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Amir Ali AhmadiPrinceton, ORFE
(Affiliated member of PACM, COS, MAE, CSML)
Caltech, CMS Colloquium
October 2018
Nonnegative Polynomialsand
Dynamical Systems
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Optimization over nonnegative polynomials
Basic semialgebraic set:
Why do this?!
Example: When is
nonnegative?nonnegative over a given basic semialgebraic set?
𝑥 ∈ ℝ𝑛 𝑔𝑖 𝑥 ≥ 0}
Ex: 𝑥13 − 2𝑥1𝑥2
4 ≥ 0𝑥14 + 3𝑥1𝑥2 − 𝑥2
6 ≥ 0
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Optimization over nonnegative polynomials
Is 𝑝 𝑥 ≥ 0 on {𝑔1 𝑥 ≥ 0,… , 𝑔𝑚 𝑥 ≥ 0}?
Optimization Statistics Control
• Lower bounds on polynomial optimization problems
• Proving infeasibility of a system of polynomial inequalities
• Fitting a polynomial to data subject to shape constraints(e.g., convexity, or monotonicity)
𝜕𝑝(𝑥)
𝜕𝑥𝑗≥ 0, ∀𝑥 ∈ 𝐵
• Stabilizing controllers
Implies that
𝑥 𝑉 𝑥 ≤ 𝛽}is in the region of attraction
𝑉 𝑥 > 0,𝑉 𝑥 ≤ 𝛽 ⇒ 𝛻𝑉 𝑥 𝑇𝑓 𝑥 < 0
𝑥 = 𝑓(𝑥)
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How to prove nonnegativity? The SOS approach
Ex:
Optimizing over set of sos polynomials is an SDP!
• A polynomial 𝑝 is a sum of squares (sos) if it can be written as
• A polynomial 𝑝 of degree 2𝑑 is sos if and only if ∃𝑄 ≽ 0 such that
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How to prove nonnegativity over a basic semialgebraic set?
Positivstellensatz: Certifies that
Putinar’s Psatz:(1993) under Archimedean condition
Stengle’s Psatz (1974)Schmudgen’s Psatz (1991)…
Search for 𝝈𝒊 is an SDP when we bound the degree.
⇓𝑝 𝑥 = 𝜎0 𝑥 + 𝑖 𝜎𝑖 𝑥 𝑔𝑖 𝑥 ,
where 𝜎𝑖 , 𝑖 = 0,… ,𝑚 are sos
All use sos polynomials…
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Outline
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Part I:Avoiding SDP in optimization over nonnegative polynomials
- LP, SOCP
- An “optimization-free” Positivstellensatz
Part II:Asymptotic stability of polynomial vector fields
- Complexity
- Computational converse Lyapunov questions
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Practical limitations of SOS
• Scalability is a nontrivial challenge!
Thm: 𝒑(𝒙) of degree 𝟐𝒅 is sos if and only if
• The size of the Gram matrix is:
• Polynomial in n for fixed d, but grows quickly
• The semidefinite constraint is expensive
• E.g., local stability analysis of a 20-state cubic vector field is typically an SDP with ~1.2M decision variables and ~200k constraints
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Simple idea…
• Let’s not work with SOS…
• Give other sufficient conditions for nonnegativity that are perhaps stronger than SOS, but hopefully cheaper
1) sums of 4th powers of polynomials
2) sums of 3 squares of polynomials
Both sets are clearly inside the SOS cone,but linear optimization over them is intractable.
Not any set inside SOS would work!
POS POS
SOS
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dsos and sdsos polynomials (1/3)
Defn. A polynomial p is diagonally-dominant-sum-of-squares(dsos) if it can be written as:
for some monomials and some nonnegative constants
Defn. A polynomial p is scaled-diagonally-dominant-sum-of-squares (sdsos) if it can be written as:
for some monomials and some constants
Note:
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dsos and sdsos polynomials (2/3)
[AAA, Majumdar, SIAM J. on Applied Algebra and Geometry]
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dsos and sdsos polynomials (3/3)
How to do better?
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Method #1: r-dsos and r-sdsos polynomials (1/2)
Defn.
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Method #1: r-dsos and r-sdsos polynomials (2/2)
• r-dsos can outperform sos!
is 1-dsos but not sos.
Theorem: Any even positive definite form is r-dsos for some r.
• Even forms include copositive programming (and all problems in NP).
• Shows that LP-based proofs of nonnegativity always possible.
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Method #2: dsos/sdsos + change of basis (1/2)
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Goal: iteratively improve 𝑧(𝑥)
[AAA, Hall, Contemporary Mathematics’17]
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Method #2: dsos/sdsos + change of basis (2/2)
LP
SOCP
Works beautifully!
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Reminder
Stability of equilibrium points
implies 𝑥 𝑉 𝑥 ≤ 𝛽} is in the region of attraction (ROA)
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Stabilizing the inverted N-link pendulum (2N states)
N=1
N=2 N=6
Runtime:
ROA volume ratio:
[Majumdar, AAA, Tedrake, CDC]
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Stabilizing ATLAS
[Majumdar, AAA, Tedrake, CDC]
Done by SDSOS Optimization
• 30 states 14 control inputs Cubic dynamics
https://github.com/spot-toolbox/spotless
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PutinarPutinar
If 𝑝 𝑥 > 0, ∀𝑥 ∈ 𝑆,then 𝑝 𝑥 = 𝜎0 𝑥 + 𝑖 𝜎𝑖 𝑥 𝑔𝑖(𝑥) ,
where 𝜎0, 𝜎𝑖 are sos
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What can DSOS/SDSOS do in theory?
𝑝 𝑥 > 0, ∀𝑥 ∈ 𝑥 ∈ ℝ𝑛 𝑔𝑖 𝑥 ≥ 0, 𝑖 = 1, … ,𝑚}
• Is there always an SOS proof?
• Is there always an SDSOS proof?
• Is there always an DSOS proof?
Yes, e.g. based on Putinar’s Psatz.(under a compactness assumption)
Yes! In fact, a much stronger statement is true.
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An optimization-free Positivstellensatz (1/2)
[AAA, Hall, Math of OR ‘18] (2018 INFORMS Young Researchers Prize)
𝑝 𝑥 > 0, ∀𝑥 ∈ 𝑥 ∈ ℝ𝑛 𝑔𝑖 𝑥 ≥ 0, 𝑖 = 1, … ,𝑚}
2𝑑 =maximum degree of 𝑝, 𝑔𝑖
⇔
∃ 𝑟 ∈ ℕ such that
𝑓 𝑣2 − 𝑤2 −1
𝑟 𝑖 𝑣𝑖
2 − 𝑤𝑖2 2 𝑑
+1
2𝑟 𝑖 𝑣𝑖
4 + 𝑤𝑖4 𝑑
⋅ 𝑖 𝑣𝑖2 + 𝑖𝑤𝑖
2 𝑟2
has nonnegative coefficients,
where 𝑓 is a form in 𝑛 +𝑚 + 3 variables and of degree 4𝑑, which can be explicitly written from 𝑝, 𝑔𝑖 and 𝑅.
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𝑝 𝑥 > 0 on 𝑥 𝑔𝑖 𝑥 ≥ 0} ⇔
∃𝑟 ∈ ℕ s. t. 𝑓 𝑣2 − 𝑤2 −1
𝑟 𝑖 𝑣𝑖
2 − 𝑤𝑖2 2 𝑑
+1
2𝑟 𝑖 𝑣𝑖
4 + 𝑤𝑖4 𝑑
⋅ 𝑖 𝑣𝑖2 + 𝑖𝑤𝑖
2 𝑟2
has ≥ 0 coefficients
Proof sketch:
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An optimization-free Positivstellensatz (2/2)
𝑝 𝑥 > 0 on 𝑥 𝑔𝑖 𝑥 ≥ 0} ⇔
∃𝑟 ∈ ℕ s. t. 𝒇 𝒗𝟐 − 𝒘𝟐 −1
𝑟 𝑖 𝑣𝑖
2 − 𝑤𝑖2 2 𝑑
+1
2𝑟 𝑖 𝑣𝑖
4 + 𝑤𝑖4 𝑑
⋅ 𝑖 𝑣𝑖2 + 𝑖𝑤𝑖
2 𝑟2
has ≥ 0 coefficients
𝑝 𝑥 > 0 on 𝑥 𝑔𝑖 𝑥 ≥ 0} ⇔
∃𝑟 ∈ ℕ s. t. 𝒇 𝒗𝟐 − 𝒘𝟐 −1
𝑟 𝑖 𝑣𝑖
2 − 𝑤𝑖2 2 𝑑
+𝟏
𝟐𝒓 𝒊 𝒗𝒊
𝟒 + 𝒘𝒊𝟒 𝒅
⋅ 𝒊𝒗𝒊𝟐 + 𝒊𝒘𝒊
𝟐 𝒓𝟐
has ≥ 𝟎 coefficients
𝑝 𝑥 > 0 on 𝑥 𝑔𝑖 𝑥 ≥ 0} ⇔
∃𝑟 ∈ ℕ s. t. 𝒇 𝒗𝟐 −𝒘𝟐 −𝟏
𝒓 𝒊 𝒗𝒊
𝟐 − 𝒘𝒊𝟐 𝟐 𝒅
+𝟏
𝟐𝒓 𝒊 𝒗𝒊
𝟒 + 𝒘𝒊𝟒 𝒅
⋅ 𝒊𝒗𝒊𝟐 + 𝒊𝒘𝒊
𝟐 𝒓𝟐
has ≥ 𝟎 coefficients
• 𝑝 𝑥 > 0 on 𝑥 𝑔𝑖 𝑥 ≥ 0} ⇔ 𝑓 is pd
• Result by Polya (1928):
𝑓 even and pd ⇒ ∃𝑟 ∈ ℕ such that 𝑓 𝑧 ⋅ 𝑖 𝑧𝑖2 𝑟
has nonnegative coefficients.
• Make 𝑓(𝑧) even by considering 𝒇 𝒗𝟐 −𝒘𝟐 . We lose positive definiteness of
𝑓 with this transformation.
• Add the positive definite term 𝟏
𝟐𝒓 𝒊 𝒗𝒊
𝟒 +𝒘𝒊𝟒 𝒅
to 𝑓(𝑣2 − 𝑤2) to make it
positive definite. Apply Polya’s result.
• The term −𝟏
𝒓 𝒊 𝒗𝒊
𝟐 −𝒘𝒊𝟐 𝟐 𝒅
ensures that the converse holds as well.
As a corollary, gives LP/SOCP-based converging hierarchies…
(Even forms with nonnegative coefficients are trivially dsos.)
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Part 2: Asymptotic Stability of
Polynomial Vector Fields
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Asymptotic stability
Example
Locally Asymp. Stable (LAS) if Globally Asymp. Stable (GAS) if
polynomial with rational coefficients
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Complexity of deciding asymptotic stability?
What if deg(f)>1? …
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Complexity of deciding asymptotic stability?
Thm: Deciding (local or global) asymptotic stability of cubic vector fields is strongly NP-hard.
[AAA]
(In particular, this rules out tests based on polynomially-sized convex programs.)
Conjecture of Arnol’d (1976): undecidable (still open)
Fact: Existence of polynomial Lyapunov functions, together
with a computable upper bound on the degree would imply decidability (e.g., by quantifier elimination)
What if deg(f)>1? …
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Thm: Deciding asymptotic stability of cubic homogeneous vector fields is strongly NP-hard.
All monomials in have the same degree
Local Asymptotic Stability = Global Asymptotic Stability
Homogeneous means:
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Proof
ONE-IN-THREE 3SAT
Thm: Deciding asymptotic stability of cubic homogeneous vector fields is strongly NP-hard.
Reduction from:
Goal: Design a cubic differential equation which is a.s. iffONE-IN-THREE 3SAT has no solution
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Proof (cont’d)
ONE-IN-THREE 3SAT
Positivity of quartic forms
Asymptotic stability ofcubic homogeneous
vector fields
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Proof (cont’d)
Thm: Let be a homogeneous polynomial. Then,is positive definite is GAS
Proof:
Apply Lyapunov’s theorem.
implies strict decrease…
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Proof:
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Nonexistence of polynomial Lyapunov functions (1/4)
System is GAS.Claim 1:
Claim 2: No polynomial Lyapunov function (of any degree) exists!
[AAA, Krstic, Parrilo, CDC’11]
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Nonexistence of polynomial Lyapunov functions (2/4)
Claim 2: No polynomial Lyapunov function (of any degree) exists!
Proof:
Impossible.
• No rational Lyapunov function either [AAA, El Khadir ’18].
• But a quadratic Lyapunovfunction locally.
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Proof:
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Nonexistence of polynomial Lyapunov functions (3/4)
System is GAS.Claim 1:
Claim 2: No polynomial Lyapunovfunction (of any degree) even locally!
[AAA, El Khadir, Systems & Control Letters’18]
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Proof idea:Suppose we had one:
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Nonexistence of polynomial Lyapunov functions (4/4)
Claim 2: No polynomial Lyapunovfunction (of any degree) even locally!
A polynomial must be constant on the unit level set of 𝑊 𝑥, 𝑦 = 𝑥4 + 𝑦4 / 𝑥2 + 𝑦2
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Let’s end on a positive note!
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Thm. A homogeneous polynomial vector field is asymptotically stable iff it admits a rational Lyapunov function of the type
where 𝑝 is a homogeneous polynomial.
[AAA, El Khadir, TAC, accepted with minor revision]
• We show that 𝑉 and − 𝑉 both have “strict SOS certificates.” 𝑉 can be found by SDP!
• Useful also for local asym. stability of non-homogeneous systems.
• We show that unlike the linear case, the degree of 𝑉 cannot be bounded as a function of the dimension and degree of 𝑓.
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Main messages
35Want to know more? aaa.princeton.edu
• SDP-free alternatives to SOS• DSOS/SDSOS (LP and SOCP)• Infeasibility certificates
based on poly-poly multiplication
POS
SOS
?
• No pseudo-poly-time algorithm for asymptotic stability of poly vector fields
• Polynomail Lyapunov function can fail even locally
• Rational Lyapunov functions deserve more attention