(semi-)algebraic geometry at ibm research ireland · 2016-11-23 · ibm research - ireland...

IBM Research - Ireland

(Semi-)Algebraic Geometryat IBM Research – Ireland

Jakub MarecekIBM Research – Ireland

with Tianran Chen, Bissan Ghaddar, Allan C. Liddell, Jie Liu, Timothy McCoy, Dhagash Mehta,

Martin Mevissen, Matthew Niemerg, and Martin Takac


AcknowledgementsWith much gratitude to my co-authors:

• Tianran Chen (Auburn University)

• Bissan Ghaddar (University of Waterloo)

• Jie Liu and Martin Takac (Lehigh University)

• Allan C. Liddell, Timothy McCoy, and Dhagash Mehta (Notre Dame)

• Martin Mevissen and Matthew Niemerg (IBM Research)

• Jonas C. Villumsen (Hitashi)

• Guanglei Wang (Pierre and Marie Curie University)

Cf.:

• JM, McCoy, Mevissen, arXiv:1412.8054

• Chen, JM, Mehta, Niemerg, to appear

• Ghaddar, JM, Mevissen, arXiv:1404.3626

• JM, Takac: arXiv:1506.08568

• Liu, JM, Takac: arXiv:1510.02171


The Agenda

An Introduction

Some structural results

Convexifications based on semidefinite programming (SDP)

SDP solvers based on tree-width decompositions, r -space sparsity, andinterior-point methods

SDP solvers based on coordinate descent

Solvers combining a convexification with Newton method on thenon-convex problem


IBM Research

• One of the world’s largest corporate reserach labs

• Driving 23 consecutive years of IBM patent leadership

• 6 Nobel laureates, 6 Turing awards, etc.

• A diversity of core academic disciplines, incl:

• Computer science

• Mathematical sciences

• Behavioural science

• Chemistry, Physics, Materials science


IBM Research – Ireland

• 1 of 12 IBM Research labs globally

• 100+ employees, plus a UCD co-lab

• 30 patents and over 100 top-tier publications in 2015

• IoT: Monitoring, forecasting, optimization, control

Highlights:

• A large body of work on power systems

• Work with water authorities across 3 continents

• Work with transportation authorities across 3 continents

• Work with airlines and a major vendor of software for airlines

• Work with a number of major car manufacturers


The Mathematical Challenges

• Power systems exhibit non-linear dynamics, which are uncertain(human behaviour) and time-varying (demand)

• Water systems exhibit non-linear dynamics, which are uncertain(human behaviour) and time-varying (demand)

• Transportation exhibits non-linear dynamics, which are uncertain(human behaviour) and time-varying (demand)

• In general, control of non-linear systems in undecidable.



• An abstraction of the dynamics in power systems is the differentialequation:

dθidt

= ωi −1

N

N∑j=1

Ki ,j sin(θi − θj), for i = 1, ...,N, (1)

where N is the number of oscillators, Ki ,j is the coupling strengthbetween the i-th and j-th oscillators. The matrix K = [Ki ,j ] may alsobe viewed as the adjacency matrix for the underlying weighted graph.Ω = (ω1, . . . , ωN) contains the natural frequencies of the N oscillators.



• Within the phase space of this system of ODEs are the steady states,where dθi

dt = 0 for all i = 1, . . . ,N.

• An analysis of the equilibria can reveal the behavior of the dynamicalsystem near the equilibria.

• Even for the steady-state problems, which are solved every couple ofminutes, there are no good solvers.

• How bad can this be?


A FERC Study: Where do Leading Solvers Fail?

Termination Conopt lpopt Knitro Minos Snopt

Exceeded time limit with aninfeasible solution

7.9% 22.2% 38.4% 0.0% 23.0%

Exceeded time limit with afeasible solution

6.5% 0.9% 6.7% 0.0% 14.4%

Early termination with aninfeasible solution

0.0% 0.4% 0.6% 11.4% 0.0%

Early termination with afeasible solution

31.2% 0.6% 1.8% 2.3% 0.2%

Normal termination with aninfeasible solution

10.5% 3.4% 0.0% 55.9% 1.6%

Percentage of feasible 81.6% 74.1% 61.0% 32.7% 75.4%

Percentage of best seen 43.8% 72.5% 52.5% 30.3% 60.8%

Source: Anya Castillo and Richard O’Neill, Staff Technical Conference, FERC, Washington DC, June 24–26, 2013. 118–3120 bus

instances, 100 runs per instance and solver, 20 min runs, Xeon E7458 2.4GHz, 64 GB RAM


The Agenda

An Introduction







Motivational Eye Candy

Figure: Smooth convex Figure: Smooth nonconvex

Figure: Nonsmooth convex Figure: Nonsmooth nonconvex


The Steady State Problems

Real-valued:

• Polar Power-Voltage pk , qk , vk , θk

• Rectangular Power-Voltage (R-PQV) pk , qk , <vk ,=vk• Polar Current Injection pk , qk , vk , θk , ik

• Rectangular Current Injection (R-CI) pk , qk , <vk ,=vk , ik• Polar Voltage-Current pk , qk , vk , θk , ilm

• Rectangular Voltage-Current (R-IV) pk , qk , <vk ,=vk , ilmFurther variants: dropping pk , qk , mixing polar and rectangular.

Complex-valued:

• With complex conjugate (PQV, CI, IV)

• Without (McCoy’s trick)

Cf. arXiv:1603.04375 for more.


Rectangular Power-Voltage Formulation (PQV)

Variables:

• vn is complex voltage at bus n

• pn is active power at bus n (the real part)

• qn is reactive power at bus n (imag. part)

Key constraints:

<vn∑m∈N

(gnm<vm − bnm=vm) + =vn∑m∈N

(gnm=vm + bnm<vm)− pn + pdn = 0

=vn∑m∈N

(gnm<vm − bnm=vm) + <vn∑m∈N

(gnm=vm + bnm<vm)− qn + qdn = 0


PQV: Derivation from the Dynamics

• Using the identities sin(θi − θj) = sin θi cos θj − sin θj cos θi

• Consider the substitution si := sin θi and ci := cos θi for alli = 1, . . . ,N − 1, and s2

i + c2i − 1 = 0, for all i = 1, . . . ,N − 1

• A system of polynomial equations:

ωi −1

N

N∑j=1

Ki ,j (sicj − sjci ) = 0

s2i + c2

i − 1 = 0,

(2)

for i = 1, . . . ,N − 1.


The Structural Results

• A reformulation of the complex conjugate

• A(2(N−1)

N−1

)bound on the number of feasible solutions considering

losses, improving upon Bezout bound of 22(N−1), and resolving aconjecture of Baillieul and Byrnes, which has been open for over threedecades

• A variety of additional structural results based on a reformulation ofthe steady-state equations to a multi-homogeneous algebraic system

• A bound on the number of feasible solutions considering losses andthe structure of the power system, based on the work of Bernstein

• Illustrations on some well-known instances, including the numbers ofroots, conditions for non-uniqueness of optima, and tree-width.


The Multi-Homogeneous Structure

• Replace all v∗n with independent variables un,

• Filter for solutions where un = v∗n once the complex solutions areobtained.

• Variables vn and un produce a multi-homogeneous structure withvariable groups vn and un:

vn∑k

Yn,kuk + un∑k

Y ∗n,kvk = 2pn n ∈ N \ G

vn∑k

Yn,kuk − un∑k

Y ∗n,kvk = 2qn n ∈ N \ G

vnun = |vn|2 n ∈ G − 0v0 = |v0|, u0 = |v0| (3)

where G be the set of slack generators for which |vn| is specified, and0 ∈ G corresponds to a reference node with phase 0.


The Multi-Homogeneous Structure

For example for the two-bus network, we obtain:

v1(Y1,0u0 + Y1,1u1) + u1(Y ∗1,0v0 + Y ∗1,1v1) = 2p1

v1(Y1,0u0 + Y1,1u1)− u1(Y ∗1,0v0 + Y ∗1,1v1) = 2q1

v0 = |v0|, u0 = |v0| (4)



Theorem

With exceptions on a parameter set of measure zero, the alternating-currentpower flow (1) has a finite number of complex solutions, which is boundedabove by: (

2n − 2

n − 1

)(5)



Proof.

Each equation in the system (1) is linear in the V variables and also inthe V ∗ variables, giving rise to a natural multi-homogeneous structure ofmult-idegree (1, 1). Since the slack bus voltage is fixed at a referencevalue, the system has 2n − 2 such equations in (n − 1, n − 1) variables.By the multi-homogeneous form of Bezout’s Theorem, the total number ofsolutions in multi-projective space CPn−1 × CPn−1 is precisely the statedbound, counting multiplicity.


An Illustration

The maximum number of steady states in a circuit with a fixed number ofbuses:

No. |N| of busesMethod 3 4 5 6 7 8 9 10 11 12 13 14

Bezout’s upper bound 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 16777216 67108864A BKK-based upper bound 8 40 192 864 3712 15488 63488 257536 1038336 4171776 16728064 67002368

Theorem 1 6 20 70 252 924 3432 12870 48620 184756 705432 2704156 10400600

Generic lower bound 6 20 70 252 924 3432 12870 48620 184756 705432 2704156 10400600



Theorem (Theorem 1 of Malajovich and Meer)

There does not exist a polynomial time algorithm to approximate the mini-mal multi-homogeneous Bezout number for a polynomial system up to anyfixed factor, unless P = NP.



Corollary

If there exists a feasible solution of the alternating-current power flow, thenthe solution has even multiplicity greater or equal to 2 or another solutionexists.



Theorem (Bernstein 1975)

Given a system of n polynomial equations in n variables, the number ofisolated complex solutions for which no variable is zero is bounded aboveby the mixed volume of the Newton polytopes of the equations.


Alternative Derivations from the Dynamics

• Using the trigonometric identitysin(θi − θj) = 1

2I(eI(θi−θj ) − e−I(θi−θj )) where I :=√−1 is the

imaginary unit

• Introduce substitution:

xi := eIθi and yi := e−Iθi ,

for all i = 1, . . . ,N − 1.

• An enlarged system of 2(N − 1) equations in 2(N − 1) variables

N∑j=1

Ki ,j

IN(xiyj − xjyi ) = ωi for i = 1, . . . ,N − 1

xiyi = 1 for i = 1, . . . ,N − 1.

(6)



For example, with N = 3, the system becomes

K1,2

3I(x1y2 − x2y1) +

K1,3

3I(x1 − y1)− ω1 = 0

K2,1

3I(x2y1 − x1y2) +

K2,3

3I(x2 − y2)− ω2 = 0

x1y1 − 1 = 0

x2y2 − 1 = 0.

(7)



• Yet another transformation based on identities

sin θi =2 tan θi

2

1 + tan2 θi2

and cos θi =1− tan

θi2

1 + tan2 θi2

, (8)

• One introduces ti := tan θi2 .



For example, with N = 3, the system becomes

K1,2

(2t1

1 + t21

1− t2

1 + t22

−2t2

1 + t22

1− t1

1 + t21

)+ K1,3

(2t1

1 + t21

1− t3

1 + t23

−2t3

1 + t23

1− t1

1 + t21

)= 3ω1

K2,1

(2t2

1 + t22

1− t1

1 + t21

−2t1

1 + t21

1− t2

1 + t22

)+ K2,3

(2t2

1 + t22

1− t3

1 + t23

−2t3

1 + t23

1− t2

1 + t22

)= 3ω1

considering ti := tan θi2 . Subsequently, one increases the degree by

multiplication.



Theorem

If there exists a choice of Ki ,j ’s and ωi ’s for which the number of nonzerocomplex solutions of (6) is the BKK bound, then for almost all choices ofcomplex Ki ,j ’s and ωi , the number of nonzero complex solutions of (6) willbe the BKK bound.


An Illustration on the Path Graph

Nodes 3 4 5 6 7 8 9 10 11 12 13 14 15

Bezout 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 16777216 67108864 268435456Binomial 6 20 70 252 924 3432 12870 48620 184756 705432 2704156 10400600 40116600

BKK for (2) 8 24 80 256 832 2688 8704 28160 91136 294912 954368 3088384 9994240BKK for (6) 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384

Generic root count 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384


More Structural Results

Remark

For the alternating-current power flows, where powers are fixed at all butthe reference bus, whenever there exists a real feasible solution, except fora parameter set of measure zero, one can enumerate all feasible solutionsin finite time.


Computations with Bertini

variable_group V0, V1; variable_group U0, U1;

I0 = V0*Yv0_0 + V1*Yv0_1; I1 = V0*Yv1_0 + V1*Yv1_1;

J0 = U0*Yu0_0 + U1*Yu0_1; J1 = U0*Yu1_0 + U1*Yu1_1;

fv0 = V0 - 1.0; fv1 = I1*U1 + J1*V1 + 7.0;

fu0 = U0 - 1.0; fu1 = -I1*U1 + J1*V1 - 7.0*I;

A Bertini encoding of ACPF on the two-bus instance of Bukhsh et al.2012, where the impedance of a single branch is 0.04 + 0.2i .


Properties of the Steady States

• One can optimise a variety of objectives over the steady states. Thecosts of real power P0 generated at the reference bus 0 by a quadraticfunction f0:

cost := f0(P0). (9)

• A norm of the vector D obtained by summing apparent powersS(u, v) + S(v , u)∀(u, v) ∈ E for:

||D||p =

( ∑(u,v)∈E

|S(u, v) + S(v , u)|p)1/p

. (10)

• The usual ||D||1 is denoted loss below.


An Illustration

Instance Source No

.|N|

of

bu

ses

No

.|E|

of

bra

nch

es

Tre

ewid

thtw

(P)

No

.|X|

of

solu

tio

ns

minx∈X

(co

st(x

))

No

.o

fm

in.

wrt

.co

st

avg

x∈X

(co

st(x

))

ma

x x∈X

(co

st(x

))

minx∈X

(lo

ss(x

))

No

.o

fm

in.

wrt

.lo

ss

avg

x∈X

(lo

ss(x

))

ma

x x∈X

(lo

ss(x

))

case2w Bukhsh et al. [?] 2 1 1 2 8.42 1 9.04 9.66 0.71 1 1.02 1.33case3KW Klos and Wojcicka [?] 3 3 2 6 -0.0 1 1250.0 1500.0 -0.0 1 -0.0 0.0case3LL Lavaei and Low [?] 3 3 2 2 1502.07 1 1502.19 1502.31 0.22 1 0.34 0.46case3w Bukhsh et al. [?] 3 2 1 2 5.88 1 5.94 6.01 0.55 1 0.58 0.61case4 McCoy et al. 4 3 1 4 1502.51 1 1505.19 1507.88 0.11 1 2.79 5.48case4ac McCoy et al. 4 4 2 6 2005.25 2 3337.5 4005.51 0.01 1 2.44 3.78case4cyc Bukhsh et al. [?] 4 4 2 4 3001.61 1 3003.56 3005.52 0.01 1 1.96 3.92case4gs Grainger and Stevenson [?] 4 4 2 10 2316.37 1 3681.07 4595.8 0.37 1 27.24 39.72case5w Lesieutre et al. [?] 5 6 2 2 2003.57 1 2004.6 2005.63 0.47 1 1.5 2.53case6ac McCoy et al. 6 6 2 23 4005.44 1 5574.23 6013.26 0.02 1 6.26 10.51case6ac2 McCoy et al. 6 6 2 22 4005.32 1 5554.03 6012.82 0.02 1 5.97 10.21case6b McCoy et al. 6 6 2 30 3005.7 5 3606.3 4508.28 0.01 1 3.9 5.88case6cyc Bukhsh et al. [?] 6 6 2 30 3005.7 4 3606.3 4508.28 0.01 1 3.9 5.88case6cyc2 McCoy et al. 6 6 2 12 1506.87 1 3131.55 4508.24 0.03 1 4.15 6.56case6cyc3 McCoy et al. 6 6 2 12 4502.42 1 4505.95 4508.01 0.02 1 3.55 5.61case7 McCoy et al. 7 7 2 2 1667.31 1 1668.36 1669.41 0.08 1 0.26 0.45case8cyc Bukhsh et al. [?] 8 8 2 60 6506.11 1 8557.72 9511.04 0.01 1 4.52 7.84case9 Chow [?] 9 9 2 16 3504.44 1 5692.02 6505.02 0.03 1 1.37 1.87case9g McCoy et al. 9 9 2 2 2604.41 1 4103.36 5602.31 0.08 1 0.26 0.45caseK4 McCoy et al. 4 6 3 8 2008.62 1 3006.32 4005.51 0.01 1 4.6 7.03caseK4sym McCoy et al. 4 6 3 6 2009.28 2 3339.29 4005.91 0.01 1 3.96 7.28caseK6 McCoy et al. 6 15 5 36 2018.2 1 5075.47 6030.0 0.02 1 17.16 27.25caseK6b McCoy et al. 6 15 5 40 6002.79 1 6017.01 6019.41 0.01 1 14.23 16.62caseK6sym McCoy et al. 6 15 5 48 6003.01 1 6017.75 6019.86 0.01 1 14.75 16.86


The Complicating Structural Property

Corollary

In the alternating-current optimal power flow problem, i.e., with s > 1,where powers are variable outside of the reference bus and there are no addi-tional inequalities, the complex solution set is empty or positive-dimensional,except for a parameter set of measure zero. When the complex solution setis positive-dimensional, if a smooth real feasible solution exists, then thereare infinitely many real feasible solutions.


The Agenda

An Introduction







The SDP Relaxation of Lavaei and Low

Lavaei and Low cast the R-PQV ACOPF without the extensions as:

minV ,PG ,QG ,

m∑l=1

fl(Pl) (11)

Pmin ≤ PG ≤ Pmax (12)

Qmin ≤ QG ≤ Qmax (13)

Vmink ≤ |Vk | ≤ Vmax

k (14)

<Vl(Vl − Vm)∗y∗lm ≤ Pmaxlm (15)

traceWY ∗eke∗k = Pk + Qk ı (16)

W 0,W = VV ∗ (17)

rank(W ) = 1 (18)


The SDP Relaxation of Lavaei and Low

Drop the rank constraint to obtain a SDP:

minX ,PG ,QG

m∑l=1

[cl2(traceYkW + PDk

)2 + cl1 traceYkW + PDk

](19)

Pmin ≤ traceYkW ≤ Pmax (20)

Qmin ≤ traceYkW ≤ Qmax (21)

(Vmink )2 ≤ traceMkW ≤ (Vmax

k )2 (22)

traceYlmW 2 + traceYlmW 2 ≤ (Smaxlm )2 (23)

traceYlmW ≤ Pmaxlm (24)

traceMlmW ≤ (Vmaxlm )2 (25)

W = XXT (26)

There is no duality gap under some spectral conditions.


How to See the SDP?

For a polynomial optimisation problem with multi-variate polynomials f , gi :

min f (x)s.t. gi (x) ≥ 0 i = 1, . . . ,m [PP]

one use the ideas of Shor (1987) and Nesterov (2000) to construct:

• The Lasserre hierarchy (2001) and its Parrilo (2003) dual

• The sparse hierarchy of Waki et al (2006), Lasserre (2006)

• The DSOS/SDSOS hierarchy of Ahmadi and Majumdar (2014)

• The DIGS hierarchy of Ghaddar et al (2015)

• The SOCP hierarchy of Kuang et al (2015)

• The variant hierarchy of Ma et al (2016)

The first levels of the hierarchies of Lasserre, Waki, Ghaddar, and Ma areequivalent to the dual of the SDP of previous slide.


The Lasserre Hierarchy

A Conic Reformulation

Given S ⊆ Rn, define Pd(S) to be the cone of polynomials of degree atmost d that are non-negative over S

max ϕ s.t. f (x)− ϕ ≥ 0 ∀ x ∈ SG ,

= max ϕ s.t. f (x)− ϕ ∈ Pd(SG).

for G = gi (x) : i = 1, . . . ,m, SG = x ∈ Rn : g(x) ≥ 0, ∀g ∈ G.

The hierarchy of Lasserre approximates Pd(SG) by sums-of-squares

KrG = Σr +

m∑i=1

gi (x)Σr−deg(gi ),

where Σd := ∑N

i=1 pi (x)2 : pi (x) ∈ Rb d2c[x ], with N =

(n+dd

)for r ≥ d .


The Lasserre Hierarchy II

The Corresponding Optimisation Problem [PP-Hr ]

maxϕ,σi (x)

ϕ s.t. f (x)− ϕ = σ0(x) +m∑i=1

σi (x)gi (x)

σ0(x) ∈ Σr ,

σi (x) ∈ Σr−deg(gi ).

Notice that this is a semidefinite programming (SDP) problem, i.e.minX∈Sn〈C ,X 〉Sn s.t. 〈Ai ,X 〉Sn = bi , i = 1, . . . ,m, X 0


The Code using YALMIP

One can define a bi-variate polynomial using, e.g.sdpvar x yp = (1 + x)4 + (1− y)2; Then:

An implementation of [PP-H1]

v = monolist([x y],degree(p)/2);

Q = sdpvar(length(v));

sos = v’*Q*v;

F = [coefficients(p-sos,[x y]) == 0, Q >= 0];

optimize(F)


The ConvergenceSummarising the results of Lasserre and others:

Proposition (arXiv:1404.3626)

Under mild assumptions, for the dense hierarchy [PP-Hr ]∗ for [PP] and therespective duals the following holds:

(a) inf [PP-Hr ] min ([PP]) as r →∞,

(b) sup [PP-Hr ]∗ min ([PP]) as r →∞,

(c) If the interior of the feasible set of [PP] is nonempty, there is no dualitygap between [PP-Hr ] and [PP-Hr ]∗.

(d) If [PP] has a unique global minimizer, x∗, then as r tends to infinitythe components of the optimal solution of [PP-Hr ] corresponding tothe linear terms converge to x∗.

See arXiv:1404.3626 for a discussion of assumptions, which are satisfied in the following applications.


The Agenda

An Introduction



SDP solvers based on tree-width decompositions, r-spacesparsity, and interior-point methods





For a 3 bus, 3 branch system:

0 5 10 15 20 25

0

2

4

6

8

10

r=2




0 50 100 150 200

0

5

10

15

20

25

30

35

r=4




0 200 400 600 800 1000 1200

0

10

20

30

40

50

60

70

80

r=6




0 500 1000 1500 2000 2500 3000 3500 4000

0

20

40

60

80

100

120

140

160

r=8


An Overview

Let us formulate a different hierarchy of SDPs using:

• Pre-processing: Drop some monomials, consider sign symmetry, etc

• Correlative sparsity pattern: The underlying network and hence thecanonical hypergraph have low tree-width

• Range-space sparsity: Any matrix can be reordered to ablock-diagonal form with a border; each block becomes an SDPconstraint, plus equalities for the border

• Facial reduction: You can pre-solve away the equalities to regain someconstraint qualification.


Correlative Sparsity [Waki et al. 2006]

The n × n Correlative Sparsity Pattern Matrix

Rij =

? for i = j

? for xi , xj in the same monomial of f

? for xi , xj in the same constraint gk

0 otherwise,

The chordal extension of its adjacency graph G have maximal cliquesIkpk=1, Ik ⊂ 1, . . . , n. The sparse approximation of Pd(S) is

KrG(I ) =

p∑k=1

Σr (Ik) +∑j∈Jk

gjΣr−deg(gj )(Ik)

,

where Σd (Ik ) is the set of all sum-of-squares polynomials of degree up to d supported on Ik and (J1, . . . , Jp) is a partitioning

of the set of polynomials gjj defining S such that for every j in Jk , the corresponding gj is supported on Ik . The support

I ⊂ 1, . . . , n of a polynomial contains i of terms xi in one of the monomials.


Correlative Sparsity II

Another Hierarchy of Relaxations ([PP-SHr ]∗) [Waki et al. 2006]

maxϕ,σk (x),σr,k (x)

ϕ

s.t. f (x)− ϕ =

p∑k=1

σk(x) +∑j∈Jk

gj(x)σj ,k(x)

σk ∈ Σr ((Ik)), σj ,k ∈ Σr−deg(gj )(Ik).

Reduction in size:

Size of k matrix inequalities

(| Ck | +r| Ck |

)vastly smaller if | Ck | n.


The ConvergenceSummarising the results of Waki, Kojima and others:

Proposition (arXiv:1404.3626)

Under mild assumptions, for the sparse hierarchy [PP-SHr ]∗ for [PP] andthe respective duals the following holds:

(a) inf [PP-SHr ] min ([PP]) as r →∞,

(b) sup [PP-SHr ]∗ min ([PP]) as r →∞,

(c) If the interior of the feasible set of [PP] is nonempty, there is no dualitygap between [PP-SHr ] and [PP-SHr ]∗.

(d) If [PP] has a unique global minimizer, x∗, then as r tends to infinitythe components of the optimal solution of [PP-SHr ] corresponding tothe linear terms converge to x∗.


Sparsity in SDPs [Kim et al. 2011]

min bTy s.t. F (y) < 0, y ∈ Rn,

where

F (y) =

1− y41 0 0 . . . 0 y1y2

0 1− y42 0 . . . 0 y2y3

0 0. . . 0 y3y4

. . . . . . . . .. . . . . . . . .

0 0 0 1− y4n−1 yn−1yn

y1y2 y2y3 y3y4 . . . yn−1yn 1− y4n

.

No correlative sparsity, but r-space sparsity!


Sparsity in SDPs II

Applying d-space and r-space conversion yields equivalent problem:

min bT y

s.t.

(1− y4

1 y1y2

y1y2 z1

) 0,(

1− y4i yiyi+1

yiyi+1 −zi−1 + zi

) 0 (i = 2, 3, . . . , n − 2),(

1− y4n−1 yn−1yn

yn−1yn 1− y4n − zn−2

) 0,

Equivalent problem satisfies correlative sparsity pattern!


What can Off-The-Shelf Solvers Do

Following much Matlab pre-processing, SeDuMi performs ok:

L1 L2Instance Bound Dim Time Bound Dim Timecase9mod 2753.23 588×168 0.6 3087.89 1792×14847 17.5case14 8081.52 888×94 0.4 8081.54 7508×66740 1420.5case14mod 7792.72 888×94 0.9 7991.07 7508×66740 904.2case30mod 576.89 4706×684 3.8 578.56 36258×49164 13740.0case39 41862.08 7282×758 2.2 41864.18 26076×215772 4359.1case57 41737.79 13366×356 3.2 * *case118 129654.62 56620×816 6.1 * *case300 719711.63 362076×1938 13.6 * *case2383wp 1.814×106 22778705×47975 3731.5 * *case2736sp 1.307×106 30019740×57408 3502.2 * *case2746wp 1.584×106 30239940×57978 3903.2 * *

See arXiv:1404.3626 for much more.


What the Off-The-Shelf Solvers cannot DoUnlike other general-purpose solvers (on L1):

Instance Time [s]Name SDP SeDuMi 1.32 SDPA 7.0 SDPLR 1.03 CSDP 6.0.1 SDPT3 4.0 Mosek 7.0case2w P 0.3 0.003 0 0.02 0.4 0.1case2w D 0.3 0.004 0 0.02 0.4 0.1case5w P 0.4 – 0 0.04 0.5 0.2case5w D 0.4 – 0 0.03 0.5 0.2case9 P 1.0 0.05 25 0.1 0.5 0.3case9 D 1.0 0.07 25 0.1 0.6 0.3case14 P 0.7 0.05 58 0.1 – 0.2case14 D 0.7 0.04 17 0.1 – 0.3case30 P 2.8 – 829 0.8 – –case30 D 6.1 – 282 1.0 – –case30Q P 2.8 – 831 0.8 – –case30Q D 6.2 – 195 1.0 – –case39 P 4.4 – 2769 1.0 – 0.7case39 D 7.7 – 723 1.3 – –case57 P 3.2 – 1930 0.5 – 0.7case57 D 4.0 – 1175 1.0 – 1.8case118 P 10.3 0.9 4400 3.4 – 1.7case118 D 17.1 – – 13.0 – –case300 P 27.5 1.7 – 109.6 – –case300 D 133.7 – – 66.7 – –


What the Off-The-Shelf Solvers cannot Do

• Run on a standard desktop

• “Good feasible solutions quick”

• Overall run-times comparable to MATPOWER, a popular heuristic

• Parallel and/or distributed computation for the SDP relaxations

• Exact tree-width decomposition fast – after all, it’s NP-Hard

• Exploitation of the structure in higher-level relaxations

• Warm start: a solution to [PP-H1] is a very good solution to [PP-H2],for an appropriate mapping of variables


The Agenda

An Introduction








0 5 10 15 20 25

0

1

2

3

4

5

6

7

case2224

case118mod

case9241pegase

case2736sp

case3w

Number of Threads

Sp

ee

du

p


The Overview

• Trivial per-iteration run-time and memory requirements, thanks to aclosed-form step for both ACOPF and general POP

• Warm start capabilities, due to the use of first-order methods

• Parallel and distributed computation easy to implement

• Overall run-times comparable to MATPOWER, a popular heuristic


Another Reformulation

min F (W ) :=∑k∈G

fk(W ) [RrBC]

s.t. tk = tr(YkW ) (27)

Pmink − Pd

k ≤tk ≤ Pmaxk − Pd

k (28)

gk = tr(YkW ) (29)

Qmink − gk ≤gk ≤ Qmax

k − gk (30)

hk = tr(MkW ) (31)

(Vmink )2 ≤hk ≤ (Vmax

k )2 (32)

ulm = tr(YlmW ) (33)

vlm = tr(YlmW ) (34)

zlm = (ulm)2 + (vlm)2 (35)

zlm ≤ (Smaxlm )2 (36)

W 0, rank(W ) ≤ r . (37)


Some Observations

O1: constraints (27–34) and (36) are box constraints,

O2: using elementary liner algebra:

W ∈ Sn s.t. W 0, rank(W ) ≤ r= RRT s.t. R ∈ Rn×r,

O3: if rank(W ∗) > 1 for the optimum W ∗ of the Lavaei-Low relaxation,there are no known methods for extracting the global optimum of[R1BC] from W , except for going higher in the hierarchies.

O4: zero duality gap at any SDP relaxation in the hierarchy of Lasserredoes not guarantee the solution of the SDP relaxation is exact for[R1BC].


Solving the SDP Relaxation

• Use the low-rank method of Burer and Monteiro, i.e. for increasingrank r , the augmented Lagrangian in variable RRT ∈ Rn×n ratherthan W ∈ Rn×n:

L(R, t, h, g, u, v, z, λt, λ

g, λ

h, λ

u, λ

v, λ

z ) :=∑k∈G

fk (RRT )

−∑k

λtk (tk − trace(YkRR

T )) + µ2

∑k

(tk − trace(YkRRT ))2

−∑k

λgk

(gk − trace(YkRRT )) + µ

2

∑k

(gk − trace(YkRRT ))2

−∑k

λhk (hk − trace(MkRR

T )) + µ2

∑k

(hk − trace(MkRRT ))2

−∑

(l,m)

λu(l,m)(u(l,m) − trace(Y(l,m)RR

T )) + µ2

∑(l,m)

(u(l,m) − trace(Y(l,m)RRT ))2

−∑

(l,m)

λv(l,m)(v(l,m) − trace(Y(l,m)RR

T )) + µ2

∑(l,m)

(v(l,m) − trace(Y(l,m)RRT ))2

−∑

(l,m)

λz(l,m)(z(l,m) − u2

(l,m) − v2(l,m)) + µ

2

∑(l,m)

(z(l,m) − u2(l,m) − v2

(l,m))2 + νR.

• Combined with a parallel coordinate descent with a closed-form step


Solving the SDP Relaxation II

1: for r = 1, 2, . . . do

2: choose R ∈ Rm×r

3: compute corresponding values of t, h, g , u, v , z4: project t, h, g , u, v , z onto the box constraints5: for k = 0, 1, 2, . . . do

6: in parallel, minimize L in t, g , h, u, v , z , coordinate-wise,7: in parallel, minimize L in R, coordinate-wise8: update Lagrange multipliers λt , λg , λh, λu, λv , λz

9: update µ10: terminate, if criteria are met11: end for12: end for


An Illustration

0 100 200 300 400

400

600

800

1000

1200

1400

1600

Iteration

Ob

jec

tive

Va

lue

1e−01

1e−02

1e−03

1e−04

1e−05

1e−06

adaptive


An Illustration

0 100 200 300 400

10−30

10−25

10−20

10−15

10−10

10−5

100

105

1e−01

1e−

02

1e−03

1e−04

1e−05

1e−06

ad

ap

tive

Iteration

In

fea

sib

ility


An Illustration

0 5 10 15 20 25

0

1

2

3

4

5

6

7

case2224

case118mod

case9241pegase

case2736sp

case3w

Number of Threads

Sp

ee

du

p


What can this Do

Let’s compare the run-times with:

• MATPOWER, a popular heuristic

• sdp pf, the pre-processor of Molzahn et al. with SeDuMi

• OPF Solver, the pre-processor of Lavaei et al. with SDPT3

MATPOWER sdp pf OPF Solver Our ApproachName Obj. T [s] Obj. T [s] Obj. T [s] Obj. T [s]case6ww 3.144+03 0.114 3.144e+03 0.74 3.144e+03 2.939 3.144e+03 0.260case14 8.082e+03 0.201 8.082e+03 0.84 – – 8.082e+03 0.031case30 5.769e+02 0.788 5.769e+02 2.70 5.765e+02 6.928 5.769e+02 0.074case39 4.189e+04 0.399 4.189e+04 3.26 4.202e+04 7.004 4.186e+04 0.885case57 4.174e+04 0.674 4.174e+04 2.69 – – 4.174e+04 0.857case118 1.297e+05 1.665 1.297e+05 6.57 – – 1.297e+05 1.967case300 7.197e+05 2.410 – 17.68 – – 7.197e+05 90.103


What can this Do

Let’s look at instances from NESTA:Instance MATPOWER sdp pf Our ApproachNESTA Obj. Time [s] Obj. Time [s] Obj. Time [s]case3 lmbd 5.812e+03 0.946 5.789e+03 2.254 5.757e+03 0.149case4 gs 1.564e+02 1.019 1.564e+02 2.392 1.564e+02 0.139case5 pjm (1.599e-01) 0.811 (1.599e-01) 2.708 2.008e+04 0.216case6 c 2.320e+01 0.825 2.320e+01 2.392 2.320e+01 0.379case6 ww 3.143e+03 0.884 3.143e+03 2.776 3.148e+03 0.242case9 wscc 5.296e+03 1.077 5.296e+03 2.621 5.296e+03 0.211case14 ieee 2.440e+02 0.762 2.440e+02 3.164 2.440e+02 0.267case30 as 8.031e+02 0.861 8.031e+02 3.916 8.031e+02 0.608case30 fsr 5.757e+02 0.898 5.757e+02 6.201 5.750e+02 0.613case30 ieee 2.049e+02 0.818 2.049e+02 4.335 2.049e+02 0.788case39 epri 9.651e+04 0.863 9.649e+04 5.676 9.651e+04 0.830case57 ieee 1.143e+03 1.119 1.143e+03 8.053 1.143e+03 0.957case118 ieee (4.433e+02) 1.181 (4.433e+02) 18.097 4.098e+03 4.032case162 dtc (3.123e+03) 0.975 (3.123e+03) 32.153 4.215e+03 8.328


What can this Do

Let’s look at API instances from NESTA:Instance MATPOWER sdp pf Our ApproachAPI Obj. Time [s] Obj. Time [s] Obj. Time [s]case3 lmbd 3.677e+02 1.113 3.333e+02 2.459 3.333e+02 0.029case5 pjm (5.953e+00) 1.002 (5.953e+00) 2.426 3.343e+03 1.059case6 c 8.144e+02 1.034 8.144e+02 2.456 8.139e+02 0.722case9 wscc 6.565e+02 1.135 6.565e+02 2.869 6.565e+02 0.708case14 ieee 3.255e+02 1.035 3.255e+02 3.569 3.245e+02 0.214case30 as 5.711e+02 1.156 5.711e+02 5.433 5.683e+02 0.615case30 fsr 3.721e+02 0.862 3.309e+02 5.276 2.194e+02 0.604case30 ieee 4.155e+02 1.076 4.155e+02 5.293 4.155e+02 0.618case39 epri (1.540e+02) 0.965 (1.540e+02) 5.109 7.427e+03 1.978case57 ieee 1.430e+03 1.041 1.429e+03 7.625 1.429e+03 2.437case162 dtc (1.502e+03) 1.215 (1.502e+03) 43.339 6.003e+03 5.833


What can this DoLet’s recall the performance of general-purpose solvers:

Instance Time [s]Name SDP SeDuMi 1.32 SDPA 7.0 SDPLR 1.03 CSDP 6.0.1 SDPT3 4.0 Mosek 7.0case2w P 0.3 0.003 0 0.02 0.4 0.1case2w D 0.3 0.004 0 0.02 0.4 0.1case5w P 0.4 – 0 0.04 0.5 0.2case5w D 0.4 – 0 0.03 0.5 0.2case9 P 1.0 0.05 25 0.1 0.5 0.3case9 D 1.0 0.07 25 0.1 0.6 0.3case14 P 0.7 0.05 58 0.1 – 0.2case14 D 0.7 0.04 17 0.1 – 0.3case30 P 2.8 – 829 0.8 – –case30 D 6.1 – 282 1.0 – –case30Q P 2.8 – 831 0.8 – –case30Q D 6.2 – 195 1.0 – –case39 P 4.4 – 2769 1.0 – 0.7case39 D 7.7 – 723 1.3 – –case57 P 3.2 – 1930 0.5 – 0.7case57 D 4.0 – 1175 1.0 – 1.8case118 P 10.3 0.9 4400 3.4 – 1.7case118 D 17.1 – – 13.0 – –case300 P 27.5 1.7 – 109.6 – –case300 D 133.7 – – 66.7 – –


The Agenda

An Introduction





Solvers combining a convexification with Newton method onthe non-convex problem


Motivation (case30)

Number of Epochs

0 200 400 600 800 1000

Infeasibility

10-25

10-20

10-15

10-10

10-5

100

105

case30 extended

CD 2

CD 4

CD 8

CD 16

CD 32

CD 64

CD 128

CD 140


Motivation (case30)

Number of Epochs

0 200 400 600 800 1000

Objective Value

400

600

800

1000

1200

1400

case30 extended

CD 2

CD 4

CD 8

CD 16

CD 32

CD 64

CD 128

CD 140


Motivation (case118)

Number of Epochs

0 500 1000 1500 2000 2500 3000

Infeasibility

10-25

10-20

10-15

10-10

10-5

100

105

case118

CD 2

CD 4

CD 8

CD 16

CD 32

CD 64

CD 128

CD 256

CD 500



Number of Epochs

0 500 1000 1500 2000 2500 3000

Objective Value

# 105

1

1.5

2

2.5

3

3.5

4case118

CD 2

CD 4

CD 8

CD 16

CD 32

CD 64

CD 128

CD 256

CD 500



Number of Epochs # 104

0 2 4 6 8 10

Infeasibility

10-20

10-15

10-10

10-5

100

105

1010

case2383wp extended

CD 4

CD 16

CD 64

CD 201




0 2 4 6 8 10

Objective Value

# 106

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5case2383wp extended

CD 4

CD 16

CD 64

CD 201


An Overview

• The quadratic convergence of Newton method is hard to beat.

• Nevertheless, without applying a convexification, Newton method canconverge to particularly poor local optima of a non-convex problem.

• Point-estimation theory of Smale offers a test, based solely onderivative-at-the-current-iterate, for inclusion in a domain ofmonotonicity of a zero of a polynomial.

• Thereby, we can combine the first-order methods for the (hierarchyof) convex Lagrangians with Newton method for (a hierarchy of)non-convex Lagrangians, while preserving convergence guaranteesassociated with the convexification.


Newton’s methodLet f : Rn → Rn be a non-linear system, Df : Rn → Rn×n its Jacobian.

Nf (x) =

x − Df (x)−1f (x) if Df (x) is invertible

x otherwise

Nkf (x) = (Nf · · · Nf )︸︷︷︸

k times

(x).

By Ralf Pfeifer:


Newton’s method

∗approximate solution, associated solution

A point x0 ∈ Rn is an approximate solution of f (x) = 0 with associatedsolution ξ if f (ξ) = 0 and

‖xk − ξ‖ ≤(

1

2

)2k−1

‖x0 − ξ‖

for xk = Nkf (x0).

An approximate solution will converge quadratically to its associatedsolution under the sequence of Newton iterations.


α-theoretic quantities

Let f : Rn → Rn be a nonlinear system, Df : Rn → Rn×n its Jacobian.

• γ = γ(f , x) := supk≥2

∥∥∥∥Df (x)−1Dkf (x)

k!

∥∥∥∥1

k−1

• β = β(f , x) := ‖x − Nf (x)‖ = ‖Df (x)−1f (x)‖• α = α(f , x) := β(f , x) · γ(f , x)


Main results of α-theory

Theorem (Blum, Cucker, Shub, Smale, 2012)

Let f : Rn → Rn be a polynomial system and x , y ∈ Rn with x 6= y .

• If x is an approximate solution of f = 0 with associated solution ξ,then ‖x − ξ‖ ≤ 2β(f , x).

• If α(f , x) < 13−3√

174 ≈ 0.157671, then x is an approximate solution

of f = 0.

• If α(f , x) < 0.03 with ‖x − y‖ · γ(f , x) < 0.05, then x and y areapproximate solutions of f = 0 with the same associated solution.


α-theoretic quantities

• γ = γ(f , x) := supk≥2

∥∥∥∥Df (x)−1Dkf (x)

k!

∥∥∥∥1

k−1

• β = β(f , x) := ‖x − Nf (x)‖ = ‖Df (x)−1f (x)‖• α = α(f , x) := β(f , x) · γ(f , x)

Dkf (x) is the symmetric tensor whose entries are the order-k partial deriva-

tives of f at x . The norm is a norm on operators from the k-fold symmetricpower of Rn to Rn.This norm is difficult to compute.


Bounding γ(f , x)Define some auxiliary quantities:

• ‖x‖21 := 1 +

∑ni=1 |xi |2

• ∆(d)(x)i ,i = d1/2i ‖x‖

di−11

• ‖g‖2 =∑|ν|≤d |gν |2

(ν1!···νn!(d−|ν|)!

d!

)(gν ∈ R, ν ∈ Nn, |ν| =

∑i νi )

• ‖f ‖2 =∑n

i=1 ‖fi‖2

• µ(f , x) = max1, ‖f ‖ · ‖Df (x)−1∆(d)(x)‖

Theorem (Shub and Smale, 1993)

If x ∈ Rn such that Df (x) is invertible, then

γ(f , x) ≤ µ(x , f )D3/2

2‖x‖1. (38)


Example

Minimizing the 2-variable Rosenbrock function

f (x , y) = (1− x)2 + 100(y − x2)2 (here (a, b) = (1, 100))

∇f (x , y) =

[−2(1− x)− 400x(y − x2)

200(y − x2)

]= 0

(x , y) = (1.001, 0.999); ‖(x , y)− (a, a)‖2 =√

2 · 10−3

γ(f , (x , y)) ≤ 3341.051306

β(f , (x , y)) ≈ 0.001858

α(f , (x , y)) ≤ 6.208154


Putting it all together

1: for k = 0, 1, 2, . . . do2: Update (x , λ) using coordinate descent for ∇L = 03: if α(∇L, x) ≤ α0 then

4: S ← (x , λ)5: for ` = 0, 1, 2, . . . do6: Update (x , λ) using Newton step on ∇L = 07: if L(x , λ) > L(S) then8: (x , λ)← S9: break

10: end if11: if T (x , λ) < ε then12: break13: end if14: end for15: end if16: end for


Illustrations (case57)

Number of Epochs

0 100 200 300 400 500 600 700 800

Infeasibility

10-25

10-20

10-15

10-10

10-5

100

105

case57

CD

Hybrid

Newton



Number of Epochs

0 100 200 300 400 500 600 700 800

Objective Value

103

104

105

106

case57

CD

Hybrid

Newton



Number of Epochs

0 200 400 600 800 1000 1200 1400 1600

Infeasibility

10-25

10-20

10-15

10-10

10-5

100

105

case118

CD

Hybrid

Newton



Number of Epochs

0 200 400 600 800 1000 1200 1400 1600

Objective Value

104

105

106

107

108

case118

CD

Hybrid

Newton




0 0.5 1 1.5 2 2.5 3

Infeasibility

10-20

10-15

10-10

10-5

100

105

1010

case300

CD

Hybrid

Newton




0 0.5 1 1.5 2 2.5 3

Objective Value

105

106

107

108

case300

CD

Hybrid

Newton




0 0.5 1 1.5 2 2.5 3

Active Set

10-2

10-1

100

case300

CD

Hybrid

Newton



Solution Time

0 10 20 30 40 50

Infeasibility

10-20

10-15

10-10

10-5

100

105

1010

case300

CD

Hybrid

Newton



Solution Time

0 10 20 30 40 50

Objective Value

105

106

107

108

case300

CD

Hybrid

Newton



Solution Time

0 10 20 30 40 50

Active Set

10-2

10-1

100

case300

CD

Hybrid

Newton



Number of Epochs

0 200 400 600 800 1000

Infeasibility

10-25

10-20

10-15

10-10

10-5

100

105

case30

CD

Hybrid

Newton



Number of Epochs

0 200 400 600 800 1000

Objective Value

101

102

103

104

105

case30

CD

Hybrid

Newton


Illustrations (case2383wp)


0 2 4 6 8 10

Infeasibility

10-20

10-15

10-10

10-5

100

105

1010

1015

case2383wp extended

CD

Hybrid

Newton




0 2 4 6 8 10

Objective Value

105

106

107

108

case2383wp extended

CD

Hybrid

Newton




0 2 4 6 8 10

Active Set

10-2

10-1

100

case2383wp extended

CD

Hybrid

Newton


More Observations

Even for polynomial optimisation, in general, one may avoid theback-tracking:

• A global (non-convex, polynomial) Lagrangian

• Taylor expansion of the square root in a square-the-square-rootabsolute value

• A local (non-convex, polynomial) Lagrangian, where tight inequalitiesare taken as equalities


References

• arXiv:1404.3626: Optimal Power Flow as a Polynomial OptimizationProblem

• arXiv:1412.8054: Power Flow as an Algebraic System

• arXiv:1506.08568: A Low-Rank Coordinate-Descent Algorithm forSemidefinite Programming Relaxations of Optimal Power Flow

• arXiv:1510.02171: Hybrid Methods in Solving Alternating-CurrentOptimal Power Flows

• arXiv:1510.06797: Alternative Linear and Second-order ConeApproximation Approaches for Polynomial Optimization

• arXiv:1601.06739: Adjustably Robust Optimal Power Flows withDemand Uncertainty via Path-based Flows

• arXiv:1603.04375: MINLP in Transmission Expansion Planning


Work in Progress

• Kuramoto Model as Algebraic Systems(with Tianran Chen, Dhagash Mehta, and Matthew Niemerg)

• Hybrid methods for generic polynomial optimisation(with Martin Takac and Jie Liu)

• Rates of convergence in SDP hierarchies(with Martin Takac and Jie Liu)

• A variant of the hierarchy of Lasserre(with Martin Mevissen and Wann-Jiun Ma)

• Control of the related non-linear systems(with Bob Shorten)

• Mixed-integer second-order cone formulations(with Horace Cheng, Andreas Grothey)

• Mixed-criticality robustness


Conclusions

• Steady state problems in power systems are an excellent motivationfor the study of polynomial optimisation

• Some hierarchies of convexifications are very hard to solve, beyond thefirst level

• Algorithm engineering matters.

• We would love to hear from you!

• Jakub Marecek: [email protected]

[email protected]


Definitions

Let n ≥ 0 be an integer and f (z) be a system of n polynomial equations inz ∈ Cn with support (A1, . . . ,An):

f1(z) =∑

α∈A1f1αz

α11 zα2

2 · · · zαnn

...fn(z) =

∑α∈An

fnαzα11 zα2

2 · · · zαnn ,

(39)

where coefficients fiα are non-zero complex numbers. It is well known thatthe polynomials define n projective hypersurfaces in a projective space CPn.


Bezout theorem

Bezout theorem states that either hypersurfaces intersect in an infinite setwith some component of positive dimension, or the number of intersectionpoints, counted with multiplicity, is equal to the product d1 · · · dn, where diis the degree of polynomial i . We call the product d1 · · · dn the usualBezout number.


Multihomogeneous Structure

Definition (Structure)

Any partition of the index set 1, . . . , n into k sets I1, . . . , Ik defines astructure. There, Zj = zi : i ∈ Ij is known as the group of variables foreach set Ij . The associated degree dij of a polynomial fi with respect togroup Zj is

dijdef= max

α∈Ai

∑l∈Ij

αl . (40)

We say that fi has multi-degree (di1, . . . , din).


Multihomogeneous Structure

• Whenever for some j , for all i , the same dij is attained for all α ∈ Ai ,we call the system homogeneous in the group of variables Zj .

• The projective space associated to the group of variables Zj in astructure has dimension

ajdef=

|Ij | − 1 if the system is homogeneous in Zj , and|Ij | otherwise.

(41)


Multi-homogeneous Bezout theorem

Definition (Multi-homogeneous Bezout Number)

Assuming n =∑k

j=1 aj , the multi-homogeneous Bezout numberBez(A1, . . . ,An; I1, . . . , Ik) is defined as the coefficient of the term∏k

j=1 ζajj , where aj is the associated dimension, within the polynomial∏n

i=1

∑kj=1 dijζj , in variables ζj , j = 1 . . . k with coefficients dij are the

associated degrees; that is (d11ζ1 + d12ζ2 + . . . + d1kζk) (d21ζ1 + d22ζ2 +. . .+ d2kζk) · · · (d2nζ1 + d2nζ2 + . . .+ dnkζk).



Consider the example of Wampler in x ∈ C3:p1(z) = x2

1 + x2 + 1,p2(z) = x1x3 + x2 + 2,p3(z) = x2x3 + x3 + 3,

(42)

with the usual Bezout number of 8. Considering the partition x1, x2,x3, where d11 = 2, d12 = 0, d21 = d22 = d31 = d32 = 1. the monomialζ2

1ζ12 is to be looked up in the polynomial 2ζ1(ζ1 + ζ2)2. The

corresponding multi-homogeneous Bezout number is hence 4 and this isthe minimum across all possible structures.



• In general, the multi-homogeneous Bezout numberBez(A1, . . . ,An; I1, . . . , Ik) is an upper bound on the number ofisolated roots in CPa1 × · · · × CPak , and thereby an upper bounds thenumber of isolated finite complex roots.

• There are a variety of additional methods for computing themulti-homogeneous Bezout number.

• In the particular case where A = A1 = · · · = An, we denote

Bez(A1, . . . ,An; I1, . . . , Ik)def=

(n

a1 a2 · · · ak

) k∏j=1

dajj , (43)

where dj = dij (equal for each i) and the multinomial coefficient(n

a1 a2 · · · ak

)def=

n!

a1! a2! · · · ak !

is the coefficient of∏k

j=1 ζakj in (ζ1 + · · ·+ ζk)n with n =

∑kj=1 aj , as

above.



• The multi-homogeneous Bezout number provides a sharper bound onthe number of isolated solutions of a system of equations than theusual Bezout number

∏ni=1 di = d1 · · · dn.

• The example of the eigenvalue problem: the Bezout number is 2n vs.structure with multi-homogeneous Bezout number of n.


Bernstein / Khovanski / Kushnirenko

• A bound on the number of isolated nonzero complex solutions whichtakes into consideration the monomials that appear in the polynomialsystem via Newton polytopes.

• A convex set is a set of points in which the line segment connectingany pair of points in the set also lie in that set.

• The convex hull of a set is the minimal convex set containing that set.

• Given a polynomial, each of its terms give rise to an exponent vector.For instance, for the term x3y2z1, the exponent vector is simply thevector whose entries are the exponents of x , y and z , respectively, i.e.,(3, 2, 1). The choice of this ordering is inconsequential as long as it iskept the same for each equation.



• The set of all exponent vectors derived from the nonzero terms of anpolynomial equation is called the support of that equation.

• For a polynomial, the convex hull of its support is known as theNewton polytope of that polynomial.

• Given n convex polytopes Q1, . . . ,Qn ⊂ Rn and positive real numbersλ1, . . . , λn Minkowski’s Theorem states that the n-dimensional volumeof the Minkowski sum λ1Q1 + · · ·+ λnQn, defined as

λ1q1 + · · ·+ λnqn | qi ∈ Qi for i = 1, . . . , n

is a homogeneous polynomial of degree n in the variables λ1, . . . , λn.



• The coefficient associated with the monomial λ1 · · ·λn in thispolynomial is known as the mixed volume of the polytopes Q1, . . . ,Qn.

• In the simplest case, the mixed volume of two line segments on theplane is precisely the area of the parallelogram spanned by translationsof these two line segments.

(semi-)algebraic geometry at ibm research ireland · 2016-11-23 · ibm research - ireland...

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