problems

  • Minimizing the maximum eigenvalue

  • Class separation with ellipsoids


Sdp able problems1
SDP-able Problems

  • Minimizing the maximum eigenvalue

  • Class separation with ellipsoids


Minimizing the maximum eigenvalue
Minimizing the Maximum Eigenvalue

Matrix M(z)

To find vector z* such that max is minimized.

Let max(M(z)) <= n

max(M(z)-nI) <= 0

min(nI - M(z)) >= 0

nI - M(z) >= 0


Minimizing the maximum eigenvalue1
Minimizing the Maximum Eigenvalue

Matrix M(z)

To find vector z* such that max is minimized.

Max -n

nI - M(z) >= 0


Sdp able problems2
SDP-able Problems

  • Minimizing the maximum eigenvalue

  • Class separation with ellipsoids


Outline3
Outline

  • Convex Optimization

    • Second Order Cone Programming (SOCP)

    • Semidefinite Programming (SDP)

  • Non-convex optimization

    • SDP relaxations

    • SOCP relaxations

  • Optimization Algorithms

    • Interior Point Method for SOCP

    • Interior Point Method for SDP


Non convex problems
Non-Convex Problems

Minimize xTQ0x + 2q0Tx + r0

Subject to xTQix + 2qiTx + ri < = 0

Qi >= 0

=> Convex

Non-Convex Quadratic Programming Problem !!!

Redefine x in homogenous coordinates.

y = (1; x)


Non convex problems1

Let’s solve this now !!!

Non-Convex Problems

Minimize xTQ0x + 2q0Tx + r0

Subject to xTQix + 2qiTx + ri < = 0

Minimize yTM0y

Subject to yTMiy < = 0

Mi = [ ri qiT; qi Qi]


Non convex problems2
Non-Convex Problems

  • Problem is NP-hard.

  • Let’s relax the problem to make it convex.

  • Pray !!!


Outline4
Outline

  • Convex Optimization

    • Second Order Cone Programming (SOCP)

    • Semidefinite Programming (SDP)

  • Non-convex optimization

    • SDP relaxations

    • SOCP relaxations

  • Optimization Algorithms

    • Interior Point Method for SOCP

    • Interior Point Method for SDP


Sdp relaxation

Bad Constraint !!!!

No donut for you !!!

SDP Relaxation

Minimize yTM0y

Subject to yTMiy < = 0

Minimize M0 Y

Subject to Mi  Y < = 0

Y = yyT


Sdp relaxation1

SDP Problem

SDP Relaxation

Minimize yTM0y

Subject to yTMiy < = 0

Minimize M0 Y

Subject to Mi  Y < = 0

Y >= 0

Nothing left to do ….

but Pray

Note that we have squared the number of variables.


Example max cut
Example - Max Cut

  • Graph: G=(V,E)

  • Maximum-Cut


Example max cut1

- xi = -1

- xi = +1

Example - Max Cut

  • Graph: G=(V,E)

  • Maximum-Cut


Example max cut2
Example - Max Cut

  • Graph: G=(V,E)

  • Maximum-Cut

Alright !!! So it’s an integer programming problem !!!

Doesn’t look like quadratic programming to me !!!


Max cut as an iqp

Max Cut problem can be written as

Max Cut as an IQP

Naah !! Let’s get it into the standard quadratic form.


Max cut as an iqp1

Max Cut problem can be written as

Max Cut as an IQP

Naah !! Let’s get it into the standard quadratic form.


Solving max cut using sdp relaxations
Solving Max Cut using SDP Relaxations

To the white board.

(You didn’t think I’ll prepare slides for this, did you??)


Outline5
Outline

  • Convex Optimization

    • Second Order Cone Programming (SOCP)

    • Semidefinite Programming (SDP)

  • Non-convex optimization

    • SDP relaxations

    • SOCP relaxations

  • Optimization Algorithms

    • Interior Point Method for SOCP

    • Interior Point Method for SDP


Socp relaxation

X - xxT >= 0

SOCP Relaxation

Minimize yTM0y

Subject to yTMiy < = 0

Remember

Y = [1 xT; x X]

Minimize M0 Y

Subject to Mi  Y < = 0

Y >= 0


Socp relaxation1
SOCP Relaxation

Say you’re given C = { C1, C2, … Cn} such that Cj >= 0

Cj (X - xxT) >= 0

(Ux)T (Ux) <= Cj X

Wait .. Isn’t this a hyperbolic constraint

Therefore, it’s SOCP-able.


Socp relaxation2
SOCP Relaxation

Minimize yTM0y

Subject to yTMiy < = 0

Minimize Q0  X + 2q0Tx + r0

Subject to Qi  X + 2qiTx + ri < = 0

Cj (X - xxT) >= 0

Cj C


Socp relaxation3
SOCP Relaxation

If C is the infinite set of all semidefinite matrices

SOCP Relaxation = SDP Relaxation

If C is finite,

SOCP relaxation is ‘looser’ than SDP relaxation.

Then why SOCP relaxation ???

Efficiency - Accuracy Tradeoff


Choice of c
Choice of C

Remember we had squared the number of variables.

Let’s try to reduce them with our choice of C.

For a general problem - Kim and Kojima

Using the structure of a specific problem -

e.g. Muramatsu and Suzuki for Max Cut


Choice of c1
Choice of C

Minimize cT x

Subject to Qi  X + 2qiTx + ri < = 0

Q X + 2qTx + r <= 0

Q = n i uiuiT

Let1 >= 2 >= …. k >=0 >= k+1 >=n


Choice of c2
Choice of C

Q+ = k i uiuiT

C =

Q X + 2qTx + r <= 0

xT Q+ x - Q+ X <= 0

xT Q+ x + k+1i uiuiT X + 2qTx + r <= 0

zi


Choice of c3
Choice of C

Q+ = k i uiuiT

C =

uiuiT i = k+1, k+2, … n

Q X + 2qTx + r <= 0

xT Q+ x + k+1i zi+ 2qTx + r <= 0

xTuiuiTx - uiuiT X <= 0


Choice of c4
Choice of C

Q+ = k i uiuiT

C =

uiuiT i = k+1, k+2, … n

Q X + 2qTx + r <= 0

xT Q+ x + k+1i zi+ 2qTx + r <= 0

xTuiuiTx - zi <= 0


Specific problem example max cut

ei eiT

i = 1, … , |V|

C =

uij uijT

(i,j)  E

vij vijT

(i,j)  E

Specific Problem Example - Max Cut

ei = [0 0 …. 1 0 …0]

uij = ei + ej

vij = ei - ej


Specific problem example max cut1
Specific Problem Example - Max Cut

Warning: Scary equations to follow.


Outline6
Outline

  • Convex Optimization

    • Second Order Cone Programming (SOCP)

    • Semidefinite Programming (SDP)

  • Non-convex optimization

    • SDP relaxations

    • SOCP relaxations

  • Optimization Algorithms

    • Interior Point Method for SOCP

    • Interior Point Method for SDP


Outline7
Outline

  • Convex Optimization

    • Second Order Cone Programming (SOCP)

    • Semidefinite Programming (SDP)

  • Non-convex optimization

    • SDP relaxations

    • SOCP relaxations

  • Optimization Algorithms

    • Interior Point Method for SOCP

    • Interior Point Method for SDP