Enclosures of Roundoff Errors using SDP Victor Magron , CNRS Jointly Certified Upper Bounds with G. Constantinides and A. Donaldson 18th French-German-Italian Conference on Optimization September 2017 New Hierarchies of SDP Relaxations for Polynomial Systems Victor Magron Enclosures of Roundoff Errors using SDP 0 / 18
Errors and Proofs G UARANTEED O PTIMIZATION Input : Linear problem (LP), geometric, semidefinite (SDP) Output : solution + certificate numeric-symbolic ❀ formal Victor Magron Enclosures of Roundoff Errors using SDP 1 / 18
Errors and Proofs G UARANTEED O PTIMIZATION Input : Linear problem (LP), geometric, semidefinite (SDP) Output : solution + certificate numeric-symbolic ❀ formal V ERIFICATION OF CRITICAL SYSTEMS Reliable software/hardware embedded codes Aerospace control molecular biology, robotics, code synthesis, . . . Victor Magron Enclosures of Roundoff Errors using SDP 1 / 18
Errors and Proofs G UARANTEED O PTIMIZATION Input : Linear problem (LP), geometric, semidefinite (SDP) Output : solution + certificate numeric-symbolic ❀ formal V ERIFICATION OF CRITICAL SYSTEMS Reliable software/hardware embedded codes Aerospace control molecular biology, robotics, code synthesis, . . . Efficient Verification of Nonlinear Systems Automated precision tuning of systems/programs analysis/synthesis Efficiency sparsity correlation patterns Certified approximation algorithms Victor Magron Enclosures of Roundoff Errors using SDP 1 / 18
Roundoff Error Bounds Real : f ( x ) : = x 1 × x 2 + x 3 Floating-point : ˆ f ( x , e ) : = [ x 1 x 2 ( 1 + e 1 ) + x 3 ]( 1 + e 2 ) Input variable constraints x ∈ X Finite precision ❀ bounds over e ∈ E : | e i | � 2 − 53 (double) Guarantees on absolute round-off error | ˆ f − f | ? ↓ Upper Bounds ↓ max ˆ max ˆ f − f f − f ↑ Lower Bounds ↑ ↓ Lower Bounds ↓ min ˆ min ˆ f − f f − f ↑ Upper Bounds ↑ Victor Magron Enclosures of Roundoff Errors using SDP 2 / 18
Nonlinear Programs Polynomials programs : + , − , × x 2 x 5 + x 3 x 6 + x 1 ( − x 1 + x 2 + x 3 − x 4 + x 5 + x 6 ) Victor Magron Enclosures of Roundoff Errors using SDP 3 / 18
Nonlinear Programs Polynomials programs : + , − , × x 2 x 5 + x 3 x 6 + x 1 ( − x 1 + x 2 + x 3 − x 4 + x 5 + x 6 ) Semialgebraic programs: | · | , √ , /, sup, inf 4 x x 1 + 1.11 Victor Magron Enclosures of Roundoff Errors using SDP 3 / 18
Nonlinear Programs Polynomials programs : + , − , × x 2 x 5 + x 3 x 6 + x 1 ( − x 1 + x 2 + x 3 − x 4 + x 5 + x 6 ) Semialgebraic programs: | · | , √ , /, sup, inf 4 x x 1 + 1.11 Transcendental programs: arctan, exp, log, . . . log ( 1 + exp ( x )) Victor Magron Enclosures of Roundoff Errors using SDP 3 / 18
Existing Frameworks Classical methods : Abstract domains [Goubault-Putot 11] F LUCTUAT : intervals, octagons, zonotopes Interval arithmetic [Daumas-Melquiond 10] G APPA : interface with C OQ proof assistant Victor Magron Enclosures of Roundoff Errors using SDP 4 / 18
Existing Frameworks Recent progress : Affine arithmetic + SMT [Darulova 14], [Darulova 17] rosa : sound compiler for reals (S CALA ) Symbolic Taylor expansions [Solovyev 15], [Chiang 17] FPTaylor : certified optimization (OC AML /H OL - LIGHT ) Guided random testing s3fp [Chiang 14] Victor Magron Enclosures of Roundoff Errors using SDP 4 / 18
Contributions Maximal Roundoff error of the program implementation of f : r ⋆ : = max | ˆ f ( x , e ) − f ( x ) | Decomposition: linear term l w.r.t. e + nonlinear term h max | l | + max | h | � r ⋆ � max | l | − max | h | Coarse bound of h with interval arithmetic Semidefinite programming (SDP) bounds for l : Victor Magron Enclosures of Roundoff Errors using SDP 5 / 18
Contributions Maximal Roundoff error of the program implementation of f : r ⋆ : = max | ˆ f ( x , e ) − f ( x ) | Decomposition: linear term l w.r.t. e + nonlinear term h max | l | + max | h | � r ⋆ � max | l | − max | h | Coarse bound of h with interval arithmetic Semidefinite programming (SDP) bounds for l : ↓ Upper Bounds ↓ ↑ Lower Bounds ↑ ↓ Lower Bounds ↓ ↑ Upper Bounds ↑ Sparse SDP relaxations Robust SDP relaxations Victor Magron Enclosures of Roundoff Errors using SDP 5 / 18
Contributions 1 General SDP framework for upper and lower bounds 2 Comparison with SMT & affine/Taylor arithmetic: ❀ Efficient optimization � Tight upper bounds + 3 Extensions to transcendental/conditional programs 4 Formal verification of SDP bounds 5 Open source tools: ↓ Upper Bounds ↓ Real2Float (in OC AML and C OQ ) ↑ Upper Bounds ↑ ↑ Lower Bounds ↑ ↓ Lower Bounds ↓ FPSDP (in M ATLAB ) Victor Magron Enclosures of Roundoff Errors using SDP 5 / 18
Introduction Semidefinite Programming for Polynomial Optimization Upper Bounds with Sparse SDP Lower Bounds with Robust SDP 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 Enclosures of Roundoff Errors using SDP 6 / 18
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 Enclosures of Roundoff Errors using SDP 6 / 18
What is Semidefinite Programming? Semidefinite Programming (SDP): ⊤ z min c z ∑ A z = d . 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 Enclosures of Roundoff Errors using SDP 6 / 18
SDP for Polynomial Optimization Prove polynomial inequalities with SDP: f ( a , b ) : = a 2 − 2 ab + b 2 � 0 . � � � � � � z 1 z 2 a Find z s.t. f ( a , b ) = a b . z 2 z 3 b � �� � � 0 Find z s.t. a 2 − 2 ab + b 2 = z 1 a 2 + 2 z 2 ab + z 3 b 2 ( A z = d ) � z 1 � � 1 � � 0 � � 0 � � 0 � z 2 0 1 0 0 = z 1 + z 2 + z 3 � z 2 z 3 0 0 1 0 0 1 0 0 � �� � � �� � � �� � � �� � F 1 F 2 F 3 F 0 Victor Magron Enclosures of Roundoff Errors using SDP 7 / 18
SDP for Polynomial Optimization Choose a cost c e.g. ( 1, 0, 1 ) and solve: ⊤ z min c z ∑ s.t. F i z i � F 0 , A z = d . i � 1 � z 1 � � z 2 − 1 Solution = � 0 (eigenvalues 0 and 2) − 1 z 2 z 3 1 � � 1 � � a � − 1 a 2 − 2 ab + b 2 = � = ( a − b ) 2 . a b − 1 1 b � �� � � 0 Solving SDP = ⇒ Finding S UMS OF S QUARES certificates Victor Magron Enclosures of Roundoff Errors using SDP 8 / 18
SDP for Polynomial Optimization NP hard General Problem : f ∗ : = min x ∈ X f ( x ) Semialgebraic set X : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } Victor Magron Enclosures of Roundoff Errors using SDP 9 / 18
SDP for Polynomial Optimization NP hard General Problem : f ∗ : = min x ∈ X f ( x ) Semialgebraic set X : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } � : = [ 0, 1 ] 2 = { x ∈ R 2 : x 1 ( 1 − x 1 ) � 0, x 2 ( 1 − x 2 ) � 0 } Victor Magron Enclosures of Roundoff Errors using SDP 9 / 18
SDP for Polynomial Optimization NP hard General Problem : f ∗ : = min x ∈ X f ( x ) Semialgebraic set X : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } � : = [ 0, 1 ] 2 = { x ∈ R 2 : x 1 ( 1 − x 1 ) � 0, x 2 ( 1 − x 2 ) � 0 } σ 0 σ 1 σ 2 � �� � g 1 g 2 f ���� ���� � � 2 x 1 x 2 + 1 1 x 1 + x 2 − 1 1 � �� � 1 � �� � ���� 8 = + x 1 ( 1 − x 1 ) + x 2 ( 1 − x 2 ) 2 2 2 2 Victor Magron Enclosures of Roundoff Errors using SDP 9 / 18
SDP for Polynomial Optimization NP hard General Problem : f ∗ : = min x ∈ X f ( x ) Semialgebraic set X : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } � : = [ 0, 1 ] 2 = { x ∈ R 2 : x 1 ( 1 − x 1 ) � 0, x 2 ( 1 − x 2 ) � 0 } σ 0 σ 1 σ 2 � �� � g 1 g 2 f ���� ���� � � 2 x 1 x 2 + 1 1 x 1 + x 2 − 1 1 � �� � 1 � �� � ���� 8 = + x 1 ( 1 − x 1 ) + x 2 ( 1 − x 2 ) 2 2 2 2 Sums of squares (SOS) σ i Victor Magron Enclosures of Roundoff Errors using SDP 9 / 18
SDP for Polynomial Optimization NP hard General Problem : f ∗ : = min x ∈ X f ( x ) Semialgebraic set X : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } � : = [ 0, 1 ] 2 = { x ∈ R 2 : x 1 ( 1 − x 1 ) � 0, x 2 ( 1 − x 2 ) � 0 } σ 0 σ 1 σ 2 � �� � g 1 g 2 f ���� ���� � � 2 x 1 x 2 + 1 1 x 1 + x 2 − 1 1 � �� � 1 � �� � ���� 8 = + x 1 ( 1 − x 1 ) + x 2 ( 1 − x 2 ) 2 2 2 2 Sums of squares (SOS) σ i Bounded degree: � � σ 0 + ∑ m Q k ( X ) : = j = 1 σ j g j , with deg σ j g j � 2 k Victor Magron Enclosures of Roundoff Errors using SDP 9 / 18
SDP for Polynomial Optimization Hierarchy of SDP relaxations : � � λ k : = sup λ : f − λ ∈ Q k ( X ) λ Convergence guarantees λ k ↑ f ∗ [Lasserre 01] Can be computed with SDP solvers ( CSDP , SDPA ) “No Free Lunch” Rule : ( n + 2 k n ) SDP variables Victor Magron Enclosures of Roundoff Errors using SDP 10 / 18
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