Semidefinite Approximations of Invariant Measure Supports and Reachable Sets for Discrete-time Polynomial Systems Victor Magron , CNRS VERIMAG Joint work with Pierre-Loïc Garoche (ONERA) Didier Henrion (LAAS) Xavier Thirioux (IRIT) CASYS-Meff Seminar 7 July 2016 Victor Magron SDP Approximations of Reachable Sets 1 / 25
The Problem Semialgebraic initial conditions X 0 : = { x ∈ R n : g 0 1 ( x ) � 0, . . . , g 0 m 0 ( x ) � 0 } Polynomial map f : R n → R n , x �→ f ( x ) : = ( f 1 ( x ) , . . . , f n ( x )) deg f = d : = max { deg f 1 , . . . , deg f n } Set of admissible trajectories X ( ∞ ) : = { ( x t ) t ∈ N : x t + 1 = f ( x t ) , ∀ t ∈ N , x 0 ∈ X 0 } t ∈ N f t ( X 0 ) ⊆ X , with X ⊂ R n a box or a X ( ∞ ) = � ball Tractable approximations of X ( ∞ ) ? Victor Magron SDP Approximations of Reachable Sets 2 / 25
The Problem Occurs in several contexts : 1 program analysis: fixpoint computation toyprogram ( x 1 , x 2 ) requires ( 0.25 � x 1 � 0.75 && 0.25 � x 2 � 0.75 ); while ( x 2 1 + x 2 2 � 1 ){ x 1 = x 1 + 2 x 1 x 2 ; x 2 = 0.5 ( x 2 − 2 x 3 1 ) ; } 2 hybrid systems, biology: Neuron Model, Growth Model 3 control: integrator, Hénon map Victor Magron SDP Approximations of Reachable Sets 2 / 25
Related work: LP relaxations 1 Contractive methods based on LP relaxations and polyhedra projection [Bertsekas 72] 2 Extension to nonlinear systems [Harwood et al. 16] 3 Bernstein/Krivine-Handelman representations [Ben Sassi- et al. 15, Ben Sassi et al. 12] � LP relaxations = ⇒ scalability + � Convex approximations of nonconvex sets = ⇒ coarse − � No convergence guarantees (very often) − Victor Magron SDP Approximations of Reachable Sets 3 / 25
Related work: SDP relaxations 1 Upper bounds of the volume of a semialgebraic set [Henrion et al. 09] 2 Tractable approximations of sets defined with quantifiers ∃ , ∀ [Lasserre 15] 3 Semidefinite characterization of region of attraction [Henrion-Korda 14] 4 Convex computation of maximum controlled invariant [Korda-Henrion-Jones 13] Victor Magron SDP Approximations of Reachable Sets 4 / 25
Related work: SDP relaxations 5 SDP approximation of polynomial images of semialgebraic sets [Magron-Henrion-Lasserre 15] X 1 : = f ( X 0 ) ⊆ X , with X ⊂ R n a box or a ball ⇒ Discrete-time system with a single iteration = Approximation of image measure supports ⇒ certified SDP over approximations of X 1 = X t : = f t ( X 0 ) � deg f t = d × t = ⇒ very expensive computation − � Would only approximate X t and not X ( ∞ ) − Victor Magron SDP Approximations of Reachable Sets 4 / 25
Contribution General framework to approximate X ( ∞ ) � No discretization is required + Victor Magron SDP Approximations of Reachable Sets 5 / 25
Contribution General framework to approximate X ( ∞ ) � No discretization is required + Infinite-dimensional LP formulation support of measures solving Liouville’s Equation Victor Magron SDP Approximations of Reachable Sets 5 / 25
Contribution General framework to approximate X ( ∞ ) � No discretization is required + Infinite-dimensional LP formulation support of measures solving Liouville’s Equation Finite-dimensional SDP relaxations X ( ∞ ) ⊆ X r : = { x ∈ X : w r ( x ) � 1 } � Strong convergence guarantees + lim r → ∞ vol ( X r \ X ( ∞ ) ) = 0 � Compute w r by solving one semidefinite program + Victor Magron SDP Approximations of Reachable Sets 5 / 25
The Problem Infinite LP Formulation for Polynomial Optimization Infinite LP Formulation for Reachability Application examples Conclusion
What is Semidefinite Programming? Linear Programming (LP): ⊤ z min c z s.t. A z � d . Linear cost c Polyhedron Linear inequalities “ ∑ i A ij z j � d i ” Victor Magron SDP Approximations of Reachable Sets 6 / 25
What is Semidefinite Programming? Semidefinite Programming (SDP): ⊤ z min c z s.t. ∑ F i z i � F 0 . i Linear cost c Symmetric matrices F 0 , F i Spectrahedron Linear matrix inequalities “ F � 0” ( F has nonnegative eigenvalues) Victor Magron SDP Approximations of Reachable Sets 7 / 25
What is Semidefinite Programming? Semidefinite Programming (SDP): ⊤ z min c z s.t. ∑ F i z i � F 0 , A z = d . i Linear cost c Symmetric matrices F 0 , F i Spectrahedron Linear matrix inequalities “ F � 0” ( F has nonnegative eigenvalues) Victor Magron SDP Approximations of Reachable Sets 8 / 25
Applications of SDP Combinatorial optimization Control theory Matrix completion Unique Games Conjecture (Khot ’02) : “A single concrete algorithm provides optimal guarantees among all efficient algorithms for a large class of computational problems.” (Barak and Steurer survey at ICM’14) Solving polynomial optimization (Lasserre ’01) Victor Magron SDP Approximations of Reachable Sets 9 / 25
Polynomial Optimization Semialgebraic set X : = { x ∈ R n : g 1 ( x ) � 0, . . . , g l ( x ) � 0 } p ∗ : = inf x ∈ X f ( x ) : NP hard Sums of squares Σ [ x ] e.g. x 2 1 − 2 x 1 x 2 + x 2 2 = ( x 1 − x 2 ) 2 � � Q ( X ) : = σ 0 ( x ) + ∑ l j = 1 σ j ( x ) g j ( x ) , with σ j ∈ Σ [ x ] REMEMBER : f ∈ Q ( X ) = ⇒ ∀ x ∈ X , f ( x ) � 0 Victor Magron SDP Approximations of Reachable Sets 10 / 25
Infinite LP Reformulation Borel σ -algebra B ( X ) (generated by the open sets of X ) M + ( X ) : set of probability measures supported on X . If µ ∈ M + ( X ) then 1 µ : B → [ 0, ∞ ) , µ ( ∅ ) = 0 i B i ) = ∑ i µ ( B i ) , for any disjoint countable ( B i ) ⊂ B ( X ) 2 µ ( � 3 Lebesgue Volume of B ∈ B ( X ) � vol B : = X λ B , with λ B ( d x ) : = 1 B ( x ) d x supp µ is the smallest set X such that µ ( R n \ X ) = 0 Victor Magron SDP Approximations of Reachable Sets 11 / 25
Infinite LP Reformulation � p ∗ = inf inf x ∈ X f ( x ) = X f d µ µ ∈M + ( X ) Victor Magron SDP Approximations of Reachable Sets 11 / 25
Primal-dual Moment-SOS [Lasserre 01] Let ( x α ) α ∈ N n be the monomial basis Definition A sequence z has a representing measure on X if there exists a finite measure µ supported on X such that � ∀ α ∈ N n . X x α µ ( d x ) , z α = Victor Magron SDP Approximations of Reachable Sets 12 / 25
Primal-dual Moment-SOS [Lasserre 01] M + ( X ) : space of probability measures supported on X Q ( X ) : quadratic module Polynomial Optimization Problems (POP) (Primal) (Dual) � inf sup X f d µ = m s.t. s.t. m ∈ R , µ ∈ M + ( X ) f − m ∈ Q ( X ) Victor Magron SDP Approximations of Reachable Sets 12 / 25
Primal-dual Moment-SOS [Lasserre 01] Finite moment sequences z of measures in M + ( X ) Truncated quadratic module Q r ( X ) : = Q ( X ) ∩ R 2 r [ x ] Polynomial Optimization Problems (POP) (Moment) (SOS) inf ∑ sup = f α z α m α s.t. M r − v j ( g j z ) � 0 , 0 � j � l , s.t. m ∈ R , z 1 = 1 f − m ∈ Q r ( X ) Victor Magron SDP Approximations of Reachable Sets 12 / 25
Semidefinite Optimization F 0 , F α symmetric real matrices, cost vector c Primal-dual pair of semidefinite programs: P : inf z ∑ α c α z α s.t. ∑ α F α z α − F 0 � 0 ( SDP ) D : sup Y Trace ( F 0 Y ) s.t. Trace ( F α Y ) = c α , Y � 0 . Freely available SDP solvers ( CSDP , SDPA , S EDUMI ) Victor Magron SDP Approximations of Reachable Sets 13 / 25
The Problem Infinite LP Formulation for Polynomial Optimization Infinite LP Formulation for Reachability Application examples Conclusion
Pushforward and Liouville’s Equation Let µ 0 ∈ M + ( X 0 ) Pushforward f # : M + ( X 0 ) → M + ( X ) : f # µ 0 ( A ) : = µ 0 ( { x ∈ X 0 : f ( x ) ∈ A } ) , ∀ A ∈ B ( X ) f # µ 0 is the image measure of µ 0 under f Victor Magron SDP Approximations of Reachable Sets 14 / 25
Pushforward and Liouville’s Equation Let µ 0 ∈ M + ( X 0 ) , α > 1 and define µ 1 : = α f # µ 0 · · · µ t : = α f # µ t − 1 t − 1 t − 1 α i f i µ : = ∑ ∑ µ i = # µ 0 i = 0 i = 0 The measures µ t , µ , µ 0 satisfy Liouville’s Equation : µ t + µ = α f # µ + µ 0 Victor Magron SDP Approximations of Reachable Sets 14 / 25
Pushforward and Liouville’s Equation Let µ t : = λ X t : Lebesgue measure restriction on X t = f t ( X 0 ) ∃ µ 0 ∈ M + ( X 0 ) s.t. µ t = α t f t # µ 0 ⇒ µ t satisfies Liouville’s Equation ! = Victor Magron SDP Approximations of Reachable Sets 14 / 25
Pushforward and Liouville’s Equation Let µ t : = λ X t : Lebesgue measure restriction on X t = f t ( X 0 ) ∃ µ 0 ∈ M + ( X 0 ) s.t. µ t = α t f t # µ 0 ⇒ µ t satisfies Liouville’s Equation ! = Proof Define µ : = ∑ t − 1 i = 0 α i f i # µ 0 . Then, µ t + µ = α f # µ + µ 0 . Victor Magron SDP Approximations of Reachable Sets 14 / 25
Pushforward and Liouville’s Equation Let µ t : = λ X t : Lebesgue measure restriction on X t = f t ( X 0 ) ∃ µ 0 ∈ M + ( X 0 ) s.t. µ t = α t f t # µ 0 ⇒ µ t satisfies Liouville’s Equation ! = Proof Define µ : = ∑ t − 1 i = 0 α i f i # µ 0 . Then, µ t + µ = α f # µ + µ 0 . Let λ X ( T ) : Lebesgue measure restriction on � T t = 0 X t ⇒ λ X ( T ) satisfies Liouville’s Equation by superposition = Victor Magron SDP Approximations of Reachable Sets 14 / 25
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