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Amir Ali AhmadiPrinceton, ORFE
(Affiliated member of PACM, COS, MAE)
CDCโ17, Tutorial LectureMelbourne
Sum of Squares Optimizationand Applications
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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?
๐ฅ โ โ๐ ๐๐ ๐ฅ โฅ 0}
Ex: ๐ฅ13 โ 2๐ฅ1๐ฅ2
4 โฅ 0๐ฅ14 + 3๐ฅ1๐ฅ2 โ ๐ฅ2
6 โฅ 0
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Optimization over nonnegative polynomials
Is ๐ ๐ฅ โฅ 0 on {๐1 ๐ฅ โฅ 0,โฆ , ๐๐ ๐ฅ โฅ 0}?
Optimization Statistics Control
โข Lower bounds on polynomial optimization problems
โข Fitting a polynomial to data subject to shape constraints(e.g., convexity, or monotonicity)
๐๐(๐ฅ)
๐๐ฅ๐โฅ 0, โ๐ฅ โ ๐ต
โข Stabilizing controllers
Implies that
๐ฅ ๐ ๐ฅ โค ๐ฝ}is in the region of attraction
๐ ๐ฅ > 0,๐ ๐ฅ โค ๐ฝ โ ๐ป๐ ๐ฅ ๐๐ ๐ฅ < 0
๐ฅ = ๐(๐ฅ)
max๐พ
๐พ
s.t. ๐ ๐ฅ โ ๐พ โฅ 0,โ๐ฅ โ {๐๐ ๐ฅ โฅ 0}
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How would you prove nonnegativity?
Ex. Decide if the following polynomial is nonnegative:
Not so easy! (In fact, NP-hard for degree โฅ 4)
But what if I told you:
โขIs it any easier to test for a sum of squares (SOS) decomposition?
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SOSSDP
where
Example coming up in Antonisโ talkFully automated in YALMIP, SOSTOOLS, SPOTLESS, GloptiPoly, โฆ
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How to prove nonnegativity over a basic semialgebraic set?
Positivstellensatz: Certifies that
Putinarโs Psatz:(1993) under Archimedean condition
Stengleโs Psatz (1974)Schmudgenโs Psatz (1991)โฆ
Search for ๐๐ is an SDP when we bound the degree.
โ๐ ๐ฅ = ๐0 ๐ฅ + ๐ ๐๐ ๐ฅ ๐๐ ๐ฅ ,
where ๐๐ , ๐ = 0,โฆ ,๐ are sos
All use sos polynomialsโฆ
[Lasserre, Parrilo]
Dynamics and Control
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Lyapunov theory with sum of squares (sos) techniques
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Lyapunov function
Ex. Lyapunovโs stability theorem.
(similar local version)
GAS
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Global stability
GASExample.
Couple lines of code in SOSTOOLS, YALMIP, SPOTLESS, etc.
Output of SDP solver:
Theoretical limitations: converse implications may fail
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โข Globally asymptotically stable.
โข But no polynomial Lyapunov function of any degree! [AAA, Krstic, Parrilo]
[AAA,Parrilo]
โข Testing asymptotic stability of cubic vector fields is strongly NP-hard. [AAA]
Converse statements possible in special cases
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1. Asymptotically stable homogeneous polynomial vector field Rational Lyapunov function with an SOS certificate.[AAA, El Khadir]
2. Exponentially stable polynomial vector field on a compact set Polynomial Lyapunov function.[Peet, Papachristodoulou]
3. Asymptotically stable switched linear system Polynomial Lyapunov function with an SOS certificate. [Parrilo, Jadbabaie]
4. Asymptotically stable switched linear system Convex polynomial Lyapunovfunction with an SOS certificate.[AAA, Jungers]
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Local stability โ SOS on the Acrobot
[Majumdar, AAA, Tedrake ](Best paper award - IEEE Conf. on Robotics and Automation)
Swing-up:
Balance:
Controller designed by SOS
(4-state system)
Statistics and Machine Learning
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Monotone regression: problem definition
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NP-hardness and SOS relaxation
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[AAA, Curmei, Hall]
Approximation theorem
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[AAA, Curmei, Hall]
Numerical experiments
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Polynomial Optimization
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Theorem: Let ๐พ๐,2๐๐ be a sequence of sets of homogeneous polynomials in
๐ variables and of degree 2๐. If:
(1) ๐พ๐,2๐๐ โ ๐๐,2๐ โ๐ and โ ๐ ๐,2๐ pd in ๐พ๐,2๐
0
(2) ๐ > 0 โ โ๐ โ โ s. t. ๐ โ ๐พ๐,2๐๐
(3) ๐พ๐,2๐๐ โ ๐พ๐,2๐
๐+1 โ๐
(4) ๐ โ ๐พ๐,2๐๐ โ ๐ + ๐๐ ๐,2๐ โ ๐พ๐,2๐
๐ , โ๐ โ [0,1]
A meta-theorem for producing hierarchies
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๐๐,2๐
๐พ๐,2๐๐
=opt. val.
min๐ฅโโ๐
๐(๐ฅ)
๐ . ๐ก. ๐๐ ๐ฅ โฅ 0, ๐ = 1,โฆ ,๐
POPmax๐พ
๐พ
๐ . ๐ก. ๐๐พ ๐ง โ1
๐๐ ๐+๐+3,4๐ ๐ง โ ๐พ๐+๐+3,4๐
๐
๐ โ โCompactness assumptions
2๐ = maximum degree of ๐, ๐๐
Then,
where ๐๐พ is a form which can be written down explicitly from ๐, ๐๐ .
Example: Artin cones ๐ด๐,2๐๐ = ๐ ๐ โ ๐ is sos for some sos ๐ of degree 2๐}
[AAA, Hall]
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An optimization-free converging hierarchy
[AAA, Hall]
๐ ๐ฅ > 0, โ๐ฅ โ ๐ฅ โ โ๐ ๐๐ ๐ฅ โฅ 0, ๐ = 1, โฆ ,๐}
2๐ =maximum degree of ๐, ๐๐
โ
โ ๐ โ โ such that
๐ ๐ฃ2 โ ๐ค2 โ1
๐ ๐ ๐ฃ๐
2 โ ๐ค๐2 2 ๐
+1
2๐ ๐ ๐ฃ๐
4 + ๐ค๐4 ๐
โ ๐ ๐ฃ๐2 + ๐๐ค๐
2 ๐2
has nonnegative coefficients,
where ๐ is a form in ๐ +๐ + 3 variables and of degree 4๐, which can be explicitly written from ๐, ๐๐ and ๐ .
(also leads to DSOS/SDSOS-based converging hierarchies for POP)
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Ongoing directions: large-scale/real-time verification
โข 30 states, 14 control inputs, cubic dynamics
โข Done with SDSOS optimization (see Georginaโs talk)
Two promising approaches:
1. LP and SOCP-based alternatives to SOS, Georginaโs talkLess powerful than SOS (Jamesโ talk), but good enough for some applications
2. Exploiting problem structure and designing customized algorithms
Antonisโ talk (next), and Pablo Parriloโs plenary (Thu. 8:30am)
Slides/references available at: aaa.princeton.edu