convex programming brookes vision reading group. huh? what is convex ??? what is programming ???...
TRANSCRIPT
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.
ExampleObjective function
Constraints
Example
Feasible region
Vertices
Objective function
Optimal solution
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
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
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
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 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
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
• 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
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
• 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
• 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.
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
Subject to Mi Y < = 0
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
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 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
• 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 !!!