complex structure of the optimal power flow problem · 2016-01-26 · complex structure of the...

Complex Structure of the Optimal Power FlowProblem

Cedric JOSZ, Daniel K. [email protected]

Talk at the University of Illinois at Urbana-Champaign

November 16th 2015

Cedric JOSZ, Daniel K. MOLZAHN [email protected] Talk at the University of Illinois at Urbana-Champaign

My research project

Page du projet MISTIS http://mistis.inrialpes.fr/

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Université Pierre et Marie CURIE - Sciences et Médecine - ... http://www.upmc.fr/fr/index.html

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Cedric Josz University of Paris VI (UPMC)

Jean Charles GilbertFrench national institute

in scientific computing (INRIA)

Jean Maeght, Patrick PanciaticiFrench transmission

system operator (RTE)

French high-voltage network: 400 and 225 kV

Benchmark network

Underlying graph

Optimal power flow

Motivations

Time (s)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Two Voltages in Steady State

Re(V)-1 -0.5 0 0.5 1

Voltages in Complex Plane: Local Optimum of 9241-bus European Network

Re(V)-1 -0.5 0 0.5 1

Voltages in Complex Plane: Local Optimum of 9241-bus European Network

Two-bus network

“g” = conductance

“b” = susceptance

Power loss minimization

Minimize

g |v1|2 − g v1v2 − g v2v1 + g |v2|2

over v1, v2 ∈ C subject to

−g − ib

2v1v2 −

g + ib

2v2v1 + g |v2|2 + pdem

b + ig

2v1v2 +

b − ig

2v2v1 − b |v2|2 + qdem

|v1|2 6 (vmax1 )2

|v2|2 6 (vmax2 )2

Quadratically-constrained quadratic programming

QCQP-C : infz∈Cn

zHH0z s.t. zHHiz 6 hi , i = 1, . . . ,m

(H0, . . . ,Hm are Hermitian matrices)

(h1, . . . , hm ∈ R)

which can also be written:

infz∈Cn

trace(H0zzH) s.t. trace(Hizz

H) 6 hi , i = 1, . . . ,m

Quadratically-constrained quadratic programming

QCQP-C : infz∈Cn

zHH0z s.t. zHHiz 6 hi , i = 1, . . . ,m

(H0, . . . ,Hm are Hermitian matrices)

(h1, . . . , hm ∈ R)

which can also be written:

infz∈Cn

trace(H0zzH) s.t. trace(Hizz

H) 6 hi , i = 1, . . . ,m

Non-commutative diagram

Conversion from complex to real numbers

Ring homomorphism:

Λ : (Cn×n,+,×) −→ (R2n×2n,+,×)

Z 7−→(<Z −=Z=Z <Z

Useful properties

1 trace[HiZ ] = 12 trace[Λ(HiZ )] = 1

2 trace [Λ(Hi )Λ(Z )]

2 Z < 0 ⇐⇒ Λ(Z ) < 0

3 Z < 0 and rank Z = 1 ⇐⇒ Λ(Z ) < 0 and rank Λ(Z ) = 2

Conversion from complex to real numbers

Ring homomorphism:

Λ : (Cn×n,+,×) −→ (R2n×2n,+,×)

Z 7−→(<Z −=Z=Z <Z

Useful properties

1 trace[HiZ ] = 12 trace[Λ(HiZ )] = 1

2 trace [Λ(Hi )Λ(Z )]

2 Z < 0 ⇐⇒ Λ(Z ) < 0

3 Z < 0 and rank Z = 1 ⇐⇒ Λ(Z ) < 0 and rank Λ(Z ) = 2

Preprocessed data

Test Number of Number ofCase Complex EdgesName Variables in Graph

GB-2224 2,053 2,581PL-2383wp 2,177 2,651PL-2736sp 2,182 2,675PL-2737sop 2,183 2,675PL-2746wop 2,189 2,708PL-2746wp 2,192 2,686PL-3012wp 2,292 2,805PL-3120sp 2,314 2,835PEGASE-89 70 185PEGASE-1354 983 1,526PEGASE-2869 2,120 3,487PEGASE-9241 7,154 12,292PEGASE-9241R 7,154 12,292

Shor relaxation (generation cost minimization)

Case SDP-R SDP-CName Val. ($/hr) Time (sec) Val. ($/hr) Time (sec)

GB-2224 1,928,194 10.9 1,928,444 6.2PL-2383wp 1,862,979 48.1 1,862,985 23.0PL-2736sp* 1,307,749 35.7 1,307,764 22.0PL-2737sop* 777,505 41.7 777,539 19.5PL-2746wop* 1,208,168 51.1 1,208,182 22.8PL-2746wp 1,631,589 43.8 1,631,655 20.0PL-3012wp 2,588,249 52.8 2,588,259 24.3PL-3120sp 2,140,568 64.4 2,140,605 25.5PEGASE-89* 5,819 1.5 5,819 0.9PEGASE-1354 74,035 11.2 74,035 5.6PEGASE-2869 133,936 38.2 133,936 20.6PEGASE-9241 310,658 369.7 310,662 136.1PEGASE-9241R 315,848 317.2 315,731 95.9

Second-order conic programming (generation cost min.)

Case SOCP-R SOCP-CName Val. ($/hr) Time (sec) Val. ($/hr) Time (sec)

GB-2224 1,855,393 3.5 1,925,723 1.4PL-2383wp 1,776,726 8.5 1,849,906 2.4PL-2736sp 1,278,926 4.8 1,303,958 1.7PL-2737sop 765,184 5.5 775,672 1.6PL-2746wop 1,180,352 5.1 1,203,821 1.7PL-2746wp 1,586,226 5.5 1,626,418 1.7PL-3012wp 2,499,097 5.9 2,571,422 2.0PL-3120sp 2,080,418 6.2 2,131,258 2.2PEGASE-89 5,744 0.5 5,810 0.4PEGASE-1354 73,102 3.4 73,999 1.5PEGASE-2869 132,520 9.0 133,869 2.7PEGASE-9241 306,050 35.3 309,309 10.0PEGASE-9241R 312,682 36.7 315,411 5.4

We need better relaxations!

Moment/sum-of-squares hierarchy

Minimize

g |v1|2 − g v1v2 − g v2v1 + g |v2|2

−g − ib

2v1v2 −

g + ib

2v2v1 + g |v2|2 + pdem

b + ig

2v1v2 +

b − ig

2v2v1 − b |v2|2 + qdem

|v1|2 6 (vmax1 )2

|v2|2 6 (vmax2 )2

Complex polynomial optimization

Minimize

f (z) :=∑α,β

fα,β zαzβ (where zα := zα1

1 . . . zαnn )

over z ∈ Cn subject to

gi (z) :=∑α,β

gi ,α,β zαzβ > 0 , i = 1, . . . ,m

All functions are real-valued complex polynomialsa.k.a.

Hermitian symmetric polynomials

Stone-Weierstrass theorem

Real version

Any continuous function on a compact set can be approximated by∑α

fn,αxα uniform convergence−−−−−−−−−−−−→

n−→+∞f ∈ C(K ⊂ Rn,R)

Complex version

Any continuous function on a compact set(((

(((((((

((hhhhhhhhhhhhcan be approximated by∑

fn,αzα

��

��XXXXXXXXX

uniform convergence−−−−−−−−−−−−→n−→+∞

f ∈ C(K ⊂ Cn,C)

Real version

n−→+∞f ∈ C(K ⊂ Rn,R)

Complex version

Any continuous function on a compact set(((

(((((((

((hhhhhhhhhhhhcan be approximated by∑

fn,αzα

��

��XXXXXXXXX

f ∈ C(K ⊂ Cn,C)

Real version

n−→+∞f ∈ C(K ⊂ Rn,R)

Complex version

Any continuous function on a compact set can be approximated by∑α,β

fn,α,β zαzβ

f ∈ C(K ⊂ Cn,C)

Moment approach

Non-convex optimization

Global optimum and value

Variable = point

Variable = interval

Average on the interval

The optimal interval

Weighted average

variable = probability distribution

Optimal probability distribution

Real moment hierarchy (Lasserre 2000)

Complex moment hierarchy (J. and Molzahn 2015)

Dual point of view

f opt := inf f subject to gi > 0 , i = 1, . . . ,m

Very simple observation:

ε > 0 =⇒ f − f opt︸︷︷︸>0

+ ε > 0 on the feasible set

Dual point of view

Very simple observation:

ε > 0 =⇒ f − f opt︸︷︷︸>0

+ ε > 0 on the feasible set

Positivstellensatz

Real version (Putinar 1993)

f (x) > 0 for all x ∈ Rn s.t. gi (x) > 0 and 1− x21 − . . .− x2

f (x) = σ0(x) +∑m

i=1 σi (x)gi (x) + σm+1(x)(1− x21 − . . .− x2

(where σi ’s are sums of squares)

Example:

x + 2 > 0 for all x ∈ R s.t. 1− x2 > 0

x + 2 =(

+ 1√2

)2(1− x2)

Positivstellensatz

f (x) = σ0(x) +∑m

i=1 σi (x)gi (x) + σm+1(x)(1− x21 − . . .− x2

Example:

x + 2 > 0 for all x ∈ R s.t. 1− x2 > 0

x + 2 =(

+ 1√2

)2(1− x2)

Positivstellensatz

f (x) = σ0(x) +∑m

i=1 σi (x)gi (x) + σm+1(x)(1− x21 − . . .− x2

Example:

x + 2 > 0 for all x ∈ R s.t. 1− x2 > 0

x + 2 =(

+ 1√2

)2(1− x2)

Positivstellensatz

Complex version (D’Angelo and Putinar 2008)

f (z) > 0 for all z ∈ Cn s.t. gi (z) > 0 and 1− |z1|2 − ..− |zn|2 = 0

f (z) = σ0(z) +∑m

i=1 σi (z)gi (z) + pm+1(z)(1− |z1|2 − . . .− |zn|2)

(where σi ’s are Hermitian sums of squares)

Example:

z + z + 3 > 0 for all z ∈ C s.t. 1− |z |2 = 0

z + z + 3 = |1|2 + |z + 1|2 + 1× (1− |z |2)

Positivstellensatz

f (z) = σ0(z) +∑m

i=1 σi (z)gi (z) + pm+1(z)(1− |z1|2 − . . .− |zn|2)

Example:

z + z + 3 > 0 for all z ∈ C s.t. 1− |z |2 = 0

z + z + 3 = |1|2 + |z + 1|2 + 1× (1− |z |2)

Positivstellensatz

f (z) = σ0(z) +∑m

i=1 σi (z)gi (z) + pm+1(z)(1− |z1|2 − . . .− |zn|2)

Example:

z + z + 3 > 0 for all z ∈ C s.t. 1− |z |2 = 0

z + z + 3 = |1|2 + |z + 1|2 + 1× (1− |z |2)

Relationship with optimization

ε > 0 =⇒ f−f opt + ε > 0 on the feasible set

ε > 0 =⇒ f − (f opt − ε) > 0 on the feasible set

ε > 0 =⇒ f − (f opt − ε)︸︷︷︸λ

> 0 on the feasible set

f opt = supλ,σi

λ subject to f − λ = σ0 +m∑i=1

(under the assumption in the Positivstellensatz)

Relationship with semidefinite programming

Real version

σ(x) =∑

α σαxα is a sum of squares

∑k(∑

γ pk,γxγ)2

⇐⇒

Matrix (σα+β)α,β is positive semidefinite

Complex version

σ(z) =∑

α,β σα,β zαzβ is a Hermitian sum of squares∑k |∑

γ pk,γzγ |2

⇐⇒

Matrix (σα,β)α,β is Hermitian positive semidefinite

Real version

σ(x) =∑

α σαxα is a sum of squares

∑k(∑

γ pk,γxγ)2

⇐⇒

Matrix (σα+β)α,β is positive semidefinite

Complex version

σ(z) =∑

α,β σα,β zαzβ is a Hermitian sum of squares∑k |∑

γ pk,γzγ |2

⇐⇒

Matrix (σα,β)α,β is Hermitian positive semidefinite

Sum-of-squares hierarchy

supλ,σi

λ subject to

{f − λ = σ0 +

∑mi=1 σigi

deg(σ0), deg(σigi ) 6 2d

Real version

One of the constraints is a ball x21 + . . .+ x2

n 6 R2

=⇒GLOBAL convergence when d −→ +∞

Complex version

One of the constraints is a sphere |z1|2 + . . .+ |zn|2 = R2

supλ,σi

λ subject to

{f − λ = σ0 +

∑mi=1 σigi

Real version

n 6 R2

Complex version

supλ,σi

λ subject to

{f − λ = σ0 +

∑mi=1 σigi

Real version

n 6 R2

Complex version

Minimize

g |v1|2 − g v1v2 − g v2v1 + g |v2|2

−g − ib

2v1v2 −

g + ib

2v2v1 + g |v2|2 + pdem

b + ig

2v1v2 +

b − ig

2v2v1 − b |v2|2 + qdem

|v1|2 6 (vmax1 )2

|v2|2 6 (vmax2 )2

Slack variable

Minimize

g |v1|2 − g v1v2 − g v2v1 + g |v2|2

over v1, v2, v3 ∈ C subject to

−g − ib

2v1v2 −

g + ib

2v2v1 + g |v2|2 + pdem

b + ig

2v1v2 +

b − ig

2v2v1 − b |v2|2 + qdem

|v1|2 6 (vmax1 )2

|v2|2 6 (vmax2 )2

|v1|2 + |v2|2 + |v3|2 = (vmax1 )2 + (vmax

Actually works in practice!

D’Angelo and Putinar (2008):

18= inf

z∈C1− 4

3|z |2 +

18|z |4 s.t. 1− |z |2 > 0

→ complex hierarchy yields −0.3333 at second and third orders

J. and Molzahn (2015):

18= inf

z1,z2∈C1− 4

3|z1|2 +

18|z1|4 s.t. 1− |z1|2 − |z2|2 = 0

→ complex hierarchy yields 0.0556 ≈ 118 at second order

Actually works in practice!

D’Angelo and Putinar (2008):

18= inf

z∈C1− 4

3|z |2 +

18|z |4 s.t. 1− |z |2 > 0

→ complex hierarchy yields −0.3333 at second and third orders

18= inf

z1,z2∈C1− 4

3|z1|2 +

18|z1|4 s.t. 1− |z1|2 − |z2|2 = 0

→ complex hierarchy yields 0.0556 ≈ 118 at second order

Successful sum of squares decomposition

1− 4

3|z1|2 +

18|z1|4 − 0.0556

0.2780|z2|2 + 0.2776|z1z2|2 + 0.6667|z2|4

(0.9444− 0.3889|z1|2 + 0.6665|z2|2)(1− |z1|2 − |z2|2)

Another example: an ellipse

Putinar and Scheiderer (2012):

1 = infz∈C

3− |z |2 s.t. |z |2 − 1

4z2 − 1

4z2 − 1 = 0

→ complex hierarchy is unbounded at second and third orders

1 = infz1,z2∈C

3− |z1|2 s.t.

{|z1|2 − 1

4z21 − 1

4 z21 − 1 = 0

3− |z1|2 − |z2|2 = 0

→ complex hierarchy yields 0.6813 at second order and 0.9699 atthird order

Another example: an ellipse

1 = infz∈C

3− |z |2 s.t. |z |2 − 1

4z2 − 1

4z2 − 1 = 0

1 = infz1,z2∈C

3− |z1|2 s.t.

{|z1|2 − 1

4z21 − 1

4 z21 − 1 = 0

3− |z1|2 − |z2|2 = 0

Nonnegative slack variable

1 = infz∈C

3− |z |2 s.t. |z |2 − 1

4z2 − 1

4z2 − 1 = 0

1 = infz1,z2∈C

3− |z1|2 s.t.

|z1|2 − 1

4z21 − 1

4 z21 − 1 = 0

3− |z1|2 − |z2|2 = 0

iz2 − iz2 = 0

z2 + z2 > 0

Complex plane

Ellipse

Sphere constraint

Projection on semi-sphere

Edge of chips

Complex vs. real hierarchy

Size: asymptotically 2d−1 times smaller at order d

Bound quality: poorer or equal at each order

Global optimality check: rank of moment matrix

Convergence guarantee: if feasible set bounded by radius R, add

x21 + . . .+ x2

n 6 R2 to real hierarchy

zn+1 and |z1|2 + . . .+ |zn+1|2 = R2 to complex hierarchy

x21 + . . .+ x2

Oscillatory polynomial optimization

Lemma for ϕ(z) =∑

α,β ϕα,β zαzβ

∀z ∈ Cn, ∀θ ∈ R, ϕ(e iθz) = ϕ(z)

⇐⇒

∀α, β, (|α| − |β|)ϕα,β = 0

Example:ϕ(z) = z2z2 with z ∈ C is oscillatoryϕ(z) = z3z2 with z ∈ C is not oscillatory

In an oscillatory problem

minimum order of the complex hierarchy=

minimum order of the real hierarchy

∀z ∈ Cn, ∀θ ∈ R, ϕ(e iθz) = ϕ(z)

⇐⇒

∀α, β, (|α| − |β|)ϕα,β = 0

∀z ∈ Cn, ∀θ ∈ R, ϕ(e iθz) = ϕ(z)

⇐⇒

∀α, β, (|α| − |β|)ϕα,β = 0

Exploiting sparsity

Moment/sum-of-squares hierarchy (power loss min.)

Case MSOS-R MSOS-CName Val. (MW) Time (sec) Val. (MW) Time (sec)

PL-2383wp 24,990 583.4 24,991 53.9PL-2736sp 18,334 44.0 18,335 17.8PL-2737sop 11,397 52.4 11,397 25.7PL-2746wop 19,210 2,662.4 19,212 124.3PL-2746wp 25,267 45.9 25,269 18.5PL-3012wp 27,642 318.7 27,644 141.0PL-3120sp 21,512 386.6 21,512 193.9PEGASE-1354 74,043 406.9 74,042 1,132.6PEGASE-2869 133,944 921.3 133,939 700.8

Do we need a sphere constraint to ensure convergence?

Minimizeg |v1|2 − g v1v2 − g v2v1 + g |v2|2

−g − ib

2v1v2 −

g + ib

2v2v1 + g |v2|2 + pdem

b + ig

2v1v2 +

b − ig

2v2v1 − b |v2|2 + qdem

|v1|2 6 (vmax1 )2

|v2|2 6 (vmax2 )2

Instrinsic property of electricity?

p(v1, v2)

(−g − ib

2v1v2 −

g + ib

2v2v1 + g |v2|2 + pdem

q(v1, v2)

(b + ig

2v1v2 +

b − ig

2v2v1 − b |v2|2 + qdem

|v1|2 + |v2|2 + Hermitian SOS + constant ???

Instrinsic property of electricity?

p(v1, v2)

(−g − ib

2v1v2 −

g + ib

2v2v1 + g |v2|2 + pdem

q(v1, v2)

(b + ig

2v1v2 +

b − ig

2v2v1 − b |v2|2 + qdem

|v1|2 + |v2|2 + Hermitian SOS + constant ???

References

RTE. “Carte du reseau de transport 400 000 et 225 000 volts,”Gestionnaire du Reseau de Transport, Centre National d’ExpertiseReseau, Depot legal : Mai 2011. [link]

University of Washington, Electrical Engineering, Power SystemsTest Case Archive. [link]

N.Z. Shor, Quadratic Optimization Problems, Sov. J. Comput.Syst. Sci., 25 (1987), pp. 1–11.

J. Lavaei and S.H. Low, Zero Duality Gap in Optimal Power FlowProblem, IEEE Trans. Power Syst., 27 (2012), pp. 92–107.

J. B. Lasserre, Global Optimization with Polynomials and theProblem of Moments, SIAM J. Optim., 11 (2001), pp. 796–817.

References

P. A. Parrilo, Structured semidefinite programs and semialgebraicgeometry methods in robustness and optimization, DoctoralThesis, California Institute of Technology, 2000.

D. K. Molzahn and Ian A. Hiskens, Sparsity-ExploitingMoment-Based Relaxations of the Optimal Power Flow Problem,IEEE Transaction on Power Systems, vol. 30, no. 6, pp.3168-3180, November 2015.

J. P. D’Angelo and M. Putinar, Polynomial Optimization onOdd-Dimensional Spheres, in Emerging Applications of AlgebraicGeometry, Springer New York, 2008.

M. Putinar and C. Scheiderer, Quillen Property of Real AlgebraicVarieties, to appear in Munster J. Math.

References

M. Putinar and C. Scheiderer, Hermitian Algebra on the Ellipse,Illinois J. Math., 56 (2012), pp. 213–220.

C. J. and D. K. Molzahn, Moment/Sum-of-Squares Hierarchy forComplex Polynomial Optimization, submitted to SIAM J. Optim.

M. Schweighofer, Optimization of Polynomials on CompactSemialgebraic Sets, SIAM J. Optim., 15 (2005), pp. 805–825.

E. J. Anderson and P. Nash, Linear Programming inInfinite-Dimensional Spaces, Theory and Applications, Wiley Int.Ser. Disc. Math. Optim., 1987.

C. D. Aliprantis and K. Border, Infinite Dimensional Analysis, AHitchhiker’s guide, Second Edition, Springer-Verlag BerlinHeidelberg, 1999.

Images

Ecociel France, Panneaux Solaires Photovoltaıques. [link]

Autolib’ va s’exporter a Lyon, L’Argus, Actualites auto, May 30th

2013. [link]

Transmission Line Monitor and Dynamic Line Rating System,Lindsey. [link]

Dispatching national, Rte France. [link]

Paint, Version 6.1, Microsoft Windows, 2009.

Jean Bernard Lasserre, Tuchan, Photograph. [link]

POV-Ray, The Persistence of Vision Raytracer, Version 3.7.

Kellogg’s achete les chips Pringles a Procter & Gamble, Le Figaro,December 15th 2012. [link]

Images

Earth at night from NASA’s Suomi National Polar-orbitingPartnership Satellite, Earth shimmers at Christmas as billions offairy lights become visible from space, The Telegraph, ScienceNews. [link]

Perry Babin, Oscilloscope, Basic Car Audio Electronics. [link]

MRI Image of Knee, Medical Media Images, Precision Color MedialImages. [link]

David Ratledge, Orion Nebula taken with a modified Canon 40D(Baader filter) from lighted polluted Lancashire!, Digital SLRImaging. [link]

Radar, Technology, Telecommunications and Media, Speyside,Corporate Relations. [link]

Oscillatory phenomena in physical systems

Schrodinger equation: HΨ = i~∂Ψ

Future directions

Enhance tractability of complex hierarchy

When are real and complex hierarchies equal?

Do power flow equations possess the Quillen property?

Thank you for your attention!

Feel free to me contact at

[email protected]

for questions or suggestions.

Backup slides

Name Active power minimization (MW)Case MSOS-R MSOS-C Matpower

PL-2383wp 24,990 24,991 24,991PL-2736sp 18,334 18,335 18,336PL-2737sop 11,397 11,397 11,397PL-2746wop 19,210 19,212 19,212PL-2746wp 25,267 25,269 25,269PL-3012wp 27,642 27,644 27,646PL-3120sp 21,512 21,512 21,513PEGASE-1354 74,043 74,042 74,043PEGASE-2869 133,944 133,939 133,945

1 = infz∈C

3− |z |2 s.t. |z |2 − 1

4z2 − 1

4z2 − 1 = 0

1 = infz1,z2∈C

3− |z1|2 s.t.

|z1|2 − 1

4z21 − 1

4 z21 − 1 = 0

3− |z1|2 − |z2|2 = 0

iz2 − iz2 = 0

z2 + z2 > 0

3rd order complex moment matrix

(0, 0) 1 0 1 2 0 1 0 2 0 1(1, 0) 0 2 0 0 2 0 4 0 2 0(0, 1) 1 0 1 2 0 1 0 2 0 1(2, 0) 2 0 2 4 0 2 0 4 0 2(1, 1) 0 2 0 0 2 0 4 0 2 0(0, 2) 1 0 1 2 0 1 0 2 0 1(3, 0) 0 4 0 0 4 0 8 0 4 0(2, 1) 2 0 2 4 0 2 0 4 0 2(1, 2) 0 2 0 0 2 0 4 0 2 0(0, 3) 1 0 1 2 0 1 0 2 0 1

Certificate of global optimality

(0, 0) 1 0 1 2 0 1 0 2 0 1(1, 0) 0 2 0 0 2 0 4 0 2 0(0, 1) 1 0 1 2 0 1 0 2 0 1(2, 0) 2 0 2 4 0 2 0 4 0 2(1, 1) 0 2 0 0 2 0 4 0 2 0(0, 2) 1 0 1 2 0 1 0 2 0 1(3, 0) 0 4 0 0 4 0 8 0 4 0(2, 1) 2 0 2 4 0 2 0 4 0 2(1, 2) 0 2 0 0 2 0 4 0 2 0(0, 3) 1 0 1 2 0 1 0 2 0 1

rank M1(y) = rank M3(y) = 2 so there are two global solutions

Complex polynomial optimization

Minimize

f (z) :=∑α,β

fα,β zαzβ (where zα := zα1

1 . . . zαnn )

over z ∈ Cn subject to

gi (z) :=∑α,β

gi ,α,β zαzβ = 0 , 1 6 i 6 m

hj(z) :=∑α,β

hj ,α,β zαzβ > 0 , 1 6 j 6 p

Quillen property of semi-algebraic varieties

Ideal generated by equality constraints:

I := Ch[z , z ]g1 + . . .+ Ch[z , z ]gm

Semiring defined by ideal:

S := I + Σ[z ] where Σ[z ] := {∑k

|pk |2 | pk ∈ C[z ] }

Module defined by equality and inequality constraints:

M := S + Σ[z ]h1 + . . .+ Σ[z ]hp

Feasible set of complex polynomial optimization:

XM := { z ∈ Cn | ∀ϕ ∈ M, ϕ(z) > 0 }

Putinar and Scheiderer (to appear in Munster J. Math.)

If R2 − |z1|2 − . . .− |zn|2 ∈ S , then: f|XM> 0 =⇒ f ∈ M

I := Ch[z , z ]g1 + . . .+ Ch[z , z ]gm

S := I + Σ[z ] where Σ[z ] := {∑k

|pk |2 | pk ∈ C[z ] }

M := S + Σ[z ]h1 + . . .+ Σ[z ]hp

XM := { z ∈ Cn | ∀ϕ ∈ M, ϕ(z) > 0 }

I := Ch[z , z ]g1 + . . .+ Ch[z , z ]gm

S := I + Σ[z ] where Σ[z ] := {∑k

|pk |2 | pk ∈ C[z ] }

M := S + Σ[z ]h1 + . . .+ Σ[z ]hp

XM := { z ∈ Cn | ∀ϕ ∈ M, ϕ(z) > 0 }

I := Ch[z , z ]g1 + . . .+ Ch[z , z ]gm

S := I + Σ[z ] where Σ[z ] := {∑k

|pk |2 | pk ∈ C[z ] }

M := S + Σ[z ]h1 + . . .+ Σ[z ]hp

XM := { z ∈ Cn | ∀ϕ ∈ M, ϕ(z) > 0 }

I := Ch[z , z ]g1 + . . .+ Ch[z , z ]gm

S := I + Σ[z ] where Σ[z ] := {∑k

|pk |2 | pk ∈ C[z ] }

M := S + Σ[z ]h1 + . . .+ Σ[z ]hp

XM := { z ∈ Cn | ∀ϕ ∈ M, ϕ(z) > 0 }

I := Ch[z , z ]g1 + . . .+ Ch[z , z ]gm

S := I + Σ[z ] where Σ[z ] := {∑k

|pk |2 | pk ∈ C[z ] }

M := S + Σ[z ]h1 + . . .+ Σ[z ]hp

XM := { z ∈ Cn | ∀ϕ ∈ M, ϕ(z) > 0 }

I := Ch[z , z ]g1 + . . .+ Ch[z , z ]gm

S := I + Σ[z ] where Σ[z ] := {∑k

|pk |2 | pk ∈ C[z ] }

M := S + Σ[z ]h1 + . . .+ Σ[z ]hp

XM := { z ∈ Cn | ∀ϕ ∈ M, ϕ(z) > 0 }

If S is Archimedean, then M has Quillen’s property.

Quillen property ⇐⇒ inf{f (z) | z ∈ XM} = sup{λ | f − λ ∈ M}

Duality bracket:〈., .〉 : Ch[z , z ]×H −→ R

(ϕ, y) 7−→∑

α,β ϕα,βyα,β

Continuous operator:A : Ch[z , z ] −→ Ch[z , z ]

ϕ 7−→ ϕ− ϕ0,0

Convex cone: M = I + Σ[z ] + Σ[z ]h1 + . . .+ Σ[z ]hp

Constants: δ0,0 ∈ H and b := Af

Linear program of infinite dimension

(P) : infϕ∈Ch[z,z] 〈ϕ, δ0,0〉 s.t. Aϕ = b and ϕ ∈ M(D) : supy∈H 〈b, y〉 s.t. δ0,0 − A∗y ∈ M∗

(ϕ, y) 7−→∑

α,β ϕα,βyα,β

ϕ 7−→ ϕ− ϕ0,0

(ϕ, y) 7−→∑

α,β ϕα,βyα,β

ϕ 7−→ ϕ− ϕ0,0

(ϕ, y) 7−→∑

α,β ϕα,βyα,β

ϕ 7−→ ϕ− ϕ0,0

(ϕ, y) 7−→∑

α,β ϕα,βyα,β

ϕ 7−→ ϕ− ϕ0,0

(ϕ, y) 7−→∑

α,β ϕα,βyα,β

ϕ 7−→ ϕ− ϕ0,0

Primal problem

ϕ ∈ Σ[z ] ⇐⇒ (ϕα,β)α,β < 0

Dual problem

M∗ =(∑

i Ch[z , z ]gi + Σ[z ] +∑

j Σ[z ]hj

⋂i (|C[z ]|2gi )⊥ ∩ (|C[z ]|2)∗ ∩

⋂j(|C[z ]|2hj)∗

For example, if φ ∈ C[z ], then:

〈|φ|2hj , y〉 =∑α,β

φαφβ〈zαzβhj(z), y〉 > 0

Primal problem

ϕ ∈ Σ[z ] ⇐⇒ (ϕα,β)α,β < 0

Dual problem

M∗ =(∑

i Ch[z , z ]gi + Σ[z ] +∑

j Σ[z ]hj

⋂i (|C[z ]|2gi )⊥ ∩ (|C[z ]|2)∗ ∩

⋂j(|C[z ]|2hj)∗

〈|φ|2hj , y〉 =∑α,β

Primal problem

ϕ ∈ Σ[z ] ⇐⇒ (ϕα,β)α,β < 0

Dual problem

M∗ =(∑

i Ch[z , z ]gi + Σ[z ] +∑

j Σ[z ]hj

⋂i (|C[z ]|2gi )⊥ ∩ (|C[z ]|2)∗ ∩

⋂j(|C[z ]|2hj)∗

〈|φ|2hj , y〉 =∑α,β

Complex moment/sum-of-squares hierarchy

f opt := infz∈Cn f (z) s.t. gi (z) = 0 and hj(z) > 0

Semidefinite programming hierarchy

infy∈Hd〈f , y〉

s.t. y0,0 = 1 and Md(y) < 0Md−d(gi )(giy) = 0, i = 1, . . . ,mMd−d(hj )(hjy) < 0, j = 1, . . . , p

supλ,r ,σ λ

s.t. f − λ = σ +∑m

i=0 rigi +∑p

j=0 σjhjλ ∈ R, σ ∈ Σd [z ], ri ∈ Rd−d(gi )[z , z ], σj ∈ Σd−d(hj )[z ]

Md−k(ϕy) is a Hermitian matrix indexed by |α|, |β| 6 d − k and:

( (Md−k(ϕy) )α,β :=∑γ,δ

ϕγ,δ yα+γ,β+δ

supλ,r ,σ λ

s.t. f − λ = σ +∑m

i=0 rigi +∑p

( (Md−k(ϕy) )α,β :=∑γ,δ

supλ,r ,σ λ

s.t. f − λ = σ +∑m

i=0 rigi +∑p

( (Md−k(ϕy) )α,β :=∑γ,δ

Algebraic geometry and measure theory

Quillen property:

∀ϕ ∈ Ch[z , z ], ϕ|XM> 0 =⇒ ϕ ∈ M

Strong moment property:

∀y ∈ H, y ∈ M∗ =⇒ yα,β =∫XM

zαzβdµ

Quillen’s property =⇒ Strong moment property

Converse true if M is Archimedean

Algebraic geometry and measure theory

Quillen property:

∀ϕ ∈ Ch[z , z ], ϕ|XM> 0 =⇒ ϕ ∈ M

Strong moment property:

∀y ∈ H, y ∈ M∗ =⇒ yα,β =∫XM

zαzβdµ

Quillen’s property =⇒ Strong moment property

Converse true if M is Archimedean

Optimization

Linear program

Quillen property ⇐⇒ val(P) = f0,0 − f opt

Strong moment property ⇐⇒ val(D) = f0,0 − f opt

Weak duality

val(P) > val(D) > f0,0 − f opt

Optimization

Linear program

Weak duality

Optimization

Linear program

Weak duality

complex structure of the optimal power flow problem · 2016-01-26 · complex structure of the...

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