Convex-programming.ppt

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

Is a linear function convex ???

Convex Set

Region above a convex function is a convex set.

Convex Set

Is the set of all positive semidefinite matrices convex??

Huh? • What is convex ???

• What is programming ??? • What is convex programming ???

Programming • Objective function to be minimized/maximized. • Constraints to be satisfied. Example

Constraints

Objective function

Example

Optimal solution Vertices

Objective function Feasible region

Huh? • What is convex ???

• What is programming ??? • What is convex programming ???

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 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 • 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 !!!

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 • || 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

x2 + y2 <= z2 Hmmm ICE CREAM !!

Second Order Cone Programming Minimize

fTx

Linear Objective Function

Subject to || Ai x+ bi || <= ciT x + di i = 1, … , L Affine mapping of SOC Constraints are SOC of ni dimensions Feasible regions are intersections of conic regions

Example

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 • Convex quadratically constrained quadratic programming • Sum of norms

• Maximum of norms • Problems with hyperbolic constraints

SOCP-able Problems • Convex quadratically constrained quadratic programming • Sum of norms

• Maximum of norms • Problems with hyperbolic constraints

QCQP Minimize

xT P0 x + 2 q0T x + r0

Pi >= 0

Subject to xT Pi x + 2 qiT x + ri

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

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 Problems • Convex quadratically constrained quadratic programming • Sum of norms

• Maximum of norms • Problems with hyperbolic constraints

Sum of Norms Minimize  || Fi x + gi || Minimize  ti Subject to || Fi x + gi || <= ti

Special Case: L-1 norm minimization

SOCP-able Problems • Convex quadratically constrained quadratic programming • Sum of norms

• Maximum of norms • Problems with hyperbolic constraints

Maximum of Norms Minimize max || Fi x + gi || Minimize t Subject to || Fi x + gi || <= t

Special Case: L-inf norm minimization

You weren’t expecting a question, were you ??

SOCP-able Problems • Convex quadratically constrained quadratic programming • Sum of norms

• Maximum of norms • Problems with hyperbolic constraints

Hyperbolic Constraints w2 <= xy

x >= 0 , y >= 0

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

Let’s see if everyone was awake !

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 Minimize C  X Subject to Ai  X = bi X >= 0 Linear Objective Function Linear Constraints Linear Programming on Semidefinite Matrices

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 • Minimizing the maximum eigenvalue

• Class separation with ellipsoids

SDP-able Problems • Minimizing the maximum eigenvalue

• Class separation with ellipsoids

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 Eigenvalue Matrix M(z) To find vector z* such that max is minimized.

Max -n nI - M(z) >= 0

SDP-able Problems • Minimizing the maximum eigenvalue

• Class separation with ellipsoids

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 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 Problems Minimize xTQ0x + 2q0Tx + r0 Subject to xTQix + 2qiTx + ri < = 0

Minimize yTM0y

Mi = [ ri qiT; qi Qi]

Subject to yTMiy < = 0 Let’s solve this now !!!

Non-Convex Problems • Problem is NP-hard.

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

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 Minimize yTM0y Subject to yTMiy < = 0

Minimize M0  Y Subject to Mi  Y < = 0

Y = yyT

Bad Constraint !!!! No donut for you !!!

SDP Relaxation Minimize yTM0y Subject to yTMiy < = 0

Minimize M0  Y Subject to Mi  Y < = 0

Y >= 0

SDP Problem

Nothing left to do …. but Pray

Note that we have squared the number of variables.

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

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

- xi = -1 - xi = +1

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

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

Max Cut as an IQP Max Cut problem can be written as

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

Solving Max Cut using SDP Relaxations To the white board. (You didn’t think I’ll prepare slides for this, did you??)

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 Minimize yTM0y Subject to yTMiy < = 0

Minimize M0  Y

Remember

Subject to Mi  Y < = 0

Y = [1 xT; x X]

Y >= 0 X - xxT >= 0

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 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 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 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 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 C C=

Q+ = k i uiuiT

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 C C=

Q+ = k i uiuiT 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 C C=

Q+ = k i uiuiT 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 = [0 0 …. 1 0 …0] uij = ei + ej vij = ei - ej

C=

ei eiT

i = 1, … , |V|

uij uijT

(i,j)  E

vij vijT

(i,j)  E

Specific Problem Example Max Cut Warning: Scary equations to follow.

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

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

Back to work now !!!