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Convex Programming Brookes Vision Reading Group

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Huh? What is convex ??? What is programming ??? What is convex programming ???

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Huh? What is convex ??? What is programming ??? What is convex programming ???

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

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Convex Function Is a linear function convex ???

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Convex Set Region above a convex function is a convex set.

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Convex Set Is the set of all positive semidefinite matrices convex??

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Huh? What is convex ??? What is programming ??? What is convex programming ???

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

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Example Feasible region Vertices Objective function Optimal solution

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Huh? What is convex ??? What is programming ??? What is convex programming ???

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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.

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Convex Programming Convex optimization function Convex feasible region Examples ??? Linear Programming Refer to Vladimir/Pushmeets reading group Second Order Cone Programming What ??? Semidefinite Programming All this sounds Greek and Latin !!!!

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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 !!!

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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

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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 x 2 + y 2 <= z 2

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Hmmm ICE CREAM !!

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Second Order Cone Programming Minimize f T x Subject to || A i x+ b i || <= c i T x + d i i = 1, …, L Linear Objective Function Affine mapping of SOC Constraints are SOC of n i dimensions Feasible regions are intersections of conic regions

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Example

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Why SOCP ?? A more general convex problem than LP – LP SOCP Fast algorithms for finding global optimum – LP - O(n 3 ) – SOCP - O(L 1/2 ) iterations of O(n 2 n i ) Many standard problems are SOCP-able

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

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

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QCQP Minimize x T P 0 x + 2 q 0 T x + r 0 Subject to x T P i x + 2 q i T x + r i P i >= 0 || P 0 1/2 x + P 0 -1/2 x || 2 + r 0 -q 0 T P 0 -1 p 0

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QCQP Minimize x T P 0 x + 2 q 0 T x + r 0 Subject to x T P i x + 2 q i T x + r i Minimize t Subject to || P 0 1/2 x + P 0 -1/2 x || < = t || P 0 1/2 x + P 0 -1/2 x || < = (r 0 -q 0 T P 0 -1 p 0 ) 1/2

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

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Sum of Norms Minimize || F i x + g i || Minimize t i Subject to || F i x + g i || <= t i Special Case: L-1 norm minimization

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

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Maximum of Norms Minimize max || F i x + g i || Minimize t Subject to || F i x + g i || <= t Special Case: L-inf norm minimization

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You werent expecting a question, were you ??

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

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Hyperbolic Constraints w 2 <= xy x >= 0, y >= 0 || [2w; x-y] || <= x+y

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Lets see if everyone was awake !

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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

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Semidefinite Programming Minimize C X Subject to A i X = b i X >= 0 Linear Objective Function Linear Constraints Linear Programming on Semidefinite Matrices

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Why SDP ?? A more general convex problem than SOCP – LP SOCP SDP Generality comes at a cost though – SOCP - O(L 1/2 ) iterations of O(n 2 n i ) – SDP - O((n i ) 1/2 ) iterations of O(n 2 n i 2 ) Many standard problems are SDP-able

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SDP-able Problems Minimizing the maximum eigenvalue Class separation with ellipsoids

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SDP-able Problems Minimizing the maximum eigenvalue Class separation with ellipsoids

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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

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Minimizing the Maximum Eigenvalue Matrix M(z) To find vector z* such that max is minimized. Max -n nI - M(z) >= 0

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SDP-able Problems Minimizing the maximum eigenvalue Class separation with ellipsoids

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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

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Non-Convex Problems Minimize x T Q 0 x + 2q 0 T x + r 0 Subject to x T Q i x + 2q i T x + r i < = 0 Q i >= 0 => Convex Redefine x in homogenous coordinates. y = (1; x) Non-Convex Quadratic Programming Problem !!!

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Non-Convex Problems Minimize x T Q 0 x + 2q 0 T x + r 0 Subject to x T Q i x + 2q i T x + r i < = 0 Minimize y T M 0 y Subject to y T M i y < = 0 M i = [ r i q i T ; q i Q i ] Lets solve this now !!!

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Non-Convex Problems Problem is NP-hard. Lets relax the problem to make it convex. Pray !!!

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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

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SDP Relaxation Minimize y T M 0 y Subject to y T M i y < = 0 Minimize M 0 Y Subject to M i Y < = 0 Y = yy T Bad Constraint !!!! No donut for you !!!

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SDP Relaxation Minimize y T M 0 y Subject to y T M i y < = 0 Minimize M 0 Y Subject to M i Y < = 0 Y >= 0 SDP Problem Nothing left to do …. but Pray Note that we have squared the number of variables.

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

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Graph: G=(V,E) Maximum-Cut Example - Max Cut - x i = -1 - x i = +1

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Graph: G=(V,E) Maximum-Cut Example - Max Cut Alright !!! So its an integer programming problem !!! Doesnt look like quadratic programming to me !!!

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Max Cut as an IQP Max Cut problem can be written as Naah !! Lets get it into the standard quadratic form.

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Max Cut as an IQP Max Cut problem can be written as Naah !! Lets get it into the standard quadratic form.

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Solving Max Cut using SDP Relaxations To the white board. (You didnt think Ill prepare slides for this, did you??)

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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

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SOCP Relaxation Minimize y T M 0 y Subject to y T M i y < = 0 Minimize M 0 Y Subject to M i Y < = 0 Y >= 0 Remember Y = [1 x T ; x X] X - xx T >= 0

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SOCP Relaxation Say youre given C = { C 1, C 2, … C n } such that C j >= 0 C j (X - xx T ) >= 0 (Ux) T (Ux) <= C j X Wait.. Isnt this a hyperbolic constraint Therefore, its SOCP-able.

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SOCP Relaxation Minimize y T M 0 y Subject to y T M i y < = 0 Minimize Q 0 X + 2q 0 T x + r 0 Subject to Q i X + 2q i T x + r i < = 0 C j (X - xx T ) >= 0 C j C

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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

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Choice of C Remember we had squared the number of variables. Lets 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

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Choice of C Minimize c T x Subject to Q i X + 2q i T x + r i < = 0 Q X + 2q T x + r <= 0 Q = n i u i u i T Let 1 >= 2 >= …. k >= 0 >= k+1 >= n

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Choice of C C = Q + = k i u i u i T Q X + 2q T x + r <= 0 x T Q + x - Q + X <= 0 x T Q + x + k+1 i u i u i T X + 2q T x + r <= 0 zizi

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Choice of C C = Q + = k i u i u i T x T Q + x + k+1 i z i + 2q T x + r <= 0 u i u i T i = k+1, k+2, … n x T u i u i T x - u i u i T X <= 0 Q X + 2q T x + r <= 0

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Choice of C C = Q + = k i u i u i T x T Q + x + k+1 i z i + 2q T x + r <= 0 u i u i T i = k+1, k+2, … n x T u i u i T x - z i <= 0 Q X + 2q T x + r <= 0

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Specific Problem Example - Max Cut e i = [0 0 …. 1 0 …0] u ij = e i + e j v ij = e i - e j C = e i e i T i = 1, …, |V| u ij u ij T (i,j) E v ij v ij T (i,j) E

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

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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

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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

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Back to work now !!!

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