Sum of Squares Optimization and Applications

Post on 16-Apr-2022

6 views 0 download

Transcript of Sum of Squares Optimization and Applications

1

Amir Ali AhmadiPrinceton, ORFE

(Affiliated member of PACM, COS, MAE)

CDC’17, Tutorial LectureMelbourne

Sum of Squares Optimizationand Applications

2

Optimization over nonnegative polynomials

Basic semialgebraic set:

Why do this?!

Example: When is

nonnegative?nonnegative over a given basic semialgebraic set?

π‘₯ ∈ ℝ𝑛 𝑔𝑖 π‘₯ β‰₯ 0}

Ex: π‘₯13 βˆ’ 2π‘₯1π‘₯2

4 β‰₯ 0π‘₯14 + 3π‘₯1π‘₯2 βˆ’ π‘₯2

6 β‰₯ 0

3

Optimization over nonnegative polynomials

Is 𝑝 π‘₯ β‰₯ 0 on {𝑔1 π‘₯ β‰₯ 0,… , π‘”π‘š π‘₯ β‰₯ 0}?

Optimization Statistics Control

β€’ Lower bounds on polynomial optimization problems

β€’ Fitting a polynomial to data subject to shape constraints(e.g., convexity, or monotonicity)

πœ•π‘(π‘₯)

πœ•π‘₯𝑗β‰₯ 0, βˆ€π‘₯ ∈ 𝐡

β€’ Stabilizing controllers

Implies that

π‘₯ 𝑉 π‘₯ ≀ 𝛽}is in the region of attraction

𝑉 π‘₯ > 0,𝑉 π‘₯ ≀ 𝛽 β‡’ 𝛻𝑉 π‘₯ 𝑇𝑓 π‘₯ < 0

π‘₯ = 𝑓(π‘₯)

max𝛾

𝛾

s.t. 𝑝 π‘₯ βˆ’ 𝛾 β‰₯ 0,βˆ€π‘₯ ∈ {𝑔𝑖 π‘₯ β‰₯ 0}

4

How would you prove nonnegativity?

Ex. Decide if the following polynomial is nonnegative:

Not so easy! (In fact, NP-hard for degree β‰₯ 4)

But what if I told you:

β€’Is it any easier to test for a sum of squares (SOS) decomposition?

5

SOSSDP

where

Example coming up in Antonis’ talkFully automated in YALMIP, SOSTOOLS, SPOTLESS, GloptiPoly, …

6

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 πˆπ’Š is an SDP when we bound the degree.

⇓𝑝 π‘₯ = 𝜎0 π‘₯ + 𝑖 πœŽπ‘– π‘₯ 𝑔𝑖 π‘₯ ,

where πœŽπ‘– , 𝑖 = 0,… ,π‘š are sos

All use sos polynomials…

[Lasserre, Parrilo]

Dynamics and Control

7

Lyapunov theory with sum of squares (sos) techniques

8

Lyapunov function

Ex. Lyapunov’s stability theorem.

(similar local version)

GAS

9

Global stability

GASExample.

Couple lines of code in SOSTOOLS, YALMIP, SPOTLESS, etc.

Output of SDP solver:

Theoretical limitations: converse implications may fail

10

β€’ Globally asymptotically stable.

β€’ But no polynomial Lyapunov function of any degree! [AAA, Krstic, Parrilo]

[AAA,Parrilo]

β€’ Testing asymptotic stability of cubic vector fields is strongly NP-hard. [AAA]

Converse statements possible in special cases

11

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]

12

Local stability – 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

13

Monotone regression: problem definition

14

NP-hardness and SOS relaxation

15

[AAA, Curmei, Hall]

Approximation theorem

16

[AAA, Curmei, Hall]

Numerical experiments

17

Polynomial Optimization

18

Theorem: Let 𝐾𝑛,2π‘‘π‘Ÿ be a sequence of sets of homogeneous polynomials in

𝑛 variables and of degree 2𝑑. If:

(1) 𝐾𝑛,2π‘‘π‘Ÿ βŠ† 𝑃𝑛,2𝑑 βˆ€π‘Ÿ and βˆƒ 𝑠𝑛,2𝑑 pd in 𝐾𝑛,2𝑑

0

(2) 𝑝 > 0 β‡’ βˆƒπ‘Ÿ ∈ β„• s. t. 𝑝 ∈ 𝐾𝑛,2π‘‘π‘Ÿ

(3) 𝐾𝑛,2π‘‘π‘Ÿ βŠ† 𝐾𝑛,2𝑑

π‘Ÿ+1 βˆ€π‘Ÿ

(4) 𝑝 ∈ 𝐾𝑛,2π‘‘π‘Ÿ β‡’ 𝑝 + πœ–π‘ π‘›,2𝑑 ∈ 𝐾𝑛,2𝑑

π‘Ÿ , βˆ€πœ– ∈ [0,1]

A meta-theorem for producing hierarchies

19

𝑃𝑛,2𝑑

𝐾𝑛,2π‘‘π‘Ÿ

=opt. val.

minπ‘₯βˆˆβ„π‘›

𝑝(π‘₯)

𝑠. 𝑑. 𝑔𝑖 π‘₯ β‰₯ 0, 𝑖 = 1,… ,π‘š

POPmax𝛾

𝛾

𝑠. 𝑑. 𝑓𝛾 𝑧 βˆ’1

π‘Ÿπ‘ π‘›+π‘š+3,4𝑑 𝑧 ∈ 𝐾𝑛+π‘š+3,4𝑑

π‘Ÿ

π‘Ÿ ↑ ∞Compactness assumptions

2𝑑 = maximum degree of 𝑝, 𝑔𝑖

Then,

where 𝑓𝛾 is a form which can be written down explicitly from 𝑝, 𝑔𝑖 .

Example: Artin cones 𝐴𝑛,2π‘‘π‘Ÿ = 𝑝 𝑝 β‹… π‘ž is sos for some sos π‘ž of degree 2π‘Ÿ}

[AAA, Hall]

20

An optimization-free converging hierarchy

[AAA, Hall]

𝑝 π‘₯ > 0, βˆ€π‘₯ ∈ π‘₯ ∈ ℝ𝑛 𝑔𝑖 π‘₯ β‰₯ 0, 𝑖 = 1, … ,π‘š}

2𝑑 =maximum degree of 𝑝, 𝑔𝑖

⇔

βˆƒ π‘Ÿ ∈ β„• such that

𝑓 𝑣2 βˆ’ 𝑀2 βˆ’1

π‘Ÿ 𝑖 𝑣𝑖

2 βˆ’ 𝑀𝑖2 2 𝑑

+1

2π‘Ÿ 𝑖 𝑣𝑖

4 + 𝑀𝑖4 𝑑

β‹… 𝑖 𝑣𝑖2 + 𝑖𝑀𝑖

2 π‘Ÿ2

has nonnegative coefficients,

where 𝑓 is a form in 𝑛 +π‘š + 3 variables and of degree 4𝑑, which can be explicitly written from 𝑝, 𝑔𝑖 and 𝑅.

(also leads to DSOS/SDSOS-based converging hierarchies for POP)

21

Ongoing directions: large-scale/real-time verification

β€’ 30 states, 14 control inputs, cubic dynamics

β€’ Done with SDSOS optimization (see Georgina’s talk)

Two promising approaches:

1. LP and SOCP-based alternatives to SOS, Georgina’s talkLess powerful than SOS (James’ talk), but good enough for some applications

2. Exploiting problem structure and designing customized algorithms

Antonis’ talk (next), and Pablo Parrilo’s plenary (Thu. 8:30am)

Slides/references available at: aaa.princeton.edu