Semidefinite Approximations of Projections and Polynomial Images of - PowerPoint PPT Presentation

Semidefinite Approximations of Projections and Polynomial Images of Semialgebraic Sets Victor Magron , CNRS VERIMAG joint work with Didier Henrion and Jean-Bernard Lasserre (LAAS) CompACS Meeting 18 January 2016 Victor Magron SDP Approximations of Semialgebraic Set Projections 1 / 30

The Problem Semialgebraic set S : = { x ∈ R n : g 1 ( x ) � 0, . . . , g l ( x ) � 0 } A polynomial map f : R n → R m , x �→ f ( x ) : = ( f 1 ( x ) , . . . , f m ( x )) deg f = d : = max { deg f 1 , . . . , deg f m } F : = f ( S ) ⊆ B , with B ⊂ R m a box or a ball Tractable approximations of F ? Victor Magron SDP Approximations of Semialgebraic Set Projections 2 / 30

The Problem Includes important special cases: 1 m = 1: polynomial optimization F ⊆ [ inf x ∈ S f ( x ) , sup f ( x )] x ∈ S 2 Approximate projections of S when f ( x ) : = ( x 1 , . . . , x m ) 3 Pareto curve approximations � � x ∈ S ( f 1 ( x ) f 2 ( x )) ⊤ For f 1 , f 2 two conflicting criteria: ( P ) min Victor Magron SDP Approximations of Semialgebraic Set Projections 2 / 30

The Problem 3 Pareto curve : set of weakly Edgeworth-Pareto optimal points � � x ∈ S ( f 1 ( x ) f 2 ( x )) ⊤ ( P ) min Definition A point ¯ x ∈ S is called a weakly Edgeworth-Pareto (EP) optimal point of Problem P , when there is no x ∈ S such that f j ( x ) < f j ( ¯ x ) , j = 1, 2. Victor Magron SDP Approximations of Semialgebraic Set Projections 2 / 30

The Problem f 1 : = ( x 1 + x 2 − 7.5 ) 2 /4 + ( − x 1 + x 2 + 3 ) 2 , g 1 : = − ( x 1 − 2 ) 3 /2 − x 2 + 2.5 , g 2 : = − x 1 − x 2 + 8 ( − x 1 + x 2 + 0.65 ) 2 + 3.85 , f 2 : = ( x 1 − 1 ) 2 /4 + ( x 2 − 4 ) 2 /4 . S : = { x ∈ R 2 : g 1 ( x ) � 0, g 2 ( x ) � 0 } . Victor Magron SDP Approximations of Semialgebraic Set Projections 2 / 30

Previous work 1 Exact description of projections with computer algebra Real quantifier elimination (QE) [Tarski 51, Collins 74, Bochnak-Coste-Roy 98] CAD: computational complexity ( sd ) 2 O ( n ) for a finite set of s polynomials Variant QE under radicality, equidimensionality [Hong-Safey 12] Victor Magron SDP Approximations of Semialgebraic Set Projections 3 / 30

Previous work 2 Scalarization methods for computing Pareto curve Numerical discretization schemes: modified Polak method [Pol 76] Iterative Eichfelder-Polak algorithm [Eich 09] Normal-boundary intersection method to find uniform spread of points [Das Dennis 98] Victor Magron SDP Approximations of Semialgebraic Set Projections 3 / 30

Contribution A unifying framework to handle projections, Pareto curve approximations and other applications No discretization is required Victor Magron SDP Approximations of Semialgebraic Set Projections 4 / 30

Contribution A unifying framework to handle projections, Pareto curve approximations and other applications No discretization is required Two different methods: 1 Existential QE: F ⊆ F 1 k : = { y ∈ B : q k ( y ) � 0 } 2 Image measure supports: F ⊆ F 2 k : = { y ∈ B : w k ( y ) � 1 } Victor Magron SDP Approximations of Semialgebraic Set Projections 4 / 30

Contribution A unifying framework to handle projections, Pareto curve approximations and other applications No discretization is required Two different methods: 1 Existential QE: F ⊆ F 1 k : = { y ∈ B : q k ( y ) � 0 } 2 Image measure supports: F ⊆ F 2 k : = { y ∈ B : w k ( y ) � 1 } Strong convergence guarantees Victor Magron SDP Approximations of Semialgebraic Set Projections 4 / 30

Contribution A unifying framework to handle projections, Pareto curve approximations and other applications No discretization is required Two different methods: 1 Existential QE: F ⊆ F 1 k : = { y ∈ B : q k ( y ) � 0 } 2 Image measure supports: F ⊆ F 2 k : = { y ∈ B : w k ( y ) � 1 } Strong convergence guarantees Compute q k or w k with Semidefinite programming (SDP) Victor Magron SDP Approximations of Semialgebraic Set Projections 4 / 30

The Problem m = 1: Polynomial Optimization Method 1: existential quantifier elimination Method 2: support of image measures Application examples Conclusion

Polynomial Optimization Semialgebraic set S : = { x ∈ R n : g 1 ( x ) � 0, . . . , g l ( x ) � 0 } p ∗ : = inf x ∈ S 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 � � σ 0 ( x ) + ∑ l Q ( S ) : = j = 1 σ j ( x ) g j ( x ) , with σ j ∈ Σ [ x ] REMEMBER : f ∈ Q ( S ) = ⇒ ∀ x ∈ S , f ( x ) � 0 Victor Magron SDP Approximations of Semialgebraic Set Projections 5 / 30

Problem reformulation Borel σ -algebra B (generated by the open sets of R n ) M + ( S ) : set of probability measures supported on S . If µ ∈ M + ( S ) then 1 µ : B → [ 0, 1 ] , µ ( ∅ ) = 0 2 µ ( � i B i ) = ∑ i µ ( B i ) , for any countable ( B i ) ⊂ B 3 � S µ ( d x ) = 1 supp ( µ ) is the smallest set S such that µ ( R n \ S ) = 0 Victor Magron SDP Approximations of Semialgebraic Set Projections 6 / 30

Problem reformulation � p ∗ = inf x ∈ S f ( x ) = S f d µ inf µ ∈M + ( S ) Victor Magron SDP Approximations of Semialgebraic Set Projections 6 / 30

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 S if there exists a finite measure µ supported on S such that � ∀ α ∈ N n . S x α µ ( d x ) , z α = Victor Magron SDP Approximations of Semialgebraic Set Projections 7 / 30

Primal-dual Moment-SOS [Lasserre 01] M + ( S ) : space of probability measures supported on S Q ( S ) : quadratic module Polynomial Optimization Problems (POP) (Primal) (Dual) � inf S f d µ = sup λ s.t. µ ∈ M + ( S ) s.t. λ ∈ R , f − λ ∈ Q ( S ) Victor Magron SDP Approximations of Semialgebraic Set Projections 7 / 30

Primal-dual Moment-SOS [Lasserre 01] Finite moment sequences z of measures in M + ( S ) Truncated quadratic module Q k ( S ) : = Q ( S ) ∩ R 2 k [ x ] Polynomial Optimization Problems (POP) (Moment) (SOS) ∑ = inf f α z α sup λ α M k − v j ( g j z ) � 0 , λ ∈ R , s.t. 0 � j � l , s.t. z 1 = 1 f − λ ∈ Q k ( S ) Victor Magron SDP Approximations of Semialgebraic Set Projections 7 / 30

Lasserre’s Hierarchy of SDP relaxations ℓ z ( q ) : q ∈ R [ x ] �→ ∑ q α z α α Moment matrix M ( z ) x α , x β : = ℓ z ( x α x β ) = z α + β Localizing matrix M ( g j z ) associated with g j M ( g j z ) x α , x β : = ℓ z ( g j x α x β ) = ∑ γ g j , γ z α + β + γ Victor Magron SDP Approximations of Semialgebraic Set Projections 8 / 30

Lasserre’s Hierarchy of SDP relaxations M k ( z ) contains ( n + 2 k n ) variables, has size ( n + k n ) Truncated matrix of order k = 2 with variables x 1 , x 2 : x 2 x 2 | | 1 x 1 x 2 x 1 x 2 1 2   | | 1 1 z 1,0 z 0,1 z 2,0 z 1,1 z 0,2 − − − − − − − −     x 1 z 1,0 | z 2,0 z 1,1 | z 3,0 z 2,1 z 1,2       x 2 z 0,1 | z 1,1 z 0,2 | z 2,1 z 1,2 z 0,3   M 2 ( z ) =   − − − − − − − − −     x 2 z 2,0 | z 3,0 z 2,1 | z 4,0 z 3,1 z 2,2   1   x 1 x 2  z 1,1 | z 2,1 z 1,2 | z 3,1 z 2,2 z 1,3    x 2 | | z 0,2 z 1,2 z 0,3 z 2,2 z 1,3 z 0,4 2 Victor Magron SDP Approximations of Semialgebraic Set Projections 8 / 30

Lasserre’s Hierarchy of SDP relaxations Consider g 1 ( x ) : = 2 − x 2 1 − x 2 2 . Then v 1 = ⌈ deg g 1 /2 ⌉ = 1. 1 x 1 x 2  2 − z 2,0 − z 0,2 2 z 1,0 − z 3,0 − z 1,2 2 z 0,1 − z 2,1 − z 0,3  1 M 1 ( g 1 z ) = 2 z 1,0 − z 3,0 − z 1,2 2 z 2,0 − z 4,0 − z 2,2 2 z 1,1 − z 3,1 − z 1,3 x 1   x 2 2 z 0,1 − z 2,1 − z 0,3 2 z 1,1 − z 3,1 − z 1,3 2 z 0,2 − z 2,2 − z 0,4 M 1 ( g 1 z )( 3, 3 ) = ℓ ( g 1 ( x ) · x 2 · x 2 ) = ℓ ( 2 x 2 2 − x 2 1 x 2 2 − x 4 2 ) = 2 z 0,2 − z 2,2 − z 0,4 Victor Magron SDP Approximations of Semialgebraic Set Projections 8 / 30

Lasserre’s Hierarchy of SDP relaxations Truncation with moments of order at most 2 k v j : = ⌈ deg g j /2 ⌉ Hierarchy of semidefinite relaxations: S f α x α µ ( d x ) = ∑ α f α z α  � inf z ℓ z ( f ) = ∑ α    M k ( z ) 0 , �  M k − v j ( g j z ) 0 , 1 � j � l , �    = z 1 1 .  Victor Magron SDP Approximations of Semialgebraic Set Projections 8 / 30

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    ∑ α F α z α − F 0 � 0 s.t.    ( 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 Semialgebraic Set Projections 9 / 30

The Problem m = 1: Polynomial Optimization Method 1: existential quantifier elimination Method 2: support of image measures Application examples Conclusion

Approximation of sets defined with “ ∃ ” Another point of view: F = { y ∈ B : ∃ x ∈ S s.t. f ( x ) = y } , Victor Magron SDP Approximations of Semialgebraic Set Projections 10 / 30

Approximation of sets defined with “ ∃ ” Another point of view: F = { y ∈ B : ∃ x ∈ S s.t. � y − f ( x ) � 2 2 = 0 } , Victor Magron SDP Approximations of Semialgebraic Set Projections 10 / 30