Lasserre’s Hierarchy of SDP relaxations M k ( y ) 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 y 1,0 y 0,1 y 2,0 y 1,1 y 0,2 − − − − − − − − x 1 y 1,0 | y 2,0 y 1,1 | y 3,0 y 2,1 y 1,2 x 2 y 0,1 | y 1,1 y 0,2 | y 2,1 y 1,2 y 0,3 M 2 ( y ) = − − − − − − − − − x 2 y 2,0 | y 3,0 y 2,1 | y 4,0 y 3,1 y 2,2 1 x 1 x 2 y 1,1 | y 2,1 y 1,2 | y 3,1 y 2,2 y 1,3 x 2 | | y 0,2 y 1,2 y 0,3 y 2,2 y 1,3 y 0,4 2 V. Magron New Applications of Moment-SOS Hierarchies 15 / 68
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 − y 2,0 − y 0,2 2 y 1,0 − y 3,0 − y 1,2 2 y 0,1 − y 2,1 − y 0,3 1 M 1 ( g 1 y ) = 2 y 1,0 − y 3,0 − y 1,2 2 y 2,0 − y 4,0 − y 2,2 2 y 1,1 − y 3,1 − y 1,3 x 1 x 2 2 y 0,1 − y 2,1 − y 0,3 2 y 1,1 − y 3,1 − y 1,3 2 y 0,2 − y 2,2 − y 0,4 M 1 ( g 1 y )( 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 y 0,2 − y 2,2 − y 0,4 V. Magron New Applications of Moment-SOS Hierarchies 15 / 68
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: K p α x α µ ( d x ) = ∑ α p α y α � inf y L y ( p ) = ∑ α M k ( y ) � 0 , M k − v j ( g j y ) � 0 , 1 � j � m , = y 1 1 . V. Magron New Applications of Moment-SOS Hierarchies 15 / 68
Semidefinite Optimization F 0 , F α symmetric real matrices, cost vector c Primal-dual pair of semidefinite programs: P : inf y ∑ α c α y α ∑ α F α y 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 ) V. Magron New Applications of Moment-SOS Hierarchies 16 / 68
Primal-dual Moment-SOS M + ( K ) : space of probability measures supported on K Polynomial Optimization Problems (POP) (Primal) (Dual) � = inf K p d µ sup λ µ ∈ M + ( K ) λ ∈ R , s.t. s.t. p − λ ∈ Q ( K ) V. Magron New Applications of Moment-SOS Hierarchies 17 / 68
Primal-dual Moment-SOS Truncated quadratic module Q k ( K ) : = Q ( K ) ∩ R 2 k [ x ] For large enough k , zero duality gap [Lasserre 01]: Polynomial Optimization Problems (POP) (Moment) (SOS) ∑ = inf p α y α sup λ α M k − v j ( g j y ) � 0 , λ ∈ R , s.t. 0 � j � m , s.t. y 1 = 1 p − λ ∈ Q k ( K ) V. Magron New Applications of Moment-SOS Hierarchies 17 / 68
Practical Computation Hierarchy of SOS relaxations: � � λ k : = sup λ : p − λ ∈ Q k ( K ) λ Convergence guarantees λ k ↑ p ∗ [Lasserre 01] If p − p ∗ ∈ Q k ( K ) for some k then: y ∗ : = ( 1, x ∗ 1 , x ∗ 2 , ( x ∗ 1 ) 2 , x ∗ 1 x ∗ 2 , . . . , ( x ∗ 1 ) 2 k , . . . , ( x ∗ n ) 2 k ) is a global minimizer of the primal SDP [Lasserre 01]. V. Magron New Applications of Moment-SOS Hierarchies 18 / 68
Practical Computation Caprasse Problem ∀ x ∈ [ − 0.5, 0.5 ] 4 , − x 1 x 3 3 + 4 x 2 x 2 3 x 4 + 4 x 1 x 3 x 2 4 + 2 x 2 x 3 4 + 4 x 1 x 3 + 4 x 2 3 − 10 x 2 x 4 − 10 x 2 4 + 5.1801 � 0. scale_pol = true : scaled on [ 0, 1 ] 4 relax_order = 2 : SOS of degree at most 4 bound_squares_variables = true : redundant constraints x 2 1 � 1, . . . , x 2 4 � 1 V. Magron New Applications of Moment-SOS Hierarchies 18 / 68
The “No Free Lunch” Rule Exponential dependency in 1 Relaxation order k (SOS degree) 2 number of variables n Computing λ k involves ( n + 2 k n ) variables At fixed k , O ( n 2 k ) variables V. Magron New Applications of Moment-SOS Hierarchies 19 / 68
Semialgebraic Extension [Lasserre-Putinar 10] Example from Flyspeck K : = [ 4, 6.3504 ] 6 ∆ ( x ) = x 1 x 4 ( − x 1 + x 2 + x 3 − x 4 + x 5 + x 6 ) + x 2 x 5 ( x 1 − x 2 + x 3 + x 4 − x 5 + x 6 ) + x 3 x 6 ( x 1 + x 2 − x 3 + x 4 + x 5 − x 6 ) − x 2 ( x 3 x 4 + x 1 x 6 ) − x 5 ( x 1 x 3 + x 4 x 6 ) ∂ 4 ∆ x = x 1 ( − x 1 + x 2 + x 3 − x 4 + x 5 + x 6 ) + x 2 x 5 + x 3 x 6 − x 2 x 3 − x 5 x 6 p ( x ) = ∂ 4 ∆ x , q ( x ) : = 4 x 1 ∆ x p ( x ) f ∗ sa : = inf � x ∈ K q ( x ) V. Magron New Applications of Moment-SOS Hierarchies 20 / 68
Semialgebraic Extension [Lasserre-Putinar 10] Example from Flyspeck: � z 1 : = q ( x ) , x ∈ K z 1 ( x ) , z 1 ( x ) , m 1 � inf M 1 � sup x ∈ K K : = { ( x , z ) ∈ R 8 : x ∈ K , h 1 ( x , z ) � 0, · · · , h 6 ( x , z ) � 0 } ˆ h 4 : = − z 2 h 1 : = z 1 − m 1 1 + q ( x ) h 2 : = M 1 − z 1 h 5 : = z 2 z 1 − p ( x ) h 3 : = z 2 1 − q ( x ) h 6 : = − z 2 z 1 + p ( x ) f ∗ sa : = inf ( x , z ) ∈ ˆ K z 2 and SOS yields λ 2 = − 0.618 < λ 3 = − 0.445. V. Magron New Applications of Moment-SOS Hierarchies 20 / 68
Introduction Moment-SOS relaxations Semialgebraic Maxplus Optimization Formal Nonlinear Optimization Approximating Pareto curves Polynomial image of semialgebraic sets Program Analysis with Polynomial Templates Conclusion
The General “Informal Framework” Given K a compact set and f a transcendental function, bound f ∗ = inf x ∈ K f ( x ) and prove f ∗ � 0 f is underestimated by a semialgebraic function f sa Reduce the problem f ∗ sa : = inf x ∈ K f sa ( x ) to a polynomial optimization problem (POP) V. Magron New Applications of Moment-SOS Hierarchies 21 / 68
Maxplus Approximation Initially introduced to solve Optimal Control Problems [Fleming-McEneaney 00] Curse of dimensionality reduction [McEaneney Kluberg, Gaubert-McEneaney-Qu 11, Qu 13]. Allowed to solve instances of dim up to 15 (inaccessible by grid methods) In our context: approximate transcendental functions V. Magron New Applications of Moment-SOS Hierarchies 22 / 68
Maxplus Approximation Definition Let γ � 0. A function φ : R n → R is said to be γ -semiconvex if the function x �→ φ ( x ) + γ 2 � x � 2 2 is convex. y par + a 2 par + a 1 arctan par − a 2 a a 1 a 2 m M par − a 1 V. Magron New Applications of Moment-SOS Hierarchies 22 / 68
Nonlinear Function Representation Exact parsimonious maxplus representations y a V. Magron New Applications of Moment-SOS Hierarchies 23 / 68
Nonlinear Function Representation Exact parsimonious maxplus representations y a V. Magron New Applications of Moment-SOS Hierarchies 23 / 68
Nonlinear Function Representation Abstract syntax tree representations of multivariate transcendental functions: leaves are semialgebraic functions of A nodes are univariate functions of D or binary operations V. Magron New Applications of Moment-SOS Hierarchies 23 / 68
Nonlinear Function Representation For the “Simple” Example from Flyspeck: + l ( x ) arctan r ( x ) V. Magron New Applications of Moment-SOS Hierarchies 23 / 68
Maxplus Optimization Algorithm First iteration: y + arctan par − l ( x ) arctan a 1 a a 1 m M r ( x ) 1 control point { a 1 } : m 1 = − 4.7 × 10 − 3 < 0 V. Magron New Applications of Moment-SOS Hierarchies 24 / 68
Maxplus Optimization Algorithm Second iteration: y + arctan par − l ( x ) arctan a 1 a a 2 a 1 m M r ( x ) par − a 2 2 control points { a 1 , a 2 } : m 2 = − 6.1 × 10 − 5 < 0 V. Magron New Applications of Moment-SOS Hierarchies 24 / 68
Maxplus Optimization Algorithm Third iteration: y + arctan par − par − a 3 l ( x ) arctan a 1 a a 2 a 3 a 1 m M r ( x ) par − a 2 3 control points { a 1 , a 2 , a 3 } : m 3 = 4.1 × 10 − 6 > 0 OK! V. Magron New Applications of Moment-SOS Hierarchies 24 / 68
Maxplus Optimization Algorithm Input: tree t , box K , SOS relaxation order k , precision p Output: bounds m and M , approximations t − 2 and t + 2 1: if t ∈ A then t − : = t , t + : = t 2: else if u : = root ( t ) ∈ D with child c then m c , M c , c − , c + : = samp _ approx ( c , K , k , p ) 3: I : = [ m c , M c ] 4: u − , u + : = unary _ approx ( u , I , c , p ) 5: t − , t + : = compose _ approx ( u , u − , u + , I , c − , c + ) 6: 7: else if bop : = root ( t ) with children c 1 and c 2 then m i , M i , c − i , c + i : = samp _ approx ( c i , K , k , p ) for i ∈ { 1, 2 } 8: t − , t + : = compose _ bop ( c − 1 , c + 1 , c − 2 , c + 2 , bop , [ m 2 , M 2 ]) 9: 10: end 11: return min _ sa ( t − , K , k ) , max _ sa ( t + , K , k ) , t − , t + V. Magron New Applications of Moment-SOS Hierarchies 25 / 68
Minimax Approximation / For Comparison The precision is an integer d The best-uniform degree- d polynomial approximation of u : h ∈ R d [ x ] � u − h � ∞ = min h ∈ R d [ x ] ( sup | u ( x ) − h ( x ) | ) min x ∈ I Implementation in Sollya [Chevillard-Joldes-Lauter 10] Interface of NLCertify with Sollya V. Magron New Applications of Moment-SOS Hierarchies 26 / 68
High-degree Polynomial Approximation + SOS i = 1 x i sin ( √ x i ) SWF : min x ∈ [ 1,500 ] n f ( x ) = − ∑ n replace sin ( √· ) by a degree- d Chebyshev polynomial Hard to combine with SOS V. Magron New Applications of Moment-SOS Hierarchies 27 / 68
High-degree Polynomial Approximation + SOS Indeed: Small d : lack of accuracy = ⇒ expensive Branch and Bound Large d : “No free lunch” rule with ( n + d n ) SDP variables V. Magron New Applications of Moment-SOS Hierarchies 27 / 68
High-degree Polynomial Approximation + SOS SWF with n = 10, d = 4: 38 min to compute a lower bound of − 430 n V. Magron New Applications of Moment-SOS Hierarchies 27 / 68
Comparison on Global Optimization Problems i = 1 x i sin ( √ x i ) x ∈ [ 1,500 ] n f ( x ) = − ∑ n min Interval Arithmetic for sin + f ∗ � − 418.9 n SOS n lower bound n lifting #boxes time − 430 n 10 0 3830 129 s − 430 n 10 2 n 16 40 s V. Magron New Applications of Moment-SOS Hierarchies 28 / 68
Comparison on Global Optimization Problems y par + n x i sin ( √ x i ) a 3 x ∈ [ 1,500 ] n f ( x ) = − ∑ min par + par + sin a 1 a 2 i = 1 √ a a 1 a 2 1 a 3 = 500 par − a 3 f ∗ � − 418.9 n par − par − a 2 a 1 n lower bound n lifting #boxes time − 430 n 10 0 3830 129 s − 430 n 10 2 n 16 40 s V. Magron New Applications of Moment-SOS Hierarchies 29 / 68
Comparison on Global Optimization Problems i = 1 x i sin ( √ x i ) x ∈ [ 1,500 ] n f ( x ) = − ∑ n min Interval Arithmetic for sin + f ∗ � − 418.9 n SOS n lower bound n lifting #boxes time − 440 n 100 0 > 10000 > 10 h − 440 n 100 2 n 274 1.9 h V. Magron New Applications of Moment-SOS Hierarchies 30 / 68
Comparison on Global Optimization Problems y par + n x i sin ( √ x i ) a 3 x ∈ [ 1,500 ] n f ( x ) = − ∑ min par + par + sin a 1 a 2 i = 1 √ a a 1 a 2 1 a 3 = 500 par − a 3 f ∗ � − 418.9 n par − par − a 2 a 1 n lower bound n lifting #boxes time − 440 n 100 0 > 10000 > 10 h − 440 n 100 2 n 274 1.9 h V. Magron New Applications of Moment-SOS Hierarchies 31 / 68
Comparison on Global Optimization Problems y par + a 3 par + par + sin a 1 n − 1 a 2 ( x i + x i + 1 ) sin ( √ x i ) a √ ∑ x ∈ [ 1,500 ] n f ( x ) = − min a 1 a 2 1 a 3 = 500 par − i = 1 a 3 par − par − a 1 a 2 n lower bound n lifting #boxes time − 967 n 1000 2 n 1 543 s − 968 n 1000 n 1 272 s V. Magron New Applications of Moment-SOS Hierarchies 32 / 68
Comparison on Global Optimization Problems y par + b 3 √ par + b �→ sin ( b ) b 1 n − 1 ( x i + x i + 1 ) sin ( √ x i ) b ∑ 1 b 1 b 2 b 3 = 500 x ∈ [ 1,500 ] n f ( x ) = − min par + par − b 2 b 3 i = 1 par − b 2 par − b 1 n lower bound n lifting #boxes time 1000 − 967 n 2 n 1 543 s 1000 − 968 n n 1 272 s V. Magron New Applications of Moment-SOS Hierarchies 33 / 68
Convergence of the Optimization Algorithm Let f be a multivariate transcendental function Let t − p be the underestimator of f , obtained at precision p Let x p opt be a minimizer of t − p over K Theorem [X. Allamigeon S. Gaubert VM B. Werner 13] Every accumulation point of the sequence ( x p opt ) is a global min- imizer of f on K . Ingredients of the proof: Convergence of Lasserre SOS hierarchy Uniform approximation schemes (Maxplus/Minimax) V. Magron New Applications of Moment-SOS Hierarchies 34 / 68
Polynomial Approximations for Semialgebraic Functions Inspired from [Lasserre - Thanh 13] Let f sa ∈ A defined on a box K ⊂ R n Let µ n be the standard Lebesgue measure on R n Best polynomial underestimator h ∈ R d [ x ] of f sa for the L 1 norm: � min K ( f sa − h ) d µ n ( P sa ) h ∈ R d [ x ] f sa − h � 0 on K . s.t. Lemma Problem ( P sa ) has a degree- d polynomial minimizer h d . V. Magron New Applications of Moment-SOS Hierarchies 35 / 68
Polynomial Approximations for Semialgebraic Functions K : = { ( x , z ) ∈ R n + p : g 1 ( x , z ) � 0, . . . , g m ( x , z ) � 0 } b.s.a.l. ˆ The quadratic module M ( ˆ K ) is Archimedean The optimal solution h d of ( P sa ) is a maximizer of: � max [ 0,1 ] n h d µ n ( P d ) h ∈ R d [ x ] ( z p − h ) ∈ M ( ˆ K ) . s.t. V. Magron New Applications of Moment-SOS Hierarchies 36 / 68
Polynomial Approximations for Semialgebraic Functions Let m d be the optimal value of Problem ( P sa ) Let h dk be a maximizer of the SOS relaxation of ( P d ) Convergence of the SOS Hierarchy The sequence ( � f sa − h dk � 1 ) k � k 0 is non-increasing and converges to m d . Each accumulation point of the sequence ( h dk ) k � k 0 is an optimal solution of Problem ( P sa ) . V. Magron New Applications of Moment-SOS Hierarchies 37 / 68
Polynomial Approximations for Semialgebraic Functions ∂ 4 ∆ x f sa ( x ) : = √ 4 x 1 ∆ x d k Upper bound of � f sa − h dk � 1 Bound 2 0.8024 -1.171 2 3 0.3709 -0.4479 2 1.617 -1.056 4 3 0.1766 -0.4493 V. Magron New Applications of Moment-SOS Hierarchies 37 / 68
Polynomial Approximations for Semialgebraic Functions rad 2 : ( x 1 , x 2 ) �→ − 64 x 2 1 + 128 x 1 x 2 + 1024 x 1 − 64 x 2 2 + 1024 x 2 − 4096 − 8 x 2 1 + 8 x 1 x 2 + 128 x 1 − 8 x 2 2 + 128 x 2 − 512 Linear and quadratic underestimators for rad 2 ( k = 3): V. Magron New Applications of Moment-SOS Hierarchies 38 / 68
Polynomial Approximations for Semialgebraic Functions rad 2 : ( x 1 , x 2 ) �→ − 64 x 2 1 + 128 x 1 x 2 + 1024 x 1 − 64 x 2 2 + 1024 x 2 − 4096 − 8 x 2 1 + 8 x 1 x 2 + 128 x 1 − 8 x 2 2 + 128 x 2 − 512 Linear and quadratic underestimators for rad 2 ( k = 3): 0.12 0.12 0.11 d = 2 d = 1 0.11 1 0 0.2 0.5 0.4 0.6 0.8 1 0 V. Magron New Applications of Moment-SOS Hierarchies 38 / 68
Contributions Published: X. Allamigeon, S. Gaubert, V. Magron, and B. Werner. Certification of inequalities involving transcendental functions: combining sdp and max-plus approximation, ECC Conference 2013. X. Allamigeon, S. Gaubert, V. Magron, and B. Werner. Certification of bounds of non-linear functions: the templates method, CICM Conference , 2013. In revision: X. Allamigeon, S. Gaubert, V. Magron, and B. Werner. Certification of Real Inequalities – Templates and Sums of Squares, arxiv:1403.5899, 2014. V. Magron New Applications of Moment-SOS Hierarchies 39 / 68
Introduction Moment-SOS relaxations Semialgebraic Maxplus Optimization Formal Nonlinear Optimization Approximating Pareto curves Polynomial image of semialgebraic sets Program Analysis with Polynomial Templates Conclusion
The General “Formal Framework” We check the correctness of SOS certificates for POP We build certificates to prove interval bounds for semialgebraic functions We bound formally transcendental functions with semialgebraic approximations V. Magron New Applications of Moment-SOS Hierarchies 40 / 68
Formal SOS bounds When q ∈ Q ( K ) , σ 0 , . . . , σ m is a positivity certificate for q Check symbolic polynomial equalities q = q ′ in C OQ Existing tactic ring [Grégoire-Mahboubi 05] Polynomials coefficients: arbitrary-size rationals bigQ [Grégoire-Théry 06] Much simpler to verify certificates using sceptical approach Extends also to semialgebraic functions V. Magron New Applications of Moment-SOS Hierarchies 41 / 68
Checking Polynomial Equalities: ring tactic Sparse Horner normal form Inductive PolC: Type := : bigQ → PolC | Pc | Pinj: positive → PolC → PolC : PolC → positive → PolC → PolC. | PX (Pc c) for constant polynomials (Pinj i p) shifts the index of i in the variables of p (PX p j q) evaluates to px j 1 + q ( x 2 , . . . , x n ) Encoding SOS certificates with Sparse Horner polynomials V. Magron New Applications of Moment-SOS Hierarchies 42 / 68
Bounding the Polynomial Remainder Normalized POP ( x ∈ [ 0, 1 ] n ) ǫ pop ( x ) : = p ( x ) − λ k − ∑ m j = 0 σ j ( x ) g j ( x ) ∀ x ∈ [ 0, 1 ] n , ǫ pop ( x ) � ǫ ∗ pop : = ∑ ǫ α ǫ α � 0 V. Magron New Applications of Moment-SOS Hierarchies 43 / 68
Formal SOS Results POP1 : ∀ x ∈ K , ∂ 4 ∆ x � − 41. POP2 : ∀ x ∈ K , ∆ x � 0. Problem n NLCertify micromega [Besson 07] POP1 6 0.08 s 9.00 s 0.09 s 0.36 s 2 POP2 3 0.39 s − 6 13.2 s − Sparse SOS relaxations = ⇒ Speedup V. Magron New Applications of Moment-SOS Hierarchies 44 / 68
Benchmarks for Flyspeck Inequalities Inequality #boxes Time Time 9922699028 39 190 s 2218 s 3318775219 338 1560 s 19136 s Comparable with Taylor interval methods in H OL - LIGHT [Hales-Solovyev 13] No free lunch: SDP informal bottleneck 22 times slower than SDP: q = q ′ formal bottleneck V. Magron New Applications of Moment-SOS Hierarchies 45 / 68
Contribution For more details on the formal side: X. Allamigeon, S. Gaubert, V. Magron and B. Werner. Formal Proofs for Nonlinear Optimization. In Revision, arxiv:1404.7282 V. Magron New Applications of Moment-SOS Hierarchies 46 / 68
Introduction Moment-SOS relaxations Semialgebraic Maxplus Optimization Formal Nonlinear Optimization Approximating Pareto curves Polynomial image of semialgebraic sets Program Analysis with Polynomial Templates Conclusion
Bicriteria Optimization Problems Let f 1 , f 2 ∈ R d [ x ] two conflicting criteria Let S : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } a semialgebraic set � � x ∈ S ( f 1 ( x ) f 2 ( x )) ⊤ ( P ) min Assumption The image space R 2 is partially ordered in a natural way ( R 2 + is the ordering cone). V. Magron New Applications of Moment-SOS Hierarchies 47 / 68
Bicriteria Optimization Problems 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 } . V. Magron New Applications of Moment-SOS Hierarchies 47 / 68
Parametric sublevel set approximation Inspired by previous research on multiobjective linear optimization [Gorissen-den Hertog 12] Workaround: reduce P to a parametric POP f ∗ ( λ ) : = min ( P λ ) : x ∈ S { f 2 ( x ) : f 1 ( x ) � λ } , V. Magron New Applications of Moment-SOS Hierarchies 48 / 68
A Hierarchy of Polynomial underestimators Moment-SOS approach [Lasserre 10]: 2 d ∑ q k / ( 1 + k ) max ( D d ) q ∈ R 2 d [ λ ] k = 0 s.t. f 2 ( x ) − q ( λ ) ∈ Q 2 d ( K ) . The hierarchy ( D d ) provides a sequence ( q d ) of polynomial underestimators of f ∗ ( λ ) . � 1 0 ( f ∗ ( λ ) − q d ( λ )) d λ = 0 lim d → ∞ V. Magron New Applications of Moment-SOS Hierarchies 49 / 68
A Hierarchy of Polynomial underestimators Degree 4 V. Magron New Applications of Moment-SOS Hierarchies 50 / 68
A Hierarchy of Polynomial underestimators Degree 6 V. Magron New Applications of Moment-SOS Hierarchies 50 / 68
A Hierarchy of Polynomial underestimators Degree 8 V. Magron New Applications of Moment-SOS Hierarchies 50 / 68
Contributions Numerical schemes that avoid computing finitely many points . Pareto curve approximation with polynomials, convergence guarantees in L 1 -norm V. Magron, D. Henrion, J.B. Lasserre. Approximating Pareto Curves using Semidefinite Relaxations. Operations Research Letters . arxiv:1404.4772, April 2014. V. Magron New Applications of Moment-SOS Hierarchies 51 / 68
Introduction Moment-SOS relaxations Semialgebraic Maxplus Optimization Formal Nonlinear Optimization Approximating Pareto curves Polynomial image of semialgebraic sets Program Analysis with Polynomial Templates Conclusion
Approximation of sets defined with “ ∃ ” Let B ⊂ R m be the unit ball and assume that F = f ( S ) ⊆ B . Another point of view: F = { y ∈ B : ∃ x ∈ S s.t. h ( x , y ) � 0 } , with h f ( x , y ) : = −� y − f ( x ) � 2 2 . Approximate F as closely as desired by a sequence of sets of the form : F k : = { y ∈ B : q k ( y ) � 0 } , for some polynomials q k ∈ R 2 k [ y ] . V. Magron New Applications of Moment-SOS Hierarchies 52 / 68
A hierarchy of outer approximations of f ( S ) Let K = S × B , g 0 : = 1 and Q k ( K ) be the k -truncated quadratic module generated by g 0 , . . . , g m : � m � ∑ Q k ( K ) = σ l ( x , y ) g l ( x ) , with σ l ∈ Σ k − v l [ x , y ] l = 0 Define h ( y ) : = sup x ∈ S h ( x , y ) Hierarchy of Semidefinite programs: � � � ρ k : = B ( q − h ) d y : q − h f ∈ Q k ( K ) min . q ∈ R 2 k [ y ] , σ l Yet another SOS program with an optimal solution q k ∈ R 2 k [ y ] ! V. Magron New Applications of Moment-SOS Hierarchies 53 / 68
A hierarchy of outer approximations of f ( S ) From the definition of q k , the sublevel sets F k : = { y ∈ B : q k ( y ) � 0 } ⊇ F , provide a sequence of certified outer approximations of F . It comes from the following: ∀ ( x , y ) ∈ K , q k ( y ) � h f ( x , y ) ⇐ ⇒ ∀ y , q k ( y ) � h ( y ) . V. Magron New Applications of Moment-SOS Hierarchies 54 / 68
Strong convergence property Theorem 1 The sequence of underestimators ( q k ) k � k 0 converges to h w.r.t the L 1 ( B ) -norm: � lim B | q k − h | d y = 0 . k → ∞ V. Magron New Applications of Moment-SOS Hierarchies 55 / 68
Strong convergence property Theorem 1 The sequence of underestimators ( q k ) k � k 0 converges to h w.r.t the L 1 ( B ) -norm: � lim B | q k − h | d y = 0 . k → ∞ 2 k → ∞ vol ( F k \ F ) = 0 . lim V. Magron New Applications of Moment-SOS Hierarchies 55 / 68
Approximation for polynomial image of semialgebraic sets Image of the unit ball S : = { x ∈ R 2 : � x � 2 2 � 1 } by f ( x ) : = ( x 1 + x 1 x 2 , x 2 − x 3 1 ) /2 F 1 V. Magron New Applications of Moment-SOS Hierarchies 56 / 68
Approximation for polynomial image of semialgebraic sets Image of the unit ball S : = { x ∈ R 2 : � x � 2 2 � 1 } by f ( x ) : = ( x 1 + x 1 x 2 , x 2 − x 3 1 ) /2 F 2 V. Magron New Applications of Moment-SOS Hierarchies 56 / 68
Approximation for polynomial image of semialgebraic sets Image of the unit ball S : = { x ∈ R 2 : � x � 2 2 � 1 } by f ( x ) : = ( x 1 + x 1 x 2 , x 2 − x 3 1 ) /2 F 3 V. Magron New Applications of Moment-SOS Hierarchies 56 / 68
Approximation for polynomial image of semialgebraic sets Image of the unit ball S : = { x ∈ R 2 : � x � 2 2 � 1 } by f ( x ) : = ( x 1 + x 1 x 2 , x 2 − x 3 1 ) /2 F 4 V. Magron New Applications of Moment-SOS Hierarchies 56 / 68
Semialgebraic set projections f ( x ) = ( x 1 , x 2 ) : projection on R 2 of the semialgebraic set 2 � 1, 1/4 − ( x 1 + 1/2 ) 2 − x 2 S : = { x ∈ R 3 : � x � 2 2 � 0, 1/9 − ( x 1 − 1/2 ) 4 − x 4 2 � 0 } F 2 V. Magron New Applications of Moment-SOS Hierarchies 57 / 68
Semialgebraic set projections f ( x ) = ( x 1 , x 2 ) : projection on R 2 of the semialgebraic set 2 � 1, 1/4 − ( x 1 + 1/2 ) 2 − x 2 S : = { x ∈ R 3 : � x � 2 2 � 0, 1/9 − ( x 1 − 1/2 ) 4 − x 4 2 � 0 } F 3 V. Magron New Applications of Moment-SOS Hierarchies 57 / 68
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