optimization over nonnegative polynomials: algorithms and...
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Amir Ali AhmadiPrinceton, ORFE
INFORMS Optimization Society Conference (Tutorial Talk)Princeton University
March 17, 2016
Optimization over Nonnegative Polynomials: Algorithms and Applications
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Optimization over nonnegative polynomials
Ex. Decide if the following polynomial is nonnegative:
Ex.
Basic semialgebraic set:
Ubiquitous in computational mathematics!
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Why would you want to do this?!
Let’s start with three broad application areas…
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1. Polynomial Optimization
Many applications:
Equivalent formulation:
Combinatorial optimization [next slide]
Game theory [next slide]
Option pricing with moment information [Merton], [Boyle], [Lo], [Bertsimas et al.],…
The optimal power flow (OPF) problem [Lavaei, Low et al.], [Bienstock et al.]…
Sensor network localization [Ye et al.],…
Low-rank matrix completion, nonnegative matrix factorization, …
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1.1. Optimality certificates in discrete optimization
One can show:
is nonnegative.
Similar algebraic formulations for other combinatorial optimization problems…
if and only if
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1.1. Infeasibility certificates in discrete optimizationPARTITION
Note that the YES answer is easy to certify.
How would you certify a NO answer?
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1.1. Infeasibility certificates in discrete optimizationPARTITION
Infeasible iff
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1.2 Payoff bounds on Nash equilibria
Upper bounds on player 1&2’s payoffs under any Nash equilibrium:
See Session TC02(Bowl 1, 1:30 PM)
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• Shape-constrained regression; e.g., monotone regression
2: Statistics and Machine Learning
(From [Gupta et al.,’15])
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Imposing monotonicity
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• Convex regression
2: Statistics and Machine Learning (Ctnd.)
• Clustering with semialgebraic sets
nonnegative
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3: Safety verification of dynamical systems
set that requires safety verification(where system currently operating)
unsafe (or forbidden) sete.g., physical collision, variable overflow, blackout, financial crisis, …
Goal: “formal certificates” of safety
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3.1: Lyapunov Barrier Certificates
needs safety verification
unsafe (or forbidden) set
Safety assured if we find a “Lyapunov function” such that:
(vector valued polynomial)
(both sets semialgebraic)
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3.2: Stability of dynamical systems
Equilibrium populations Spread of epidemicsDynamics of prices Robotics
Control
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Lyapunov’s theorem for asymptotic stability
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Existence of a (Lyapunov) function
such that
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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 ≥ 4) (Have seen the proof already!)
But what if I told you:
Natural questions:•Is it any easier to test for a sum of squares (SOS) decomposition?•Is every nonnegative polynomial SOS?
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Sum of Squares and Semidefinite Programming
Q. Is it any easier to decide sos?
Yes! Can be reduced to a semidefinite program (SDP)
A broad generalization of linear programs
Can be solved to arbitrary accuracy in polynomial time(e.g., using interior point algorithms)
Can also efficiently search and optimize over sos polynomials
This enables numerous applications…
[Lasserre], [Nesterov], [Parrilo]
[Nesterov, Nemirovski], [Alizadeh]
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SOSSDP
where
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Example
Fully automated in YALMIP, SOSTOOLS, SPOTLESS, GloptiPoly, …
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Automated search for Lyapunov functions
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Lyapunov function
Ex. Lyapunov’s stability theorem.
(similar local version)
GAS
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Global stability
GASExample.
Output of SDP solver:
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Motzkin (1967):
Robinson (1973):
Hilbert’s 1888 Paper
n,d 2 4 ≥6
1 yes yes yes
2 yes yes no
3 yes no no
≥4 yes no no
n,d 2 4 ≥6
1 yes yes yes
2 yes yes yes
3 yes yes no
≥4 yes no no
PolynomialsForms
(homog. polynomials)
Q. SOS
?Nonnegativity
Homogenization:
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The Motzkin polynomial
• How to show it’s nonnegative but not sos?!
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The good news
In relatively small dimensions and degrees, it seems difficult to construct nonnegative polynomials that are not sos
Especially true if additional structure is required
For example, the following is OPEN:
Construct a convex, nonnegative polynomial that is not sos
Empirical evidence from various domains over the last decade:
SOS is a very powerful relaxation.
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Hilbert’s 17th Problem (1900)
p nonnegative ?
Q.
Artin (1927): Yes!
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Positivstellensatz: a complete algebraic proof system
Let’s motivate it with a toy example:
Consider the task of proving the statement:
Short algebraic proof (certificate):
The Positivstellensatz vastly generalizes what happened here:
Algebraic certificates of infeasibility of any system of polynomial inequalities (or algebraic implications among them)
Automated proof system (via semidefinite programming)
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Positivstellensatz: a generalization of Farkas lemma
Farkas lemma (1902):
is infeasible
(The S-lemma is also a theorem of this type for quadratics)
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PositivstellensatzStengle (1974):
Comments:
Hilbert’s 17th problem is a straightforward corollary
Other versions due to Shmudgen and Putinar (can look simpler)
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Parrilo/Lasserre SDP hierarchies
Recall POP:
Idea:
Comments:
Each fixed level of the hierarchy is an SDP of polynomial size
Originally, Parrilo’s hierarchy is based on Stengle’s Psatz, whereas Lasserre’s is based on Putinar’s Psatz
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Lyapunov Barrier Certificates
needs safety verification
unsafe (or forbidden) set
Safety assured if we find a “Lyapunov function” such that:
(vector valued polynomial)
(both sets semialgebraic)
(extends to uncertain or hybrid systems, stochastic differential equations and probabilistic guarantees [Prajna et al.])
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How did I plot this?
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This stuff actually works! (SOS on Acrobot)
[Majumdar, AAA, Tedrake ](Best paper award - IEEE Conf. on Robotics and Automation, ’13)
Swing-up:
Balance:
Controller designed by SOS
(4-state system)
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Some recent algorithmic developments
Approximating sum of squares programs with LPs and SOCPs
• Do we really need semidefinite programming?• Tradeoffs between speed and accuracy of approximation?
(DSOS and SDSOS Optimization)
• Order of magnitude speed-up in practice.• New applications at larger scale now within reach.• Potential for real-time algebraic optimization.
Highlights:
[AAA, Majumdar][AAA, Hall][AAA, Dash, Hall][Majumdar, AAA, Tedrake]
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Some recent algorithmic developments
Stabilization/ collision avoidance by SDSOS Optimization
TC02, Bowl 1, at 1:30 PM
(w/ Majumdar)
(w/ Majumdar, Tedrake)
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Want to know more?
Thank you!aaa.princeton.edu
• Next tutorial: the Lasserre hierarchy• Etienne de Klerk
• After lunch: follow-up session• TC02, Bowl 1, at 1:30 PM
• Tomorrow (FA02)• Javad Lavaei, polynomial optimization in energy
applications• Tomorrow (FD01)
• Russ Tedrake’s plenary talk, applications in robotics