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FtsHLonDnaKGroLGroS

Chaperones

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Heat stabilizes32

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Collaborators and contributors(partial list)

Theory: Parrilo, Carlson, Paganini, Papachristodoulo, Prajna, Goncalves, Fazel, Lall, D’Andrea, Jadbabaie, many current and former students, …

Web/Internet: Low, Willinger, Vinnicombe,Kelly, Zhu,Yu, Wang, Chandy, Effros, …

Biology: Csete,Yi, Arkin, Simon, AfCS, Borisuk, Bolouri, Kitano, Kurata, Khammash, El-Samad, Gross, Endelman, Sauro, Hucka, Finney, …

Physics: Mabuchi, Doherty, Barahona, Reynolds, Asimakapoulos,…Turbulence: Bamieh, Dahleh, Bobba, Gharib, Marsden, …Engineering CAD: Ortiz, Murray, Schroder, Burdick, …Disturbance ecology: Moritz, Carlson, Robert, …Finance: Martinez, Primbs, Yamada, Giannelli,… Caltech faculty

Other Caltech

For more details

www.cds.caltech.edu/~doylewww.aut.ee.ethz.ch/~parrilo

And thanks to Carla Gomes for helpful discussions.

Subthemes of this program

• Scalability of algorithms and protocols– Large network and physical problems– Decentralized, asynchronous, multiscale– Computational complexity: P/NP/coNP

• Approaches– Duality– Randomness

• Workshop II part of this program• Workshop last week on “Phase Transitions of

Algorithmic Complexity”

The Internet hourglass

Web FTP Mail News Video Audio ping napster

Applications

TCP SCTP UDP ICMP

Transport protocols

Ethernet 802.11 SatelliteOpticalPower lines BluetoothATM

Link technologies

The Internet hourglass

Web FTP Mail News Video Audio ping napster

Applications

TCP SCTP UDP ICMP

Transport protocols

Ethernet 802.11 SatelliteOpticalPower lines BluetoothATM

Link technologies

From Hari Balakrishnan

Everythingon IP

IP oneverything

Towards a theory of the Internet• The well-known original design principles are a

rudimentary “theory of the Internet.” • This is a nearly pure robustness theory (little else is being

optimized).• Can we provide a “deep,” complete, and coherent theory

of internetworking? (Like standard comms and controls.)• If we can’t say something systematic about the Internet

protocols, we’re probably kidding ourselves about our ability to treat more complex problems.

• Nevertheless this is just a “warm-up” for a theory of ubiquitous embedded software, protocols, and networks for real-time control of everything, everywhere.

Network protocols.

packetspacketspacketspacketspacketspackets

RoutingProvisioning

Network protocols.

RoutingProvisioning

Network protocols.

RoutingProvisioning

Horizontal decompositionEach level is decentralized and asynchronous

RoutingProvisioning

Horizontal decomposition

on• “Breaks” standard communications and control theories. • Coherent, complete theory is missing but possible. First cut nearly done.• In what sense, if any, is this optimal?• What needs to be done to fix it?

Key elements of new theory• Primal/dual vertical and horizontal decomposition

(Kelly et al, Low et al) • Source coding into mice and elephants. (Appears to be

“universal” but needs more study.)• Congestion control for bandwidth utilization and minimal

delay. Proofs use relaxations (but still handcrafted).• How bad is short path (low delay for mice) routing for

elephants in a “well-provisioned” network? Conjecture: Not bad.

• Vertical and horizontal integration can be made “nearly” optimal in an asymptotic sense. (In what sense?)

• Lots of people here are working out details (the IPAM team!). Stay tuned.

• Networking protocols• Multiscale physics• Biological networks• Business, finance, econ organization• Unifying theoretical framework?

What’s next?

• Scalable, integrated robustness analysis and software/protocol verification for hybrid control of nonlinear systems.

• New extensions to robust control using sum-of-squares and semidefinite programming (SOS/SDP) offers extraordinary promise.

• Already demonstrated on wide array of complex problems (controls, maxcut, quantum entanglement).

• Potentially deep connections between verification and robustness.

• Huge implications for biology and physics.• That’s the good news.

Compute

Communicate Communicate

StoreCommunicate

Communications and computing

Compute

EnvironmentEnvironment

StoreCommunicate

Computation

Devices

Dynamical SystemsDynamical Systems

DevicesCommunication Communication

Control

From• Software to/from human• Human in the loop

To• Software to Software• Full automation• Integrated control,

comms, computing• Closer to physical

substrateCompute

Communicate

Computation

Devices

Communication Communication

Control

• New capabilities & robustness• New fragilities & vulnerabilities

Good new, bad news, good news• Good: Powerful new capabilities enabled by “embedded,

everywhere”• Bad: Frightening new potentials for massive cascading

failure events• Good: Need for new math tools for verifying robustness

of embedded networking.– Embedded: Ubiquitous, sensing, actuating– Networking: Connected, distributed, asynchronous

• This represents an enormous change, the impact of which is not fully appreciated

• Robustness and verifiability of highly autonomous control systems with embedded software is the central challenge

• Until recently, there were no promising methods for addressing this full problem

• Even very special cases have had limited theoretical support for systematic verification of robustness

• Everything has changed!

Compute

Communicate

Computation

Devices

Communication Communication

Control• New capabilities & robustness• New fragilities & vulnerabilities

RoutingProvisioning

on• “Breaks” standard communications and control theories. • Duality as a method for decomposition• Distributed and asynchronous control• Other applications

• Robustness analysis• A posteriori error bounds for PDEs

Robust hybrid/nonlinear systems theory of embedded networks?

“Theory” without scalable algorithms.

Linear theory plus bounds, with scalable algorithms.

Hacking. (Scalable algorithms without theory.)

Theory with scalable

algorithms?

Most research: Not scalable, no theory.

Provably robust, scalable protocols for

control over embedded networks.

Hacking.

Theory with scalable

algorithms.

Robustness verification of embedded control software/hardware.

Provably robust, scalable Internet

protocols.

1. Robustness/Fragility: Uncertainty in components, environment, and modeling, assumptions, and computational approximations

2. Verifiability: Short proofs of robustness3. Complexity: Extreme, highly structured

internal complexity is typically needed to produce verifiably robust behavior

4. Scarce resources: All tradeoffs are aggravated by efficiency and scarce resources

Key issues

Robustness, evolvability/scalability,

verifiability

Ideal performance

Robustness

Evolvability

Verifiability

Typical design IP

• Relative to“nominal” performance under ideal conditions, robust performance typically requires– greater internal complexity– some loss of nominal performance

• Tradeoffs between robustness, evolvability, and verifiability seem less severe (e.g. IP)

Robustness, evolvability/scalability

, verifiability

Ideal performance

Robustness

Evolvability

Verifiability

• That a system is not merely robust, but verifiably so, is an important engineering requirement and major research challenge

• There is much anecdotal evidence and some new theoretical support as well for the compatibility of robustness, evolvability, and verifiability

• Verifiability in forward engineering translates into comprehensibility in reverse engineering of biological systems

• This research direction may be good news for understanding complex biological processes

Computational complexity

Assume you already know:• P/NP and NP complete• SAT and 3-SAT

…but not necessarily• NP vs coNP• Duality and relaxations

Typically NP hard.

• If true, there is always a short proof.• Which may be hard to find.

( ) ?nx R F x

Typically coNP hard.

• More important problem.• Short proofs may not exist.

Fundamental asymmetries* • Between P and NP• Between NP and coNP

* Unless they’re the same…

( ) ?nx R F x

What makes a

problem “harder”?What makes a

problem “harder”?

( ) ?nx R F x

Easy to find solutions?

Satisfiable or feasible

( ) ?nx R F x

Easy to find proofs?

Unsatisfiable or infeasible

Trivially sharp "phase transition" at

max ( )x

Complexity?

Example: Satisfiability

• SAT: Given a formula in propositional calculus, is there an assignment to its variables making it true?

• We consider clausal form, e.g.:• (a OR (NOT b) OR c) AND (b OR d) AND (b OR

(NOT d) OR a)• a, b, c, and d are Boolean (True/False) variables.

• Problem is NP-Complete. (Cook 1971)

• Shows surprising “power” of SAT for encoding computational problems.

Generating Hard Random Formulas

• Key: Use fixed-clause-length model.– (Mitchell, Selman, and Levesque 1992)

• Critical parameter: ratio of the number of clauses to the number of variables.

• Hardest 3SAT problems at ratio = 4.3

Hardness of 3SAT

02 3 4 5

Ratio of Clauses-to-Variables

DP Calls

50 var 40 var 20 var

EasyEasy

The 4.3 Point

0.02 3 4 5

50% sat

Mitchell, Selman, and Levesque 1991

• At low ratios:– few clauses

(constraints)– many assignments– easily found

• At high ratios:– many clauses– inconsistencies

easily detected

• Refer to as a – SAT transition– Complexity transition

• Is SAT transition either necessary or sufficient for complexity transition?

• Connections with phase transitions in statistical physics?

• Are transitions “sharp” in large size limit?

0.02 3 4 5

50% sat

Mitchell, Selman, and Levesque 1991

Theoretical Status Of Threshold

• Very challenging problem ...

• Current status:– 3SAT threshold lies between 3.45 and 4.6

(Motwani et al. 1994, Achlioptas et al. 2001,

Kirousis 2002, Broder and Suen 1993, Dubois

2000; Achlioptas and Beame 2001, Friedgut 1997,

• Other problems better characterized (NPP)

SAT Phase transitions

Complexity

Quasigroups or Latin Squares

Quasigroup or Latin Square

(Order 4)

32% preassignment

Gomes and Selman 96

A quasigroup is an n-by-n matrix such that each row and column is a permutation of the

same n colors

Quasigroup with Holes (QWH)Quasigroup with Holes (QWH)

• Given a full quasigroup, “punch” holes into it

32% holes

• Always completable (satisfiable), so no SAT transition.• Appears to have a complexity transition (easy-hard-easy).

Complexity

plexity? ?

Lots of problems with statistical physics story.

Why may it be reasonable that math, algorithms, and randomness

are so effective?

• Robust systems are verifiably so?

• Do only robust systems persist as coherent, structured objects of study (universes, solar systems, planets, life forms, protocols, …)?

• If so, then mostly robust (and verifiably so) systems are around for us to study.

What can we do with lattices that will be easy to understand, yet relevant to the “real” computational complexity problems that we most care about?

Key abstractions:1. Robustness/Fragility2. Verifiability3. Complexity

Lattice models?

.2 .4 .6 .8

Density = fraction of occupied sites (black)

Not connected Connected

Focus on “horizontal” paths.

Some (nonstandard) definitions

“Vertical” paths in empty sites are allowed to connect through corners or edges. (8 neighbors)

“Horizontal” paths connect only on edges. (4 neighbors.Ordinary

square site percolation.)

vertical paths horizontal paths

Critical phase transition at density = .59…

.2 .4 .6 .8

Not connected Connected

• Robustness is provided by barriers in some state space. These prevent cascading failure events.

• Lattices offer a crude abstraction, in that paths can be thought of as barriers, with robustness to perturbations in the lattice.

• Verifiability complexity is measured in the length of the proof required to verify robustness.

• Lattices can offer a variety of crude abstractions to this as well. The length of minimal paths would be a simple measure of “proof length.”

Caution: potential source of confusion.

Very special features:• Dual and primal problems are “essentially” the same.• There is no duality gap.

Barriers in 1d lattices are 0d cuts.

Barriers in 3d lattices are 2d

path fragments

barrier

In general, barriers are d-1 dimensional (dual) cuts stopping 1-dim (primal) paths in a d-dim lattice.

Critical phase transition at density = .59…

Lattices offer pedagogically useful but potentially dangerously misleading simplifications, which are thus both strengths and weaknesses:

1. Internal complexity2. Computational complexity3. Duality

1. Internal vs external complexity: Real biology and technology uses extremely complex hierarchical organization in order to create robust and verifiably (simple) behavior. Lattices allow no distinction between complex organization and complex behavior. This can be very misleading.

2. Computational complexity: Most lattice computational problems are in P and thus easily explored, but fail to illustrate the P/NP asymmetry. We will rely on notions of complexity that are good analogies, but not precisely comparable.

3. Duality: Duality is greatly simplified and transparent. This makes exposition easy but hides the NP/coNP asymmetry which is central to the general problem.

Lattices offer enormous (and potentially dangerous) simplifications:• Robustness problem= existence of horizontal path• Verification = prove existence of horizontal path• Complexity = minimum horizontal path length (of proof)• Model fragility = minimum number of site changes to break all horizontal paths (= create a vertical path)

Note: I’m going to draw small lattices and rely on your imagination for what large lattices would look like.

.2 .4 .6 .8

Alternative definition of “complexity:”• The “computer” is you, looking at the lattice and determining by inspection whether there is a path or not. • This can be easy or hard, depending on the density. • This is not exactly the same as minimal path length, but close enough for now.• Do a very informal story, and then make it rigorous.

Exist horizontal

No Yes

For random lattices, there are 4 regimes, with all combinations of Easy/Hard and Yes/No. The hard cases correspond to lattices that are of intermediate density, near the critical point. Easy cases are either high or low densities, which always correspond to Yes or No, respectively.

No Yes

It is much easier to see with all the clusters colored. But that’s cheating, because determining the clusters is essentially the computational problem.

The orthodox

story:

No Yes

Hard problems are associated in

some way with the phase

transition.

The counter-examples

Low/Yes High/No

Easy &Robust

Hard & Fragile

No Yes

Exactly the oppositeof criticality

• Yes or no• Easy or hard• High or low density• Robust or fragile (to perturbations)

Low/Yes High/No

Easy &Robust

Hard & Fragile

1. Yes or no2. Easy or hard3. Low or high density4. Robust or fragile (to

perturbations)

16 different possible combinations

Low/Yes High/No

Easy &Robust

Hard & Fragile

1. Yes or no2. Easy or hard3. Low or high density4. Robust or fragile (to

perturbations)

16 different possible combinations

Low Density (but connected)

Robust Fragile

High density

Hard implies fragile (we’ll prove this later). So only 6 of the 8 possibilities exist, and the critical density is nothing special. We will prove that these and only these implications hold.

Low Density

Robust Fragile

High density

Robust Fragile

Low Density

Robust Fragile

All interesting real world problems are in this regime, with efficient, highly structured, rare configurations, using scarce (limited) resources.

Cruise control

Electronic ignition

Temperature control

Electronic fuel injection

Anti-lock brakes

Electronic transmissionElectric power steering (PAS)

Air bags

Active suspension

EGR control

Low Density

Robust Fragile

Impossible.

Improbable in random lattices.

Low Density

Robust Fragile

High density

Theorem:

Fragility Complexity Scarcity

Proof tonite.

Low Density High density

Robust Fragile

Theorem:

Random lattices are complex (and fragile) only at critical phase

transition.

OccupiedEmpty

MinPath

n = length of side = densityl = MinPath length

Definitions. Assume there is a connected (horizontal) path of minimal length l .

Typical “minimal” path

OccupiedEmpty

MinPath

Typical “minimal” cutn = length of side = densityl = MinPath lengthb = MinCut barrier length

Definitions. Assume there is a connected path of minimal length l .

Typical “minimal” path

n = length of side = densityl = MinPath lengthb = MinCut barrier length

Definitions. Assume there is a connected path of minimal length l .

Vertical path

n = length of side = densityl = MinPath lengthb = MinCut barrier length

Assume a path exists. (Otherwise L=F=.) Necessarily 1/n, n2 l n and define

log Resource scarcity 0 log( )

log = Path Length Complexity 0 log( )

log = Path Fragility 0 log( )

lL L n

nF F n

l = MinPath lengthb = MinCut barrier length

lL L n

nF F n

Theorem: F L S

l = MinPath lengthb = MinCut barrier length

lL L n

nF F n

Theorem: F L S

Proof (Vinnicombe&sushi): To provide robustness to b changes, there must be at least b independent paths, which by assumption have minimum length l. Necessarily n2 lb, or n/b l/n. Take log of both sides.

lL L n

nF F n

Theorem: F L S

Lattices and paths can be:1. Resources: Scarce or rich2. Existence of path: Yes or no3. Complexity: Hard or easy4. Perturbations: Fragile or robust

Anything is possible, consistent with the theorem.

This is “maximally tight” in the sense that:

lL L n

nF F n

Theorem: F L S

Lattices and paths can be:1. Existence: Yes or no2. Resources: Scarce or rich3. Perturbations: Fragile or robust4. Complexity: Hard or easy

Anything is possible, consistent with the theorem.

We’ll just consider the 8 cases with paths.

-S=log()

Scarce Rich

Robust

Fragile

Theorem: F L S

Scarce Rich

Robust

Fragile

Theorem: F L S

Scarce Rich

Robust

Fragile

Theorem: F L S

Scarce

Robust

Fragile

HardTheorem: F L S

Scarce

Robust

Fragile

OccupiedEmpty

MinPath

F=S, L=0

Theorem: F L S

Scarce

Robust

Fragile

F=S, L=0

Theorem: F L S

Most robust

possible.

Scarce

Robust

Fragile

Easy and Fragile

F=log(n)>S, L=0

Theorem: F L S

Scarce

Robust

Fragile

OccupiedEmpty

MinPath

Theorem: F L S

( 2 1)

( 1) ( )

( 2 1)

for 1 ( )

n m b d d

l n m n b

n n m d n b

n dl n n b

nb nd n nb n dn bd d

n n nb nb nd dn bd d

n n nb bd d n

b d b db d n l

To construct asymptotically tight cases where n2 = lb, consider the lattice below.

= densityb = MinCut barrier lengthl = MinPath lengthn = length of sidem = # of “cells”d = width of open regions

Now take limits:

Consider the limit where

( 1) ( )

b d n l

n d nm

b d b d

n n m d n b n mdn

d bn n

b d b d

n nl lb n

By constructing lattices as below, with n>>m>>1, it is possible to find lattices such that any n2 lb, with <1 is achievable.

Scarce

Robust

Fragile

Theorem: F L S

Scarce

Robust

Fragile

Theorem: F L S

The Fragile Face

Scarce

Robust

Fragile

Theorem: F L S

The Four Corners

Scarce

Robust

Fragile

Theorem: F L S

Most RobustF=S

Most FragileF>>S

Scarce

Robust

Fragile

Theorem: F L S

Random

Scarce

Robust

Fragile

Cruise control

Electronic ignition

Temperature control

Anti-lock brakes

Air bags

Active suspension

EGR control

Efficient and Efficient and robust is far from robust is far from

randomrandom

• How general is this?• Seems to hold in all theory where it has been

investigated.• Extensive literature on ill-conditioning in LPs and

numerical linear algebra.• Anecdotally, seems to capture essence of many

complexity problems.• Needs to be combine with laws constraining net

system fragility.

Phase transitions

Complexity

Bad news and good news

• Bad news? Some hoped-for connections between phase transitions and complexity are not there.

• Good news?: Ideas still interesting.

• Lots more really good news!

• The alternative is much richer and useful, and connects in interesting ways with phase transitions

• New algorithms, new mathematics, new practical applications,…

• And deep implications for physics.

Phase transitions

Complexity

?• Internet traffic and topology• Biological and ecological

networks• Evolution and extinction• Earthquakes and forest fires• Finance and economics• Social and political systems

Physics and the edge of chaocritiplexity

Cruise control

Electronic ignition

Temperature control

Anti-lock brakes

Air bags

Active suspension

EGR control

Phase transitions

Complexity

?• Internet traffic and topology• Biological and ecological

networks• Evolution and extinction• Earthquakes and forest fires• Finance and economics• Social and political systems

Rich new unifying theory of complex

control, communication, and computing systems

• Ubiquity of power laws• Coherent structures in shear

flow turbulence• Macro dissipation and

irreversibility vs. micro reversibility.

• Quantum entanglement, measurement, and the QM/Classical transition

• Growing group of physicists and experimentalists are joining this effort (Carlson, Mabuchi, Doherty, Gharib,…)

Rich new unifying theory of complex

control, communication, and computing systems

Semialgebraic geometry

convex optimization (SDP)

More powerful bounds for the co-NP side

• Polynomial time computation.

• Never worse than the standard.

• Exhausts co-NP.

Polynomial functions: NP-hard problem.

( ) ?nx R F x

Trivially sharp "phase transition" at

max ( )x

Complexity?

/ 2 1| 1 0

M x R x Ax x b c

c bM x R x

Special case: Scalar QP

Assume for nontriviality that 0.A

nM x R x Ax x b c

Special case: Scalar QP

1 / 4b A b c

1) "phase transition" when / 4

2) complexity depends only on

3) 1) and 2) are only trivially related

c b A b

• Polynomial functions: NP-hard problem.

• A “simple” relaxation (Shor): find the minimum γsuch that γ- F(x) is a sum of squares (SOS).

• Upper bound on the global maximum.

• Solvable using SDP, in polynomial time.

• A concise proof of nonnegativity.

• Surprisingly effective (Parrilo & Sturmfels 2001).

( ) ?nx R F x

• Exactly as in QP case, SAT “phase transition” does not imply complexity.

• SOS/SDP relaxations much faster than standard algebraic methods (QE,GB, etc.).

• Before SOS/SDP, might have conjectured that this was an example of phase transition induced complexity.

• SOS/SDP gives certified upper bound in polynomial time.• If exact, can recover an optimal feasible point.• Surprisingly effective:

– In more than 10000 “random” problems, always the correct solution…

• Bad examples do exist (otherwise NP=co-NP), but “rare.”– Variations of the Motzkin polynomial.– Reductions of hard problems (e.g. NPP is nice)– None could be found using random search…

A sufficient condition for nonnegativity:

Sums of squares (SOS)

2( ) : ( ) ( ) ?i ii

f x p x f x • Convex condition (Shor, 1987)

• Efficiently checked using SDP (Parrilo).

Write: ( ) , 0Tp x z Qz Q

where z is a vector of monomials. Expanding and equating sides, obtain linear constraints among the Qij. Finding a PSD Q subject to these conditions is exactly a semidefinite program (LMI).

Nested families of SOS (Parrilo)

( ) SOS, ( ) :

( ) ( ) ( ) ?

g x f x

g x p x f x

2( ) . . kk ig x e g xNested families

exhaust co-NP

, ( ) ?

where is a multivariate polynomial

nx R f x

Conjectures on why such a boring “phase transition:”• One polynomial is generically robust, therefore no complexity. • QPs capture the essence of this.• Can make up other “phase transitions” which create fragilities, and thus the possibility of complexity

Search for counterexample

nested family of model sets

nested family of proof sets

Search for proof

cone( ), ideal( ) :

1 0i if f g g

{ : ( ) 0, ( ) 0}ni ix R f x g x

• Convex, but infinite dimensional.• Efficient (P time) search subsets (relaxations) using SOS/SDP (Parrilo)• Guaranteed to converge

Search for proof

Positivstellensatz

Search for simple counterexample

Search for short proof

| 0, 0, 0

M x R Ax b

P R b A

cone( ), ideal( ) : 1 0i if f g g f g

{ : ( ) 0, ( ) 0}ni ix R f x g x

( ) ( ) | 0

( ) 1 0 0, 1 0

Cone Ax b Ax b

Ax b A b

Special case: LP

?partition

Choose random n-bit integer complex fragilekv

kdata vNPP

kv S S

k kS S

Fragile = large changes in solution from small changes in data

2 2 2{ : 0, 0}n

k k kx R x x v

22 2 2{ : 0}n

k k kx R x x v

fragile

Choose random n-bit integer complex fragilekv

kdata vNPP

, ( ) ?

where is a multivariate polynomial

nx R f x

Random f

2 2 2{ : 0, 0}n

k k kx R x x v

2 2 2{ , : 0}n n

k k k kR R x R x x v

0 0 0 0 00 1 1 0 0

{ , 0}

0 1 1 0 0

Very unlikely to be feasible. Contrast with random polynomial.

nM x R x Ax x b c

1) "phase transition" when / 4

2) complexity depends only on

3) 1) and 2) are only trivially related

c b A b

• Complexity is caused by fragility (ill-conditioning).• Another example: Purely satisfiable QCP• Phase transitions are, in general, unrelated to complexity• Random scalar QP problems are generically robust (well-conditioned) and thus simple

Phase transitions

Complexity

Semialgebraic geometry

convex optimization (SDP)

More powerful bounds for the co-NP side

• Polynomial time computation.

• Never worse than the standard.

• Exhausts co-NP.

Think of LMIs as quadratic forms, not as matrices.

LMIs: quadratic forms, that are positive definite.

A key insight

• General forms , not necessarily quadratic.

• Instead of nonnegativity (NP-hard), use sum of squares.

SOS: multivariable forms, that are sum of squares.

Search for proof

• Models describe sets of possible (uncertain) behaviors intersected with sets of unacceptable behaviors (failures)• Thus verification of robustness (of protocols, embedded, dynamics, etc) involves showing that a set is empty.• Searching for an element x M is in NP, since checking whether a given x M is typically in P.• Proving that M is empty is in coNP and there may not be short proofs.

cone( ), ideal( ) :

1 0i if f g g

{ : ( ) 0, ( ) 0}ni ix R f x g x

• Convex, but infinite dimensional.• Efficient (P time) search subsets (relaxations) using SOS/SDP• Guaranteed to converge

Seach for proof

| 0, 0, 0

M x R Ax b

P R b A

cone( ), ideal( ) : 1 0i if f g g f g

{ : ( ) 0, ( ) 0}ni ix R f x g x

( ) ( ) | 0

( ) 1 0 0, 1 0

Cone Ax b Ax b

Ax b A b

Special case: LP

Failure to find short proof implies some relaxed model is nonempty (which is bad).

A sufficient condition for nonnegativity:

Sums of squares (SOS)

2( ) : ( ) ( ) ?i ii

f x p x f x • Convex condition (Shor, 1987)

• Efficiently checked using SDP (Parrilo).

Write: ( ) , 0Tp x z Qz Q

where z is a vector of monomials. Expanding and equating sides, obtain linear constraints among the Qij. Finding a PSD Q subject to these conditions is exactly a semidefinite program (LMI).

Nested families of SOS (Parrilo)

( ) SOS, ( ) :

( ) ( ) ( ) ?

g x f x

g x p x f x

2( ) . . kk ig x e g xNested families

exhaust co-NP

, ( ) ?F

A Few Applications

• Nonlinear dynamical systems– Lyapunov function computation– Bendixson-Dulac criterion– Robust bifurcation analysis

• Continuous and combinatorial optimization– Polynomial global optimization– Graph problems: e.G. Max cut– Problems with mixed continuous/discrete vars.

• Hybrid???

Let’s see some examples…

Continuous Global Optimization

• Polynomial functions: NP-hard problem.

• A “simple” relaxation (Shor): find the maximum γsuch that f(x) – γ is a sum of squares.

• Lower bound on the global optimum.

• Solvable using SDP, in polynomial time.

• A concise proof of nonnegativity.

• Surprisingly effective (Parrilo & Sturmfels 2001).

• Much faster than exact algebraic methods (QE,GB, etc.).• Provides a certified lower bound.• If exact, can recover an optimal feasible point.• Surprisingly effective:

– In more than 10000 “random” problems, always the correct solution…

• Bad examples do exist (otherwise NP=co-NP), but “rare.”– Variations of the Motzkin polynomial.– Reductions of hard problems.– None could be found using random search…

A model co-NP problem:

Check emptiness of semialgebraic sets.

Obtain LMI sufficient conditions.

Can be made arbitrarily tight, with more computation.

Polynomial time checkable certificates.

More general framework

Semialgebraic Sets

• Semialgebraic: finite number of polynomial equalities and inequalities.

• Continuous, discrete, or mixture of variables.• Is a given semialgebraic set empty?

– Feasibility of polynomial equations: NP-hard…

• Search for bounded-complexity emptiness proofs, using SDP. (Parrilo 2000)

0)(,0)( xgxf ii

• Stengle, 1974

• Generalizes Hilbert’s Nullstellensatz and LP duality

• Infeasibility certificates of polynomial equations over the real field.

• Parrilo: Bounded degree solutions computed via SDP!

Nested family of polytime relaxations: for quadratics, the first level is the S-procedure…

01:)(ideal),(cone gfggff ii

Positivstellensatz (Real Nullstellensatz)

empty is}0)(,0)(:{ xgxfRx iin

if and only if

Combinatorial optimization: MAX CUT

•Given a graph

•Partition the nodes in two subsets

•To maximize the number of edges between the two subsets.

A mathematical formulation: )1(max,

}1,1{ji

Hard combinatorial problem (NP-complete).

Compute upper bounds using convex relaxations.

Standard semidefinite relaxation:

tracemax

WYiiYY

tracemin1,0

Dual problems

Tighter bounds are obtained.

Never worse than the standard relaxation.

In some cases (n-cycle, Petersen graph), provably better.

Still polynomial time.

This is just a first step. We can do better!

The new tools provide higher order relaxations.

MAX CUT on the Petersen graph

The standard SDP upper bound: 12.5

Second relaxation bound: 12.

The improved bound is exact. A corresponding coloring.

Finding Lyapunov functions

• Ubiquitous, fundamental problem

• Algorithmic LMI solution

After optimization: coefficients of V.

A Lyapunov function V, that proves stability.

Test using SOS and SDP.

Convex, but still NP hard.

Finding Lyapunov functions

• Ubiquitous, fundamental problem

• Algorithmic LMI solution0 fVV

Given:

Propose:

iij yxcyxV

After optimization: coefficients of V.

A Lyapunov function V, that proves stability.

Conclusion: a certificate of global stability

-10 -5 0 5 10 -10 -5 0 5-10 -5 0 5 10

c=1 c=2.164 c=10

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

Global stability of a switching system using 4th order MLFs defined in 6 equiangular partitions

DS applications: Bendixson-Dulac

• In 2D rules out periodic orbits.

• Higher dimensional generalizations (Rantzer) provide

• Weaker stability criterion than Lyapunov (allowing a zero-measure set of divergent trajectories).

• Convexity for synthesis.

• How to search for ρ ?

DS applications: Bendixson-Dulac

• Restrict to polynomial (or rational) solutions, use SOS.

• As for Lyapunov, now a fully algorithmic procedure.

Given:22 yxyxy

Propose: cybxa

After optimization: 1,3,32

23 )( f 0

x ' = y

y ' = - x - y + x 2 + y 2

-3 -2 -1 0 1 2 3

yConclusion: a certificate of the inexistence of periodic orbits

x ' = y

yConclusion: a certificate of the inexistence of periodic orbits

stable

saddle

Stronger μ upper bounds

• Structured singular value µ is NP-hard (as general QP)

• Standard µ upper bound can be interpreted:

•As a computational scheme.

•As an intrinsic robustness analysis question (time-varying uncertainty).

•As the first step in a hierarchy of convex relaxations.

•For the four-block Morton & Doyle counterexample:

Standard upper bound: 1

Second relaxation: 0.895

Exact µ value: 0.8723

What is the message ?

Even if short proofs are not guaranteed to exist,

in many cases they do.

What happens in the broader setting of

robustness and verification?

Line of Attack

• Want to decouple – System complexity– Complexity of verification.

“bad” region

Nominal

System

• Even for extremely complex systems, there may exist simple robustness proofs. Try to look for those first…

What is the message ?

Even if short proofs are not guaranteed to exist,

in many cases they do.

• Robustness, verifiability, and complexity are inextricably linked.

• Lots of circumstantial evidence:– All our previous experience in robustness analysis and

optimization: µ upper bounds, etc.– Hard mathematical results, linking complexity with distance to

set of ill-posed instances (Smale, etc).

• Partly intrinsic (as in optimization problems), but can also be a consequence of design.

Admit multiple interpretations:– Alternative reformulations (perhaps more natural).– Relaxation of assumptions (LTI -> LTV,

commutativity, etc.)– Purely computational schemes.– Bounded depth derivations.

What are these short proofs?

• Everything discussed is for analysis or verification.

• Synthesis is a much more complicated beast.

About synthesis…

( )?x P x

( , )?y x P x y

• In general, in higher complexity classes, harder than NP-hard (Tierno & Doyle 1995).

• Alternating quantifiers, relativized Turing machines: the polynomial time hierarchy.

The polynomial time hierarchy

......

NPCo-NP

PAnalysis

Synthesis

( )?x P x

( , )?y x P x y

( , ) 0y x P x y

Why are LMIs ubiquitous?

• In general is Π2-hard.

• No current hope of solving this efficiently.

• But when P(x,y) is quadratic in x and affine in y…

• Drops two levels to P, polynomial time !

( , ) 0y x P x y

P(x,y) is quadratic in x and affine in y

- -11 -1State feedback synthesis: (A+BK)X A +B

Output feedback: eg H , multiobjective

LPV synthesis

Lyapunov fcn for nonlinear 1-dim systems

Backstepping: Lower triangular struc e

• Synthesis results depend on hand-crafted “tricks” that we don’t fully understand yet.

• Until recently we could say the same about analysis, where custom techniques abound.

• For analysis, there’s a method in the madness, earlier results unified and expanded.

Entangled Quantum States(Doherty, Parrilo, Spedalieri 2001)

• Entangled states are one of the most important distinguishing features of quantum physics.

• Bell inequalities: hidden variable theories must be non-local.

• Teleportation: entanglement + classical communication.• Quantum computing: some computational problems may

have lower complexity if entangled states are available.

How to determine whether or not a given state is entangled ?

• QM state described by psd Hermitian matrices ρ• States of multipartite systems are described by operators on the tensor product of vector spaces

• Product states:• each system is in a definite state

• Separable states: • a convex combination of product states.

•Entangled states: those that cannot be written as a convex combination of product states.

11 1 1A B

1,0, iiiii ppp

Decision problem: find a decomposition of as a convex combination of product states or prove that no such decomposition exists.

Theorem (Horodecki 1996): a state is entangled if and only if there exists a Hermitian Z such that:

* * * *

, Tr 0jij kl i k l

x y xx yy Z Z x x y y

(Hahn-Banach Theorem)

Separablestates

Z is an “entanglement witness,”a generalization of Bell’s inequalities

First RelaxationRestrict attention to a special type of Z:

22* * *;

m ik m ikj kij kl i k l i k iZ Z x x y y G x y H x y

x y xG x y

y H x y

The bihermitian form Z is a sum of squared magnitudes.

2Tminimize Tr

subject to 0, 0

If minimum is lessthan zero, is entangled

First Relaxation

2Tminimize Tr

subject to 0, 0

If minimum is lessthan zero, is entangled

• Equivalent to known condition• Peres-Horodecki Criterion, 1996• Known as PPT (Positive Partial Transpose)• Exact in low dimensions• Counterexamples in higher dimensions

Further relaxations

Broaden the class of allowed Z to those for which

lkikliji yyxxZx j

is a sum of squared magnitudes.

Also a semidefinite program.

Strictly stronger than PPT.

Can directly generate a whole hierarchy of tests.

Second Relaxation

jiijkm

jkijkmjkiijkmlkikliji

xyxHxyxGyyxxZx

If the minimum is less than zero then is entangled.Detects all the non-PPT entangled states tried…

minimize ZTrsubject to 0G 0H

xyxKHGxyxxyxIZxyx T 21T**

Quantum entanglement and Robust control

Entanglement Robustness

Exact problem

is hardIs entangled? Robust?

Known sufficient condition

PPT Upper bound

Exact in low dimensions

Counterexamples Horodecki-Choi Morton/Packard/Doyle

Quantum entanglement and Robust control

Entanglement Robustness

Exact problem

is hardIs entangled? Robust?

Known sufficient condition Equivalent

Counterexamples Horodecki-Choi Morton/Packard/Doyle

Higher order relaxations Equivalent

Higher order relaxations

• Nested family of SDPs

• Necessary: Guaranteed to converge to true answer

• No uniform bound (or P=NP)

• Tighter tests for entanglement

• Improved upper bounds in robust control

• Special cases of general approach

• All of this is the work of Pablo Parrilo (PhD, Caltech, 2000, now Professor at ETHZ)

• My contribution: I kept out of his way.

Summary

• Single framework with substantial advances in – Testing entanglement– MaxCut– Global continuous optimization– Finding Lyapunov functions for nonlinear systems– Improved robustness analysis upper bounds– Many other applications

• This is just the tip of a big iceberg

Nested relaxations and SDP

Robustness coNP hard problem

Exact problem

is hardRobust?

Known sufficient condition

Standard upper bound

Usually a known

relaxation

( )?x P x

Higher order relaxations

Sharper sufficient conditions.

Converges to exact solution.

• Huge breakthroughs…

• …but also a “natural” culmination of more than 2 decades of research in robust control.

• Initial applications focus has been CS and physics,

• … but substantial promise for “persistent mysteries” in controls and dynamical systems

• Completely changes the possibilities for – robust hybrid/nonlinear control – interactions with CS and physics

• Unique opportunities for controls community– Resolve old difficulties within controls– Unify and integrate fragmented disciplines within– Unify and integrate without: comms and CS– Impact on physics and biology

• Unique capabilities of controls community– New tools, but built on robust control machinery– Unique talent and training

0 parameters

( , , ) dynamics

noise, disturbances

x f x p w

RProblem: In general, computation grows exponentially with m and n.

Key idea: systematic search for short proofs.

Chemical oscillator (Prajna, Papachristodoulou)

constant

X A A BX Y X

x a x x y

y b x y

Nondimensional state equations

x a x x y

y b x y

Limit cycle for

b a b a

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 0.1 0.2 0.3 0.4 0.5 0.6

a = 0.1, b = 0.13

0 0.2 0.4 0.6 0.8 1 1.2 1.4

x a x x y

y b x y

a x x y

equilibrium

x a x x x x y y

y b x x y y

a x x y

x a x x x x y y

y b x x y y

For what , does there

exist ( , ) such that

0 0.2 0.4 0.6 0.8 1 1.2 1.4

4 order ( , ) th V x y

1 1.5 2 2.5

a = 0.6, b = 1.1

2.2 2.6 3 3.4

1a = 1, b = 2

Features of new approach (Parrilo)

• SOS/SDP: Based on Sum-of-square (SOS) and semidefinite programming (SDP)

• Exist “gold standard” relaxation algorithms for canonical coNP hard problems, such as– MaxCut– Quantum entanglement– Robustness () upper bound

• All special cases of first step of SOS/SDP• Further steps (all in P) converge to answer• No uniform bound (or P=NP)

• Standard tools of robust (linear) control – Unmodeled dynamics, nonlinearities, and IQCs– Noise and disturbances– Real parameter variations– D-K iteration for -synthesis

• Are all treated much better…

• And generalized to – Nonlinear– Hybrid– DAEs– Constrained

Caveats

• Inherits difficulties from robust control

• High state dimension and large LMIs

• Must find ways to exploit structure, symmetries, sparseness

• Note: many researchers don’t want to get rid of the ad hoc, handcrafted core of their approaches to control (why take the fun out of it?)

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