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Convex Programming

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Convex Programming

Brookes Vision Reading Group

- What is convex ???
- What is programming ???
- What is convex programming ???

- What is convex ???
- What is programming ???
- What is convex programming ???

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

Is a linear function convex ???

Region above a convex function is a convex set.

Is the set of all positive semidefinite matrices convex??

- What is convex ???
- What is programming ???
- What is convex programming ???

Objective function

Example

Constraints

- Objective function to be minimized/maximized.
- Constraints to be satisfied.

Optimal solution

Vertices

Objective function

Feasible region

- What is convex ???
- What is programming ???
- What is 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 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 !!!!

- 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 !!!

- 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

- || 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

Hmmm

ICE CREAM !!

Linear Objective Function

Affine mapping of SOC

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

- 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

- Convex quadratically constrained quadratic programming
- Sum of norms
- Maximum of norms
- Problems with hyperbolic constraints

- Convex quadratically constrained quadratic programming
- Sum of norms
- Maximum of norms
- Problems with hyperbolic constraints

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

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

- Convex quadratically constrained quadratic programming
- Sum of norms
- Maximum of norms
- Problems with hyperbolic constraints

Minimize || Fi x + gi ||

Minimize ti

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

Special Case: L-1 norm minimization

- Convex quadratically constrained quadratic programming
- Sum of norms
- Maximum of norms
- Problems with hyperbolic constraints

Minimize max || Fi x + gi ||

Minimize t

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

Special Case: L-inf norm minimization

- Convex quadratically constrained quadratic programming
- Sum of norms
- Maximum of norms
- Problems with hyperbolic constraints

x >= 0 , y >= 0

w2 <= xy

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

- 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

Linear Objective Function

Linear Constraints

Minimize C X

Subject to Ai X = bi

X >= 0

Linear Programming on Semidefinite Matrices

- 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

- Minimizing the maximum eigenvalue
- Class separation with ellipsoids

- Minimizing the maximum eigenvalue
- Class separation with ellipsoids

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

Matrix M(z)

To find vector z* such that max is minimized.

Max -n

nI - M(z) >= 0

- Minimizing the maximum eigenvalue
- Class separation with ellipsoids

- 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

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)

Let’s solve this now !!!

Minimize xTQ0x + 2q0Tx + r0

Subject to xTQix + 2qiTx + ri < = 0

Minimize yTM0y

Subject to yTMiy < = 0

Mi = [ ri qiT; qi Qi]

- Problem is NP-hard.
- Let’s relax the problem to make it convex.
- Pray !!!

- 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

Bad Constraint !!!!

No donut for you !!!

Minimize yTM0y

Subject to yTMiy < = 0

Minimize M0 Y

Subject to Mi Y < = 0

Y = yyT

SDP Problem

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.

- Graph: G=(V,E)
- Maximum-Cut

- xi = -1

- xi = +1

- Graph: G=(V,E)
- Maximum-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 problem can be written as

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

Max Cut problem can be written as

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

To the white board.

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

- 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

X - xxT >= 0

Minimize yTM0y

Subject to yTMiy < = 0

Remember

Y = [1 xT; x X]

Minimize M0 Y

Subject to Mi Y < = 0

Y >= 0

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.

Minimize yTM0y

Subject to yTMiy < = 0

Minimize Q0 X + 2q0Tx + r0

Subject to Qi X + 2qiTx + ri < = 0

Cj (X - xxT) >= 0

Cj C

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

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

Minimize cT x

Subject to Qi X + 2qiTx + ri < = 0

Q X + 2qTx + r <= 0

Q = n i uiuiT

Let1 >= 2 >= …. k >=0 >= k+1 >=n

Q+ = k i uiuiT

C =

Q X + 2qTx + r <= 0

xT Q+ x - Q+ X <= 0

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

zi

Q+ = k i uiuiT

C =

uiuiT i = k+1, k+2, … n

Q X + 2qTx + r <= 0

xT Q+ x + k+1i zi+ 2qTx + r <= 0

xTuiuiTx - uiuiT X <= 0

Q+ = k i uiuiT

C =

uiuiT i = k+1, k+2, … n

Q X + 2qTx + r <= 0

xT Q+ x + k+1i zi+ 2qTx + r <= 0

xTuiuiTx - zi <= 0

ei eiT

i = 1, … , |V|

C =

uij uijT

(i,j) E

vij vijT

(i,j) E

ei = [0 0 …. 1 0 …0]

uij = ei + ej

vij = ei - ej

Warning: Scary equations to follow.

- 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

- 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