[PPT] - Spatial Branch-and-Cut for QCQP with Complex Bounded Variables Chen PowerPoint Presentation

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Aussois 2016 SBC for QCQP – 1

Spatial Branch-and-Cut for QCQP with Complex Bounded Variables

Chen Chen Alper Atamt¨ urk Shmuel S. Oren

January 4, 2016

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Introduction

Introduction (Nonconvex) Complex Bounded QCQP Why Complex? Spatial Branch-and-Cut Experiments Conclusion

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(Nonconvex) Complex Bounded QCQP

Introduction (Nonconvex) Complex Bounded QCQP Why Complex? Spatial Branch-and-Cut Experiments Conclusion

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min x∗Q0x + Re(c∗

0x) + b0

s.t. x∗Qix + Re(c∗

i x) + bi ≤ 0, i = 1, ..., m

ℓ ≤ x ≤ u x ∈ Cn

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(Nonconvex) Complex Bounded QCQP

Introduction (Nonconvex) Complex Bounded QCQP Why Complex? Spatial Branch-and-Cut Experiments Conclusion

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min x∗Q0x + Re(c∗

0x) + b0

s.t. x∗Qix + Re(c∗

i x) + bi ≤ 0, i = 1, ..., m

ℓ ≤ x ≤ u x ∈ Cn

Applications:

AC Optimal Power Flow
Signal Processing e.g. [Waldspurger, D’Aspremont, Mallat

2015]

Control Theory [Ben Tal, Nemirovski, Roos 2003]

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Why Complex?

Introduction (Nonconvex) Complex Bounded QCQP Why Complex? Spatial Branch-and-Cut Experiments Conclusion

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Real QCQP to complex QCQP: set imaginary components to zero Complex to real by defining 2x real variables: x := w + ıt. We use the complex structure to exploit the angle valid inequality:

Lij(wiwj + titj) ≤ tiwj − witj ≤ Uij(wiwj + titj)

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Why Complex?

Introduction (Nonconvex) Complex Bounded QCQP Why Complex? Spatial Branch-and-Cut Experiments Conclusion

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Real QCQP to complex QCQP: set imaginary components to zero Complex to real by defining 2x real variables: x := w + ıt. We use the complex structure to exploit the angle valid inequality:

Lij(wiwj + titj) ≤ tiwj − witj ≤ Uij(wiwj + titj)

It has a clear interpretation in polar coordinates:

θi := arctan

Im(xi)

Re(xi)

arctan(Lij) ≤ θi − θj ≤ arctan(Uij)

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Spatial Branch-and-Cut

Introduction Spatial Branch-and-Cut Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities Valid Inequalities cont. Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion

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Overview

Introduction Spatial Branch-and-Cut Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities Valid Inequalities cont. Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion

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Spatial branching, e.g. bound bisection:

(L + U)/2 ≤ x ∨ (L + U)/2 ≥ x

Requires a lower bound — we shall use a SDP relaxation. Our contributions:

Valid inequalities to strengthen the relaxation
Branching rules
Bound tightening procedures

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SDP Relaxation

Introduction Spatial Branch-and-Cut Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities Valid Inequalities cont. Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion

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Standard approach of [Shor 1987], [Lovasz & Schrijver 1991] (or moment relaxation ala Lasserre):

min Q0X + Re(c∗

0x) + b0

(1) s.t. Qi, X + Re(c∗

i x) + bi ≤ 0,

i = 1, ..., m

(2)

ℓ ≤ x ≤ u

(3)

Y := 1 x∗ x X

0,

(4) where Y is Hermitian. X = xx∗ iff Y has rank one or zero.

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A Valid Nonconvex Set

Introduction Spatial Branch-and-Cut Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities Valid Inequalities cont. Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion

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Let X := W + ıT . Every bounded complex QCQP has the following:

Lii ≤ Wii ≤ Uii,

(5)

Ljj ≤ Wjj ≤ Ujj,

(6)

LijWij ≤ Tij ≤ UijWij,

(7)

WiiWjj = W 2

ij + T 2 ij.

(8)

(5)-(6) follow directly from boundedness assumption
(7) is the angle constraint (can be shown)
(8) comes from X = xx∗

This gives us bounds for every complex entry of X!

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Valid Inequalities

Introduction Spatial Branch-and-Cut Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities Valid Inequalities cont. Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion

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The convex hull of (5)-(8) is given by the SDP relaxation (X 0, (5)-(7)) and

π0 + π1Wii + π2Wjj + π3Wij + π4Tij ≥ UjjWii + UiiWjj − UiiUjj, π0 + π1Wii + π2Wjj + π3Wij + π4Tij ≥ LjjWii + LiiWjj − LiiLjj,

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Valid Inequalities cont.

Introduction Spatial Branch-and-Cut Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities Valid Inequalities cont. Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion

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The coefficients π are defined as

π0 := −

LiiLjjUiiUjj,

π1 := −

LjjUjj,

π2 := −

LiiUii,

π3 := (

Lii +
Uii)(
Ljj +
Ujj)1 − f(Lij)f(Uij)

1 + f(Lij)f(Uij), π4 := (

Lii +
Uii)(
Ljj +
Ujj) f(Lij) + f(Uij)

1 + f(Lij)f(Uij).

where

f(x) :=

(

√ 1 + x2 − 1)/x, x = 0 0, x = 0 .

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Properties of VIs

Introduction Spatial Branch-and-Cut Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities Valid Inequalities cont. Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion

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If L = U then VI + SDP gives an exact solution, i.e. convergence
Can be added in a sparse manner for sparse SDPs
Can be interpreted as complex analogues of the RLT

inequalities

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Branching Rules

Introduction Spatial Branch-and-Cut Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities Valid Inequalities cont. Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion

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Consider a strategy of bisecting matrix bounds Lij, Uij. The question then is: which i, j entry to choose?

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Branching Rules

Introduction Spatial Branch-and-Cut Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities Valid Inequalities cont. Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion

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Consider a strategy of bisecting matrix bounds Lij, Uij. The question then is: which i, j entry to choose?

Objective-based approach, e.g. reliability branching

[Achterberg, Koch, Martin, 2005]. We apply a spatial branching-adapted rule of Belotti et al. as a benchmark (rb-int-br), call it RBEB.

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Branching Rules

Introduction Spatial Branch-and-Cut Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities Valid Inequalities cont. Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion

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Consider a strategy of bisecting matrix bounds Lij, Uij. The question then is: which i, j entry to choose?

Objective-based approach, e.g. reliability branching

[Achterberg, Koch, Martin, 2005]. We apply a spatial branching-adapted rule of Belotti et al. as a benchmark (rb-int-br), call it RBEB.

Our proposal: violation-based branching.

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Violation-Based Branching

Introduction Spatial Branch-and-Cut Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities Valid Inequalities cont. Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion

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Recall that we want a rank-one matrix in the SDP solution. However, rank is a global measure. For n > 1 a nonzero Hermitian positive semidefinite n × n matrix

X has rank one iff all of its 2 × 2 principal minors are zero.

Proposal: find the 2 × 2 submatrix with greatest minimum eigenvalue. Three options to branch: {i, i}, {j, j}, {i, j}. Two proposals to select among the three: MVSB: strong branching (try all three) MVWB: use a worst-case approximation that is easily computable

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Bound tightening

Introduction Spatial Branch-and-Cut Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities Valid Inequalities cont. Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion

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Standard procedure is to minimize/maximize a variable over a

relaxation. With SDP relaxation this is expensive!

We propose two closed-form procedures. First, consider

ax2 + xy + c ≤ 0, ℓ ≤ y ≤ u.

By use of aggregating variables, many quadratic inequalities can be put in such a form. Bounds on x can be inferred from the bounds on

y.

Second, tightening based on the principle that the sum of differences around a cycle must equal to zero. Defining

δij := xi − xj, then for some cycle of indices x, say {1, 2, 3}, we

have δ12 + δ23 + δ31 = 0. The arctangents of Lij, Uij represent such differences.

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Experiments

Introduction Spatial Branch-and-Cut Experiments Setup SDP Alone Does Not Converge In A Nutshell BoxQP Results Conclusion

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Setup

Introduction Spatial Branch-and-Cut Experiments Setup SDP Alone Does Not Converge In A Nutshell BoxQP Results Conclusion

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YALMIP , MOSEK, IPOPT, laptop. Spatial branch-and-cut implemented with depth-first search. Modified IEEE cases (9 to 118 node networks), BoxQP instances (20 to 60 variables).

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SDP Alone Does Not Converge

Introduction Spatial Branch-and-Cut Experiments Setup SDP Alone Does Not Converge In A Nutshell BoxQP Results Conclusion

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Table 1: Comparison of branching rules using CSDP for ACOPF

case rgap

MVSB RBEB

nodes depth time egap nodes depth time egap

g9 0.36% 10077* 101* 4654 0.36% 6481 101* 5421* 0.36% g14 0.16% 8235 101* 5427* 0.16% 10041* 101* 5021 0.16% g30 0.18% 4247 101* 5440* 0.18% 5821 101* 5425* 0.18% g57 2.31% 379 101* 6348* 2.31% 1765 101* 5475* 2.31% Average 0.75% 5735 101 5467 0.75% 6027 101 5335 0.75%

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In A Nutshell

Introduction Spatial Branch-and-Cut Experiments Setup SDP Alone Does Not Converge In A Nutshell BoxQP Results Conclusion

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Average across 9 instances. Average root gap is 5.6%.

config nodes depth time egap

No BT RBEB 5081 86 3737 4.79% MVSB 3713 30 2740 1.50% MVWB 1639 50 848 0.60% BT RBEB 5482 82 3735 4.64% MVSB 487 23 1083 0.60% MVWB 765 35 525 0.60%

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BoxQP Results

Introduction Spatial Branch-and-Cut Experiments Setup SDP Alone Does Not Converge In A Nutshell BoxQP Results Conclusion

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BoxQP (average over nontrivial instances):

config nodes depth time egap

SDP+VI RBEB 257 45 1229 0.17% MVSB 58 8 879 0.33% MVWB 195 15 851 0.29% SDP+RLT RBEB 43 14 407 0.00% MVSB 13 3 201 0.00% MVWB 28 4 122 0.00%

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Conclusion

Introduction Spatial Branch-and-Cut Experiments Conclusion Wrap-Up

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Wrap-Up

Introduction Spatial Branch-and-Cut Experiments Conclusion Wrap-Up

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Valid inequalities to strengthen the complex SDP relaxation
Branching rules based on rank-one violation
Closed-form bound tightening procedures

Future (current): cuts!

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Fin. Merci!

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Contact

Introduction Spatial Branch-and-Cut Experiments Conclusion

Fin. Merci!

Contact

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