A “Simple” Example In the computational part: Semialgebraic functions: composition of polynomials with | · | , √ , + , − , × , /, sup, inf, . . . p ( x ) : = ∂ 4 ∆ x q ( x ) : = 4 x 1 ∆ x � r ( x ) : = p ( x ) / q ( x ) 2 + 1.6294 − 0.2213 ( √ x 2 + √ x 3 + √ x 5 + √ x 6 − 8.0 ) + l ( x ) : = − π 0.913 ( √ x 4 − 2.52 ) + 0.728 ( √ x 1 − 2.0 ) Victor Magron Certified Optimization for System Verification 11 / 45
A “Simple” Example In the computational part: Transcendental functions T : composition of semialgebraic functions with arctan, exp, sin, + , − , × , . . . Victor Magron Certified Optimization for System Verification 11 / 45
A “Simple” Example In the computational part: Feasible set K : = [ 4, 6.3504 ] 3 × [ 6.3504, 8 ] × [ 4, 6.3504 ] 2 Lemma 9922699028 from Flyspeck: � p ( x ) � ∀ x ∈ K , arctan + l ( x ) � 0 � q ( x ) Victor Magron Certified Optimization for System Verification 11 / 45
Existing Formal Frameworks Formal proofs for Global Optimization: Bernstein polynomial methods [Zumkeller’s PhD 08] SMT methods [Gao et al. 12] Interval analysis and Sums of squares Victor Magron Certified Optimization for System Verification 12 / 45
Existing Formal Frameworks Interval analysis Certified interval arithmetic in C OQ [Melquiond 12] Taylor methods in HOL Light [Solovyev thesis 13] Formal verification of floating-point operations robust but subject to the Curse of Dimensionality Victor Magron Certified Optimization for System Verification 12 / 45
Existing Formal Frameworks Lemma 9922699028 from Flyspeck: � ∂ 4 ∆ x � √ 4 x 1 ∆ x ∀ x ∈ K , arctan + l ( x ) � 0 Dependency issue using Interval Calculus: One can bound ∂ 4 ∆ x / √ 4 x 1 ∆ x and l ( x ) separately Too coarse lower bound: − 0.87 Subdivide K to prove the inequality K 3 K K 0 K 1 K 4 = ⇒ K 2 Victor Magron Certified Optimization for System Verification 12 / 45
Existing Formal Frameworks Sums of squares (SOS) techniques Formalized in H OL - LIGHT [Harrison 07] C OQ [Besson 07] Precise methods but scalability and robustness issues (numerical) powerful: global optimality certificates without branching but not so robust: handles moderate size problems Restricted to polynomials Victor Magron Certified Optimization for System Verification 12 / 45
Existing Formal Frameworks 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. Decompose the polynomial as SOS of degree at most 4 Gives a nonnegative bound! Victor Magron Certified Optimization for System Verification 12 / 45
Existing Formal Frameworks Approximation theory: Chebyshev/Taylor models mandatory for non-polynomial problems hard to combine with SOS techniques (degree of approximation) Victor Magron Certified Optimization for System Verification 12 / 45
Existing Formal Frameworks Can we develop a new approach with both keeping the respective strength of interval and precision of SOS? Proving Flyspeck Inequalities is challenging: medium-size and tight Victor Magron Certified Optimization for System Verification 12 / 45
Contribution: Publications and Software M., Allamigeon, Gaubert, Werner. Formal Proofs for Nonlinear Optimization, Journal of Formalized Reasoning 8(1):1–24, 2015 . Hales, Adams, Bauer, Dang, Harrison, Hoang, Kaliszyk, M., Mclaughlin, Nguyen, Nguyen, Nipkow, Obua, Pleso, Rute, Solovyev, Ta, Tran, Trieu, Urban, Vu & Zumkeller, Forum of Mathematics, Pi , 5 2017 Software Implementation NLCertify : 15 000 lines of OC AML code 4000 lines of C OQ code M. NLCertify: A Tool for Formal Nonlinear Optimization, ICMS , 2014. Victor Magron Certified Optimization for System Verification 12 / 45
SDP for Nonlinear Optimization SDP for Characterizing Values/Curves/Sets Semialgebraic Maxplus Optimization Roundoff Error Bounds Pareto Curves Polynomial Images of Semialgebraic Sets Reachable Sets of Polynomial Systems Invariant Measures of Polynomial Systems Exact Polynomial Optimization Conclusion
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 under-approximated by a semialgebraic function f sa Reduce the problem f ∗ sa : = inf x ∈ K f sa ( x ) to a polynomial optimization problem (POP) Victor Magron Certified Optimization for System Verification 13 / 45
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 Victor Magron Certified Optimization for System Verification 14 / 45
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 Victor Magron Certified Optimization for System Verification 14 / 45
Nonlinear Function Representation Exact parsimonious maxplus representations y a Victor Magron Certified Optimization for System Verification 15 / 45
Nonlinear Function Representation Exact parsimonious maxplus representations y a Victor Magron Certified Optimization for System Verification 15 / 45
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 Victor Magron Certified Optimization for System Verification 15 / 45
Nonlinear Function Representation For the “Simple” Example from Flyspeck: + l ( x ) arctan r ( x ) Victor Magron Certified Optimization for System Verification 15 / 45
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 Victor Magron Certified Optimization for System Verification 16 / 45
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 Victor Magron Certified Optimization for System Verification 16 / 45
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! Victor Magron Certified Optimization for System Verification 16 / 45
SDP for Nonlinear Optimization SDP for Characterizing Values/Curves/Sets Semialgebraic Maxplus Optimization Roundoff Error Bounds Pareto Curves Polynomial Images of Semialgebraic Sets Reachable Sets of Polynomial Systems Invariant Measures of Polynomial Systems Exact Polynomial Optimization Conclusion
Roundoff Error Bounds Exact: f ( x ) : = x 1 x 2 + x 3 x 4 Floating-point: ˆ f ( x , e ) : = [ x 1 x 2 ( 1 + e 1 ) + x 3 x 4 ( 1 + e 2 )]( 1 + e 3 ) | e i | � 2 − p x ∈ X , p = 24 (single) or 53 (double) Victor Magron Certified Optimization for System Verification 17 / 45
Roundoff Error Bounds Input: exact f ( x ) , floating-point ˆ f ( x , e ) Output: Bounds for f − ˆ f f ( x , e ) = ∑ 1: Error r ( x , e ) : = f ( x ) − ˆ r α ( e ) x α α 2: Decompose r ( x , e ) = l ( x , e ) + h ( x , e ) , l linear in e 3: Bound h ( x , e ) with interval arithmetic 4: Bound l ( x , e ) with S PARSE S UMS OF S QUARES Victor Magron Certified Optimization for System Verification 17 / 45
Roundoff Error Bounds Sparse SDP Optimization [Waki, Lasserre 06] Correlative sparsity pattern (csp) of vars x 2 x 5 + x 3 x 6 − x 2 x 3 − x 5 x 6 + x 1 ( − x 1 + x 2 + x 3 − x 4 + x 5 + x 6 ) 6 5 1 4 3 2 Victor Magron Certified Optimization for System Verification 17 / 45
Roundoff Error Bounds Sparse SDP Optimization [Waki, Lasserre 06] Correlative sparsity pattern (csp) of vars x 2 x 5 + x 3 x 6 − x 2 x 3 − x 5 x 6 + x 1 ( − x 1 + x 2 + x 3 − x 4 + x 5 + x 6 ) 6 5 1 4 3 2 C 1 : = { 1, 4 } C 2 : = { 1, 2, 3, 5 } 1 Maximal cliques C 1 , . . . , C l C 3 : = { 1, 3, 5, 6 } 2 Average size κ ❀ ( κ + 2 k κ ) vars Dense SDP: 210 vars Sparse SDP: 115 vars Victor Magron Certified Optimization for System Verification 17 / 45
Contributions l ( x , e ) = ∑ m i = 1 s i ( x ) e i Maximal cliques correspond to { x , e 1 } , . . . , { x , e m } M., Constantinides, Donaldson. Certified Roundoff Error Bounds Using Semidefinite Programming, Trans. Math. Soft. , 2016 Victor Magron Certified Optimization for System Verification 17 / 45
SDP for Nonlinear Optimization SDP for Characterizing Values/Curves/Sets Semialgebraic Maxplus Optimization Roundoff Error Bounds Pareto Curves Polynomial Images of Semialgebraic Sets Reachable Sets of Polynomial Systems Invariant Measures of Polynomial Systems Exact Polynomial Optimization Conclusion
Bicriteria Optimization Problems Let f 1 , f 2 ∈ R [ 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). Victor Magron Certified Optimization for System Verification 18 / 45
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 } . Victor Magron Certified Optimization for System Verification 18 / 45
Parametric Sublevel Set Approximations 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 } , variable ( x , λ ) ∈ K = S × [ 0, 1 ] Victor Magron Certified Optimization for System Verification 19 / 45
A Hierarchy of Polynomial Approximations Moment-SOS approach [Lasserre 10]: 2 k ∑ q i / ( 1 + i ) max ( D k ) q ∈ R 2 k [ λ ] i = 0 s.t. f 2 ( x ) − q ( λ ) ∈ Q 2 k ( K ) . The hierarchy ( D k ) provides a sequence ( q k ) of polynomial under-approximations of f ∗ ( λ ) . � 1 0 ( f ∗ ( λ ) − q k ( λ )) d λ = 0 lim d → ∞ Victor Magron Certified Optimization for System Verification 20 / 45
A Hierarchy of Polynomial Approximations Degree 4 Victor Magron Certified Optimization for System Verification 21 / 45
A Hierarchy of Polynomial Approximations Degree 6 Victor Magron Certified Optimization for System Verification 21 / 45
A Hierarchy of Polynomial Approximations Degree 8 Victor Magron Certified Optimization for System Verification 21 / 45
Contributions Numerical schemes that avoid computing finitely many points . Pareto curve approximation with polynomials, convergence guarantees in L 1 -norm M., Henrion, Lasserre. Approximating Pareto Curves using Semidefinite Relaxations. Operations Research Letters , 2014. Victor Magron Certified Optimization for System Verification 22 / 45
SDP for Nonlinear Optimization SDP for Characterizing Values/Curves/Sets Semialgebraic Maxplus Optimization Roundoff Error Bounds Pareto Curves Polynomial Images of Semialgebraic Sets Reachable Sets of Polynomial Systems Invariant Measures of Polynomial Systems Exact Polynomial Optimization Conclusion
Polynomial Images of Semialgebraic Sets 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 Certified Optimization for System Verification 23 / 45
Polynomial Images of Semialgebraic Sets 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 Certified Optimization for System Verification 23 / 45
Support of Image Measures Pushforward f # : M ( S ) → M ( B ) : f # µ 0 ( A ) : = µ 0 ( { x ∈ S : f ( x ) ∈ A } ) , ∀ A ∈ B ( B ) , ∀ µ 0 ∈ M ( S ) f # µ 0 is the image measure of µ 0 under f Victor Magron Certified Optimization for System Verification 24 / 45
Support of Image Measures � p ∗ : = B µ 1 sup µ 0 , µ 1 , ˆ µ 1 µ 1 + ˆ µ 1 = λ B , s.t. µ 1 = f # µ 0 , µ 0 ∈ M + ( S ) , µ 1 , ˆ µ 1 ∈ M + ( B ) . Lebesgue measure on B is λ B ( d y ) : = 1 B ( y ) d y Victor Magron Certified Optimization for System Verification 24 / 45
Support of Image Measures � p ∗ : = sup B µ 1 µ 0 , µ 1 , ˆ µ 1 µ 1 + ˆ µ 1 = λ B , s.t. µ 1 = f # µ 0 , µ 0 ∈ M + ( S ) , µ 1 ∈ M + ( B ) . µ 1 , ˆ Lemma Let µ ∗ 1 be an optimal solution of the above LP. 1 = λ F and p ∗ = vol F . Then µ ∗ Victor Magron Certified Optimization for System Verification 24 / 45
Method 2: Primal-dual LP Formulation Primal LP Dual LP � � p ∗ : = d ∗ : = inf sup µ 1 w ( y ) λ B ( d y ) v , w µ 0 , µ 1 , ˆ µ 1 v ( f ( x )) � 0, ∀ x ∈ S , s.t. µ 1 + ˆ µ 1 = λ B , s.t. w ( y ) � 1 + v ( y ) , ∀ y ∈ B , µ 1 = f # µ 0 , w ( y ) � 0, ∀ y ∈ B , µ 0 ∈ M + ( S ) , v , w ∈ C ( B ) . µ 1 ∈ M + ( B ) . µ 1 , ˆ Victor Magron Certified Optimization for System Verification 25 / 45
Method 2: Strong Convergence Property Strengthening of the dual LP: d ∗ ∑ w β z B k : = inf β v , w β ∈ N m 2 k s.t. v ◦ f ∈ Q kd ( S ) , w − 1 − v ∈ Q k ( B ) , w ∈ Q k ( B ) , v , w ∈ R 2 k [ y ] . Victor Magron Certified Optimization for System Verification 26 / 45
Method 2: Strong Convergence Property Theorem ◦ F � = ∅ and Q k ( S ) is Archimedean, Assuming that 1 The sequence ( w k ) converges to 1 F w.r.t the L 1 ( B ) -norm: � lim B | w k − 1 F | d y = 0 . k → ∞ Victor Magron Certified Optimization for System Verification 27 / 45
Method 2: Strong Convergence Property Theorem ◦ F � = ∅ and Q k ( S ) is Archimedean, Assuming that 1 The sequence ( w k ) converges to 1 F w.r.t the L 1 ( B ) -norm: � lim B | w k − 1 F | d y = 0 . k → ∞ 2 Let F k : = { y ∈ B : w k ( y ) � 1 } . Then, k → ∞ vol ( F k \ F ) = 0 . lim Victor Magron Certified Optimization for System Verification 27 / 45
Polynomial Image of the Unit Ball 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 Victor Magron Certified Optimization for System Verification 28 / 45
Polynomial Image of the Unit Ball 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 Victor Magron Certified Optimization for System Verification 28 / 45
Polynomial Image of the Unit Ball 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 Victor Magron Certified Optimization for System Verification 28 / 45
Polynomial Image of the Unit Ball 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 Victor Magron Certified Optimization for System Verification 28 / 45
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 Victor Magron Certified Optimization for System Verification 29 / 45
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 Victor Magron Certified Optimization for System Verification 29 / 45
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 4 Victor Magron Certified Optimization for System Verification 29 / 45
Approximating Pareto Curves Back on our previous nonconvex example: F 1 Victor Magron Certified Optimization for System Verification 30 / 45
Approximating Pareto Curves Back on our previous nonconvex example: F 2 Victor Magron Certified Optimization for System Verification 30 / 45
Approximating Pareto Curves Back on our previous nonconvex example: F 3 Victor Magron Certified Optimization for System Verification 30 / 45
Approximating Pareto Curves “Zoom” on the region which is hard to approximate: F 4 Victor Magron Certified Optimization for System Verification 31 / 45
Approximating Pareto Curves “Zoom” on the region which is hard to approximate: F 5 Victor Magron Certified Optimization for System Verification 31 / 45
Semialgebraic Image of Semialgebraic Sets Image of the unit ball S : = { x ∈ R 2 : � x � 2 2 � 1 } by f ( x ) : = ( min ( x 1 + x 1 x 2 , x 2 1 ) , x 2 − x 3 1 ) /3 F 1 Victor Magron Certified Optimization for System Verification 32 / 45
Semialgebraic Image of Semialgebraic Sets Image of the unit ball S : = { x ∈ R 2 : � x � 2 2 � 1 } by f ( x ) : = ( min ( x 1 + x 1 x 2 , x 2 1 ) , x 2 − x 3 1 ) /3 F 2 Victor Magron Certified Optimization for System Verification 32 / 45
Semialgebraic Image of Semialgebraic Sets Image of the unit ball S : = { x ∈ R 2 : � x � 2 2 � 1 } by f ( x ) : = ( min ( x 1 + x 1 x 2 , x 2 1 ) , x 2 − x 3 1 ) /3 F 3 Victor Magron Certified Optimization for System Verification 32 / 45
Semialgebraic Image of Semialgebraic Sets Image of the unit ball S : = { x ∈ R 2 : � x � 2 2 � 1 } by f ( x ) : = ( min ( x 1 + x 1 x 2 , x 2 1 ) , x 2 − x 3 1 ) /3 F 4 Victor Magron Certified Optimization for System Verification 32 / 45
Contributions M., Henrion, Lasserre. Semidefinite approximations of projections and polynomial images of semialgebraic sets. SIAM Opt. , 2015. Victor Magron Certified Optimization for System Verification 33 / 45
Reachable Sets of Polynomial Systems Iterations x t + 1 = f ( x t ) Uncertain x t + 1 = f ( x t , u ) Converging SDP hierarchies Image measure Liouville equation (conservation) µ t + µ = f # µ + µ 0 Victor Magron Certified Optimization for System Verification 34 / 45
Reachable Sets of Polynomial Systems Iterations x t + 1 = f ( x t ) Uncertain x t + 1 = f ( x t , u ) Converging SDP hierarchies Image measure Liouville equation (conservation) µ t + µ = f # µ + µ 0 M., Garoche, Henrion, Thirioux. Semidefinite Approximations of Reachable Sets for Discrete-time Polynomial Systems, 2017. Victor Magron Certified Optimization for System Verification 34 / 45
Invariant Measures of Polynomial Systems x t + 1 = f ( x t ) = ⇒ f # µ − µ = 0 Discrete x = f ( x ) = ⇒ div f µ = 0 Continuous ˙ Converging SDP hierarchies measures with density in L p singular measures = ⇒ chaotic attractors Victor Magron Certified Optimization for System Verification 35 / 45
Invariant Measures of Polynomial Systems x t + 1 = f ( x t ) = ⇒ f # µ − µ = 0 Discrete x = f ( x ) = ⇒ div f µ = 0 Continuous ˙ Converging SDP hierarchies measures with density in L p singular measures = ⇒ chaotic attractors M., Forets, Henrion. Semidefinite Characterization of Invariant Measures for Polynomial Systems. In Progress , 2018. Victor Magron Certified Optimization for System Verification 35 / 45
SDP for Nonlinear Optimization SDP for Characterizing Values/Curves/Sets Exact Polynomial Optimization Conclusion
Exact Polynomial Optimization [Lasserre/Parrilo 01] Numerical solvers compute σ i Semidefinite programming (SDP) � approximate certificates f = 4 X 4 1 + 4 X 3 1 X 2 − 7 X 2 1 X 2 2 − 2 X 1 X 3 2 + 10 X 4 2 2 ) 2 + ( 4 2 ) 2 + ( 2 1 + X 1 X 2 − 8 3 X 1 X 2 + 3 f ≃ σ = ( 2 X 2 3 X 2 2 X 2 7 X 2 2 ) 2 Victor Magron Certified Optimization for System Verification 36 / 45
Exact Polynomial Optimization [Lasserre/Parrilo 01] Numerical solvers compute σ i Semidefinite programming (SDP) � approximate certificates f = 4 X 4 1 + 4 X 3 1 X 2 − 7 X 2 1 X 2 2 − 2 X 1 X 3 2 + 10 X 4 2 2 ) 2 + ( 4 2 ) 2 + ( 2 1 + X 1 X 2 − 8 3 X 1 X 2 + 3 f ≃ σ = ( 2 X 2 3 X 2 2 X 2 7 X 2 2 ) 2 f = σ + 8 2 − 2 2 + 983 9 X 2 1 X 2 3 X 1 X 3 1764 X 4 2 Victor Magron Certified Optimization for System Verification 36 / 45
Exact Polynomial Optimization [Lasserre/Parrilo 01] Numerical solvers compute σ i Semidefinite programming (SDP) � approximate certificates f = 4 X 4 1 + 4 X 3 1 X 2 − 7 X 2 1 X 2 2 − 2 X 1 X 3 2 + 10 X 4 2 2 ) 2 + ( 4 2 ) 2 + ( 2 1 + X 1 X 2 − 8 3 X 1 X 2 + 3 f ≃ σ = ( 2 X 2 3 X 2 2 X 2 7 X 2 2 ) 2 f = σ + 8 2 − 2 2 + 983 9 X 2 1 X 2 3 X 1 X 3 1764 X 4 2 → ≃ = The Question of Exact Certification How to go from approximate to exact certification? Victor Magron Certified Optimization for System Verification 36 / 45
One Answer when K = R n Hybrid S YMBOLIC /N UMERIC methods [Peyrl-Parrilo 08] [Kaltofen et. al 08] f ( X ) ≃ v DT ( X ) ˜ Q v D ( X ) 0 � ˜ Q ∈ R D × D v D ( X ) = ( 1, X 1 , . . . , X n , X 2 1 , . . . , X D n ) Victor Magron Certified Optimization for System Verification 37 / 45
One Answer when K = R n Hybrid S YMBOLIC /N UMERIC methods [Peyrl-Parrilo 08] [Kaltofen et. al 08] f ( X ) ≃ v DT ( X ) ˜ Q v D ( X ) 0 � ˜ Q ∈ R D × D v D ( X ) = ( 1, X 1 , . . . , X n , X 2 1 , . . . , X D n ) → ≃ = ˜ Q Rounding Q Projection ∏ ( Q ) f ( X ) = v DT ( X ) ∏ ( Q ) v D ( X ) ∏ ( Q ) � 0 when ε → 0 Victor Magron Certified Optimization for System Verification 37 / 45
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