convex programming

Convex Programming

Brookes Vision Reading Group

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?




Programming

• Objective function to be minimized/maximized.

• Constraints to be satisfied.

ExampleObjective function

Constraints

Example

Feasible region

Vertices

Objective function

Optimal solution

Huh?




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




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

HmmmICE CREAM !!

Second Order Cone Programming

Minimize fTx

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

i = 1, … , L

Linear Objective Function

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

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

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


• Sum of norms



Sum of Norms

Minimize || Fi x + gi ||

Minimize ti

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

Special Case: L-1 norm minimization


• Sum of norms



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 ??


• Sum of norms



Hyperbolic Constraints

w2 <= xy x >= 0 , y >= 0

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

Let’s see if everyone was awake !

Outline




Semidefinite Programming

Minimize C XSubject to Ai X = bi

X >= 0Linear 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∑ni

2)

• Many standard problems are SDP-able

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 -nnI - M(z) >= 0

SDP-able Problems

• Minimizing the maximum eigenvalue

• Class separation with ellipsoids

Outline




Non-Convex Problems

Minimize xTQ0x + 2q0Tx + r0

Subject to xTQix + 2qiTx + ri < = 0

Qi >= 0

=> Convex

Redefine x in homogenous coordinates.

y = (1; x)

Non-Convex Quadratic Programming Problem !!!

Non-Convex Problems

Minimize xTQ0x + 2q0Tx + r0

Subject to xTQix + 2qiTx + ri < = 0

Minimize yTM0y

Subject to yTMiy < = 0

Mi = [ ri qiT; qi Qi]

Let’s solve this now !!!

Non-Convex Problems

• Problem is NP-hard.

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

• Pray !!!

Outline




SDP Relaxation

Minimize yTM0y


Minimize M0 Y

Subject to Mi Y < = 0

Y = yyT

Bad Constraint !!!!

No donut for you !!!

SDP Relaxation

Minimize yTM0y


Minimize M0 Y


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

• Graph: G=(V,E)

• Maximum-Cut

Example - Max Cut

- xi = -1

- xi = +1

• Graph: G=(V,E)

• Maximum-Cut

Example - Max 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.

Solving Max Cut using SDP Relaxations

To the white board.(You didn’t think I’ll prepare slides for this, did you??)

Outline




SOCP Relaxation

Minimize yTM0y


Minimize M0 Y


Y >= 0

Remember

Y = [1 xT; x X]

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


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 matricesSOCP 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 uiui

T

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 uiui

T

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

uiuiT i = k+1, k+2, … n

xTuiuiTx - uiui

T X <= 0

Q X + 2qTx + r <= 0

Choice of C

C = Q+ = k i uiui

T

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

uiuiT i = k+1, k+2, … n

xTuiuiTx - zi <= 0

Q X + 2qTx + r <= 0

Specific Problem Example - Max Cut

ei = [0 0 …. 1 0 …0]

uij = ei + ej

vij = ei - ej

C = ei ei

T i = 1, … , |V|

uij uijT (i,j) E

vij vijT (i,j) E

Specific Problem Example - Max Cut

Warning: Scary equations to follow.

Outline




Back to work now !!!

convex programming

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