Title: John Doyle
1Robustness and Complexity
John Doyle Control and Dynamical
Systems BioEngineering Electrical
Engineering Caltech
2Collaborators and contributors(partial list)
- Theory Parrilo, Carlson, Paganini,
Papachristodoulo, Prajna, Goncalves, Fazel, Lall,
DAndrea, 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
Other
3For more details
www.cds.caltech.edu/doyle www.aut.ee.ethz.ch/par
rilo
And thanks to Carla Gomes for helpful discussions.
4Subthemes 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
5The Internet hourglass
IP
6The Internet hourglass
Everything on IP
IP
From Hari Balakrishnan
7Towards 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 cant say something systematic about the
Internet protocols, were 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.
8Network protocols.
Files
HTTP
TCP
IP
packets
packets
packets
packets
packets
packets
Routing Provisioning
9Network protocols.
HTTP
TCP
Vertical decomposition Protocol Stack
IP
Routing Provisioning
10Network protocols.
HTTP
TCP
IP
Horizontal decomposition Each level is
decentralized and asynchronous
Routing Provisioning
11- 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?
HTTP
TCP
Vertical decomposition
IP
Horizontal decomposition
Routing Provisioning
12Key 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.
13- Networking protocols
- Multiscale physics
- Biological networks
- Business, finance, econ organization
- Unifying theoretical framework?
Vertical decomposition
Horizontal decomposition
14Whats 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.
- Thats the good news.
15Communications and computing
Store
Communicate
Compute
Communicate
Communicate
16Store
Communicate
Compute
Communicate
Communicate
Act
Environment
17Computation
Communication
Communication
Devices
Devices
Dynamical Systems
18- From
- Software to/from human
- Human in the loop
- To
- Software to Software
- Full automation
- Integrated control, comms, computing
- Closer to physical substrate
Computation
- New capabilities robustness
- New fragilities vulnerabilities
Communication
Communication
Devices
Devices
Control
Dynamical Systems
19Good 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
DeborahWorld
20- 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!
- 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
Computation
- New capabilities robustness
- New fragilities vulnerabilities
Communication
Communication
Devices
Devices
Control
Dynamical Systems
21- 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
HTTP
TCP
Vertical decomposition
IP
Horizontal decomposition
Routing Provisioning
22Robust hybrid/nonlinear systems theory of
embedded networks?
Linear theory plus bounds, with scalable
algorithms.
Theory without scalable algorithms.
Hacking. (Scalable algorithms without theory.)
Theory with scalable algorithms?
Most research Not scalable, no theory.
23Provably robust, scalable Internet protocols.
Robustness verification of embedded control
software/hardware.
Hacking.
Theory with scalable algorithms.
24Key issues
- Robustness/Fragility Uncertainty in components,
environment, and modeling, assumptions, and
computational approximations - Verifiability Short proofs of robustness
- Complexity Extreme, highly structured internal
complexity is typically needed to produce
verifiably robust behavior - Scarce resources All tradeoffs are aggravated by
efficiency and scarce resources
25Robustness, evolvability/scalability,
verifiability
Robustness
Ideal performance
- Relative tonominal 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)
26Robustness, 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
27Computational complexity
?
- Assume you already know
- P/NP and NP complete
- SAT and 3-SAT
- but not necessarily
- NP vs coNP
- Duality and relaxations
28Typically NP hard.
?
29Typically coNP hard.
- Fundamental asymmetries
- Between P and NP
- Between NP and coNP
?
- More important problem.
- Short proofs may not exist.
Unless theyre the same
30?
What makes a problem harder?
31Easy to find solutions?
?
Satisfiable or feasible
32?
Easy to find proofs?
Unsatisfiable or infeasible
33?0
Complexity?
34Example 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.
35Generating 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
36Hardness of 3SAT
4000
50 var
40 var
20 var
3000
DP Calls
Hard
2000
Easy
1000
Easy
0
2
3
4
5
6
7
8
Ratio of Clauses-to-Variables
371.0
- At low ratios
- few clauses (constraints)
- many assignments
- easily found
- At high ratios
- many clauses
- inconsistencies easily detected
50 sat
0.8
0.6
Probability
0.4
0.2
0.0
2
3
4
5
6
7
8
Ratio of Clauses-to-Variables
Mitchell, Selman, and Levesque 1991
The 4.3 Point
38- 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?
39Theoretical 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, - etc.)
- Other problems better characterized (NPP)
40SAT Phase transitions
?
?
Complexity
41Quasigroups or Latin Squares
A quasigroup is an n-by-n matrix such that each
row and column is a permutation of the same n
colors
Quasigroup or Latin Square (Order 4)
32 preassignment
Gomes and Selman 96
42Quasigroup with Holes (QWH)
- Given a full quasigroup, punch holes into it
- Always completable (satisfiable), so no SAT
transition. - Appears to have a complexity transition
(easy-hard-easy).
43SAT Phase transitions
?
?
Complexity
44Lots of problems with statistical physics story.
45Why 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.
46Lattice models?
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
- Robustness/Fragility
- Verifiability
- Complexity
47.2
.4
.6
.8
Density fraction of occupied sites (black)
Focus on horizontal paths.
48Vertical 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.)
Focus on horizontal paths.
Some (nonstandard) definitions
49Critical phase transition at density .59
50.2
.4
.6
.8
Density fraction of occupied sites (black)
Focus on horizontal paths.
51- 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.
52- Very special features
- Dual and primal problems are essentially the
same. - There is no duality gap.
Caution potential source of confusion.
53Barriers in 3d lattices are 2d cuts.
Barriers in 1d lattices are 0d cuts.
path fragments
barrier
In general, barriers are d-1 dimensional (dual)
cuts stopping 1-dim (primal) paths in a d-dim
lattice.
54Critical phase transition at density .59
55- Lattices offer pedagogically useful but
potentially dangerously misleading
simplifications, which are thus both strengths
and weaknesses - Internal complexity
- Computational complexity
- Duality
Focus on horizontal paths.
56- 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. - 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. - Duality Duality is greatly simplified and
transparent. This makes exposition easy but hides
the NP/coNP asymmetry which is central to the
general problem.
57- 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)
Focus on horizontal paths.
58Note Im going to draw small lattices and rely
on your imagination for what large lattices would
look like.
59- 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.
.2
.4
.6
.8
Density fraction of occupied sites (black)
60No
Yes
Easy
Exist horizontal path?
Hard
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.
61No
Yes
No
Yes
Easy
Easy
Hard
Hard
It is much easier to see with all the clusters
colored. But thats cheating, because determining
the clusters is essentially the computational
problem.
62The orthodox story
No
Yes
Easy
Hard problems are associated in some way with the
phase transition.
Hard
63The counter-examples
Exactly the opposite of criticality
No
Yes
- Yes or no
- Easy or hard
- High or low density
- Robust or fragile (to perturbations)
Easy
Hard
64The counter-examples
Exactly the opposite of criticality
- Yes or no
- Easy or hard
- Low or high density
- Robust or fragile (to perturbations)
16 different possible combinations
65The counter-examples
Exactly the opposite of criticality
8
- Yes or no
- Easy or hard
- Low or high density
- Robust or fragile (to perturbations)
16 different possible combinations
66Robust
Fragile
Robust
Fragile
Easy
Easy
Hard
Hard
Low Density (but connected)
High density
Hard implies fragile (well 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.
67Robust
Fragile
Robust
Fragile
Easy
Easy
Hard
Hard
Low Density
High density
Robust
Fragile
Random
Easy
Hard
68Robust
Fragile
All interesting real world problems are in this
regime, with efficient, highly structured, rare
configurations, using scarce (limited) resources.
Easy
Hard
Low Density
Robust
Fragile
Random
Easy
Hard
69Robust
Fragile
Easy
Impossible.
Hard
Low Density
Robust
Fragile
Improbable in random lattices.
Random
Easy
Hard
70Robust
Fragile
Robust
Fragile
Easy
Easy
Hard
Hard
Low Density
High density
Proof tonite.
71Random lattices are complex (and fragile) only
at critical phase transition.
Low Density
High density
Robust
Fragile
Easy
Hard
72Definitions. Assume there is a connected
(horizontal) path of minimal length l .
n length of side r density l MinPath length
Occupied
Empty
MinPath
Typical minimal path
73Definitions. Assume there is a connected path of
minimal length l .
n length of side r density l MinPath
length b MinCut barrier length
Typical minimal cut
Occupied
Empty
MinPath
Typical minimal path
b
74Definitions. Assume there is a connected path of
minimal length l .
n length of side r density l MinPath
length b MinCut barrier length
Vertical path
b
75n length of side r density l MinPath
length b MinCut barrier length
Assume a path exists. (Otherwise LF?.)
Necessarily ? ? 1/n, n2 ? l ? n and define
76l MinPath length b MinCut barrier length
77l MinPath length b MinCut barrier length
Proof (Vinnicombesushi) 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.
78This is maximally tight in the sense that
- Lattices and paths can be
- Resources Scarce or rich
- Existence of path Yes or no
- Complexity Hard or easy
- Perturbations Fragile or robust
79- Lattices and paths can be
- Existence Yes or no
- Resources Scarce or rich
- Perturbations Fragile or robust
- Complexity Hard or easy
Anything is possible, consistent with the theorem.
Well just consider the 8 cases with paths.
80Fragile
Hard
Robust
Easy
Rich
Scarce
-Slog(?)
81Fragile
Hard
Robust
Easy
Rich
Scarce
82Fragile
Hard
Robust
Easy
Rich
Scarce
83Hard
Fragile
Scarce
Rich
Easy
Robust
84Easy
Occupied
Empty
MinPath
FS, L0
85Easy
Most robust possible.
FS, L0
86Easy and Fragile
Flog(n)gtS, L0
87Hard
Fragile
FSL
Scarce
Rich
Easy
m
d
Robust
b
Occupied
Empty
MinPath
88r density b MinCut barrier length l MinPath
length n length of side m of cells d
width of open regions
To construct asymptotically tight cases where ?n2
lb, consider the lattice below.
m
d
b
b
d
89Now take limits
By constructing lattices as below, with ngtgtmgtgt1,
it is possible to find lattices such that any ?n2
? lb, with ?lt1 is achievable.
90FSL
Hard
Fragile
Scarce
Rich
Easy
Robust
91The Fragile Face
Hard
Fragile
Scarce
Rich
Easy
Robust
92The Four Corners
Hard
Fragile
Scarce
Rich
Easy
Robust
93FSL
Fragile
Most Fragile FgtgtS
Scarce
Most Robust FS
Easy
Robust
94Random
Hard
Fragile
Scarce
Rich
Easy
Robust
95Efficient and robust is far from random
96- 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.
97Phase transitions
Complexity
98Bad 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.
99Physics and the edge of chaocritiplexity
Phase transitions
- Internet traffic and topology
- Biological and ecological networks
- Evolution and extinction
- Earthquakes and forest fires
- Finance and economics
- Social and political systems
?
Complexity
100(No Transcript)
101Physics and the edge of chaocritiplexity
Phase transitions
- 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
Complexity
102Physics and the edge of chaocritiplexity
- 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
103More powerful bounds for the co-NP side
Semialgebraic geometry convex optimization
(SDP)
- Polynomial time computation.
- Never worse than the standard.
- Exhausts co-NP.
104- Polynomial functions NP-hard problem.
105?0
Complexity?
106Special case Scalar QP
107Special case Scalar QP
108- Polynomial functions NP-hard problem.
- A simple relaxation (Shor) find the minimum
?such that ?- 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).
109- 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 NPco-NP), but
rare. - Variations of the Motzkin polynomial.
- Reductions of hard problems (e.g. NPP is nice)
- None could be found using random search
110Sums of squares (SOS)
A sufficient condition for nonnegativity
- Convex condition (Shor, 1987)
- Efficiently checked using SDP (Parrilo). Write
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).
111Nested families of SOS (Parrilo)
exhaust co-NP
112- 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
?0
113Search for counterexample
coNP
NP
Search for proof
114Positivstellensatz
Search for counterexample
Search for proof
- Convex, but infinite dimensional.
- Efficient (P time) search subsets (relaxations)
using SOS/SDP (Parrilo) - Guaranteed to converge
115Search for simple counterexample
Search for short proof
116Special case LP
117NPP
?
Fragile large changes in solution from small
changes in data
118NPP
119Random f
120Very unlikely to be feasible. Contrast with
random polynomial.
121- 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
122Phase transitions
Complexity
123More powerful bounds for the co-NP side
Semialgebraic geometry convex optimization
(SDP)
- Polynomial time computation.
- Never worse than the standard.
- Exhausts co-NP.
124A key insight
Think of LMIs as quadratic forms, not as
matrices. LMIs quadratic forms, that are
positive definite.
- General forms , not necessarily quadratic.
- Instead of nonnegativity (NP-hard), use sum of
squares.
SOS multivariable forms, that are sum of
squares.
125Search for counterexample
- 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.
Search for proof
126Search for counterexample
Seach for proof
- Convex, but infinite dimensional.
- Efficient (P time) search subsets (relaxations)
using SOS/SDP - Guaranteed to converge
127Search for simple counterexample
Search for short proof
128Special case LP
129Search for simple counterexample
Search for short proof
130Search for simple counterexample
Failure to find short proof implies some relaxed
model is nonempty (which is bad).
Search for short proof
131Sums of squares (SOS)
A sufficient condition for nonnegativity
- Convex condition (Shor, 1987)
- Efficiently checked using SDP (Parrilo). Write
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).
132Nested families of SOS (Parrilo)
exhaust co-NP
133?0
134A 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???
Lets see some examples
135Continuous Global Optimization
- Polynomial functions NP-hard problem.
- A simple relaxation (Shor) find the maximum
?such that f(x) ? 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).
136- 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 NPco-NP), but
rare. - Variations of the Motzkin polynomial.
- Reductions of hard problems.
- None could be found using random search
137More general framework
- 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.
138Semialgebraic 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)
139Positivstellensatz (Real Nullstellensatz)
if and only if
- Stengle, 1974
- Generalizes Hilberts 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
140Combinatorial 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
Hard combinatorial problem (NP-complete).
Compute upper bounds using convex relaxations.
141Standard semidefinite relaxation
Dual problems
This is just a first step. We can do better! The
new tools provide higher order relaxations.
- Tighter bounds are obtained.
- Never worse than the standard relaxation.
- In some cases (n-cycle, Petersen graph),
provably better. - Still polynomial time.
142MAX CUT on the Petersen graph
The standard SDP upper bound 12.5 Second
relaxation bound 12. The improved bound is
exact. A corresponding coloring.
143Finding Lyapunov functions
- Ubiquitous, fundamental problem
- Algorithmic LMI solution
Convex, but still NP hard.
Test using SOS and SDP.
After optimization coefficients of V.
A Lyapunov function V, that proves stability.
144Finding Lyapunov functions
- Ubiquitous, fundamental problem
- Algorithmic LMI solution
Given
Propose
After optimization coefficients of V.
A Lyapunov function V, that proves stability.
145Conclusion a certificate of global stability
146Flow of f
(x) with the corresponding Lyapunov function
1
5
c1
c2.164
c10
0
-5
-10
-15
-10
-5
0
5
10
-10
-5
0
5
10
-10
-5
0
5
1472
1.5
1
0.5
0
2
x
-0.5
-1
-1.5
-2
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x
1
Global stability of a switching system using 4th
order MLFs defined in 6 equiangular partitions
148(No Transcript)
149DS 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 ? ?
150DS applications Bendixson-Dulac
- Restrict to polynomial (or rational) solutions,
use SOS. - As for Lyapunov, now a fully algorithmic
procedure.
Given
Propose
After optimization
151Conclusion a certificate of the inexistence of
periodic orbits
x ' y
2
2
y ' - x - y x
y
3
2
1
0
y
-1
-2
-3
-3
-2
-1
0
1
2
3
x
152Conclusion a certificate of the inexistence of
periodic orbits
x ' y
saddle
y