Two-player games between polynomial optimizers and semidefinite - PowerPoint PPT Presentation

Two-player games between polynomial optimizers and semidefinite solvers Victor Magron , CNRS–LAAS Joint work with Jean-Bernard Lasserre (CNRS–LAAS) Mohab Safey El Din (Sorbonne Université) SIAM AG, Bern, 11 July 2019 f Σ Victor Magron Two-player games between polynomial optimizers & SDP solvers 0 / 21

SDP for Polynomial Optimization NP-hard NON CONVEX Problem f ⋆ = inf f ( x ) Theory (Primal) (Dual) � inf f d µ sup λ µ proba ⇒ ⇐ with p − λ � 0 with INFINITE LP Victor Magron Two-player games between polynomial optimizers & SDP solvers 1 / 21

SDP for Polynomial Optimization NP-hard NON CONVEX Problem f ⋆ = inf f ( x ) Practice (Primal Relaxation ) (Dual Strengthening ) � x α d µ f − λ = sum of squares moments finite number ⇒ SDP ⇐ fixed degree d ↑ f ∗ L ASSERRE ’ S H IERARCHY of CONVEX P ROBLEMS f ⋆ [Lasserre/Parrilo 01] degree d ⇒ ( n + d = n ) SDP VARIABLES n vars Numeric = ⇒ Approx Certificate Solvers Victor Magron Two-player games between polynomial optimizers & SDP solvers 1 / 21

Success Stories: Lasserre’s Hierarchy M ODELING P OWER : Cast as ∞ -dimensional LP over measures S TATIC Polynomial Optimization Optimal Powerflow n ≃ 10 3 [Josz et al 16] Roundoff Error n ≃ 10 2 [Magron et al 17] D YNAMICAL Polynomial Optimization Regions of attraction [Henrion et al 14] Reachable sets [Magron et al 19] △ ! APPROXIMATE O PTIMIZATION B OUNDS ! Victor Magron Two-player games between polynomial optimizers & SDP solvers 2 / 21

Two-player Games: Optimizers vs Solvers M OTZKIN POLYNOMIAL f sums of squares = Σ f = 1 Σ 27 + x 2 y 4 + x 4 y 2 − x 2 y 2 ∈ Σ f � 0 but f / Victor Magron Two-player games between polynomial optimizers & SDP solvers 3 / 21

Two-player Games: Optimizers vs Solvers M OTZKIN POLYNOMIAL f sums of squares = Σ f = 1 Σ 27 + x 2 y 4 + x 4 y 2 − x 2 y 2 ∈ Σ f � 0 but f / √ 3 f ⋆ = ( x , y ) ∈ R 2 f ( x , y ) = 0 for | x ⋆ | = | y ⋆ | = min 3 Lasserre’s hierarchy: order 3 � f ⋆ 3 = − ∞ unbounded SDP = ⇒ f / ∈ Σ Victor Magron Two-player games between polynomial optimizers & SDP solvers 3 / 21

Two-player Games: Optimizers vs Solvers M OTZKIN POLYNOMIAL f sums of squares = Σ f = 1 Σ 27 + x 2 y 4 + x 4 y 2 − x 2 y 2 ∈ Σ f � 0 but f / √ 3 f ⋆ = ( x , y ) ∈ R 2 f ( x , y ) = 0 for | x ⋆ | = | y ⋆ | = min 3 Lasserre’s hierarchy: order 3 � f ⋆ 3 = − ∞ unbounded SDP = ⇒ f / ∈ Σ 4 = − ∞ order 4 � f ⋆ Victor Magron Two-player games between polynomial optimizers & SDP solvers 3 / 21

Two-player Games: Optimizers vs Solvers M OTZKIN POLYNOMIAL f sums of squares = Σ f = 1 Σ 27 + x 2 y 4 + x 4 y 2 − x 2 y 2 ∈ Σ f � 0 but f / √ 3 f ⋆ = ( x , y ) ∈ R 2 f ( x , y ) = 0 for | x ⋆ | = | y ⋆ | = min 3 Lasserre’s hierarchy: order 3 � f ⋆ 3 = − ∞ unbounded SDP = ⇒ f / ∈ Σ 4 = − ∞ order 4 � f ⋆ order 5 � f ⋆ 5 ≃ − 0.4 Victor Magron Two-player games between polynomial optimizers & SDP solvers 3 / 21

Two-player Games: Optimizers vs Solvers M OTZKIN POLYNOMIAL f sums of squares = Σ f = 1 Σ 27 + x 2 y 4 + x 4 y 2 − x 2 y 2 ∈ Σ f � 0 but f / √ 3 f ⋆ = ( x , y ) ∈ R 2 f ( x , y ) = 0 for | x ⋆ | = | y ⋆ | = min 3 Lasserre’s hierarchy: order 3 � f ⋆ 3 = − ∞ unbounded SDP = ⇒ f / ∈ Σ 4 = − ∞ order 4 � f ⋆ order 5 � f ⋆ 5 ≃ − 0.4 8 ≃ − 10 − 8 ⊕ extraction of x ⋆ , y ⋆ Paradox ?! order 8 � f ⋆ Victor Magron Two-player games between polynomial optimizers & SDP solvers 3 / 21

Two-player Games: Optimizers vs Solvers A PPROXIMATE SOLUTIONS sum of squares of a 2 − 2 ab + b 2 ? ( 1.00001 a − 0.99998 b ) 2 ! a 2 − 2 ab + b 2 ≃ ( 1.00001 a − 0.99998 b ) 2 a 2 − 2 ab + b 2 � = 1.0000200001 a 2 − 1.9999799996 ab + 0.9999600004 b 2 → = ? ≃ Victor Magron Two-player games between polynomial optimizers & SDP solvers 4 / 21

SDP for Polynomial Optimization Optimization Game Certification Game

Inaccurate SDP do Robust Optimization f ⋆ = inf ∑ f α x α α Moment matrix M d ( y ) α , β = y α + β Accurate SDP Relaxations (Primal Relaxation ) (Dual Strengthening ) y ∑ inf f α y α sup λ α s.t. M d ( y ) � 0 f − λ = σ y 0 = 1 σ ∈ Σ d Victor Magron Two-player games between polynomial optimizers & SDP solvers 5 / 21

Inaccurate SDP do Robust Optimization f ⋆ = inf ∑ f α x α α Moment matrix M d ( y ) α , β = y α + β M d ( y ) = ∑ B α y α α Accurate SDP Relaxations (Primal Relaxation ) (Dual Strengthening ) y ∑ inf f α y α sup λ α s.t. M d ( y ) � 0 f α − λ 1 α = 0 = � B α , Q � y 0 = 1 Q � 0 Victor Magron Two-player games between polynomial optimizers & SDP solvers 5 / 21

Inaccurate SDP do Robust Optimization f ⋆ = inf ∑ f α x α α M d ( y ) α , β = y α + β Moment matrix M d ( y ) = ∑ B α y α α Inaccurate SDP Relaxations (Primal Relaxation ) (Dual Strengthening ) sup λ | f α − λ 1 α = 0 − � B α , Q � | � ε Q � − η I Victor Magron Two-player games between polynomial optimizers & SDP solvers 5 / 21

Inaccurate SDP do Robust Optimization f ⋆ = inf ∑ f α x α α M d ( y ) α , β = y α + β Moment matrix M d ( y ) = ∑ B α y α α Inaccurate SDP Relaxations (Primal Relaxation ) (Dual Strengthening ) y ∑ inf f α y α + η � M d ( y ) , I � + ε � y � 1 sup λ α s.t. M d ( y ) � 0 | f α − λ 1 α = 0 − � B α , Q � | � ε y 0 = 1 Q � − η I Victor Magron Two-player games between polynomial optimizers & SDP solvers 5 / 21

Priority to Trace Equalities: ε = 0 f = f + η ∑ ˜ x 2 β β Inaccurate SDP Relaxations (Primal Relaxation ) (Dual Strengthening ) y ∑ inf f α y α + η � M d ( y ) , I � sup λ α s.t. M d ( y ) � 0 f α − λ 1 α = 0 − � B α , Q � = 0 y 0 = 1 Q � − η I Victor Magron Two-player games between polynomial optimizers & SDP solvers 6 / 21

Priority to Trace Equalities: ε = 0 f = f + η ∑ ˜ x 2 β β Inaccurate SDP Relaxations (Primal Relaxation ) (Dual Strengthening ) y ∑ inf f α y α + η � M d ( y ) , I � sup λ α s.t. M d ( y ) � 0 f α − λ 1 α = 0 − � B α , Q − η I � = 0 y 0 = 1 Q � 0 Victor Magron Two-player games between polynomial optimizers & SDP solvers 6 / 21

Priority to Trace Equalities: ε = 0 f = f + η ∑ ˜ x 2 β β Inaccurate SDP Relaxations (Primal Relaxation ) (Dual Strengthening ) ˜ y ∑ inf f α y α sup λ α ˜ s.t. M d ( y ) � 0 f − λ = σ y 0 = 1 σ ∈ Σ d Victor Magron Two-player games between polynomial optimizers & SDP solvers 6 / 21

Priority to Trace Equalities: ε = 0 x 2 β : | θ | � η } B ∞ ( f , η ) : = { f + θ ∑ β Victor Magron Two-player games between polynomial optimizers & SDP solvers 7 / 21

Priority to Trace Equalities: ε = 0 x 2 β : | θ | � η } B ∞ ( f , η ) : = { f + θ ∑ β Theorem [Lasserre-Magron 19] Inaccurate SDP relaxations of the robust problem ˜ f ( x ) max min ˜ x f ∈ B ∞ ( f , η ) Victor Magron Two-player games between polynomial optimizers & SDP solvers 7 / 21

Priority to Trace Equalities: ε = 0 Theorem [Lasserre 06] For fixed n , any f � 0 can be approximated arbitrarily closely by SOS polynomials. Victor Magron Two-player games between polynomial optimizers & SDP solvers 8 / 21

Priority to Trace Equalities: ε = 0 Theorem [Lasserre 06] For fixed n , any f � 0 can be approximated arbitrarily closely by SOS polynomials. f ˜ f f = f + η ∑ ˜ x 2 β Σ Σ | β | � d Victor Magron Two-player games between polynomial optimizers & SDP solvers 8 / 21

Priority to Trace Equalities: ε = 0 Theorem [Lasserre 06] For fixed n , any f � 0 can be approximated arbitrarily closely by SOS polynomials. f ˜ f f = f + η ∑ ˜ x 2 β Σ Σ | β | � d At fixed η , when d ր , ˜ f ∈ Σ ! f + 10 − 7 ∑ x 2 β ∈ Σ | β | � 4 Paradox Explanation Victor Magron Two-player games between polynomial optimizers & SDP solvers 8 / 21

Priority to SDP Inequalities: η = 0 Inaccurate SDP Relaxations (Primal Relaxation ) (Dual Strengthening ) y ∑ inf f α y α + ε � y � 1 sup λ α s.t. M d ( y ) � 0 | f α − λ 1 α = 0 − � B α , Q � | � ε y 0 = 1 Q � 0 Victor Magron Two-player games between polynomial optimizers & SDP solvers 9 / 21

Priority to SDP Inequalities: η = 0 B ∞ ( f , ε ) : = { ˜ f : � ˜ f − f � ∞ � ε } Inaccurate SDP Relaxations (Primal Relaxation ) (Dual Strengthening ) y ∑ f α y α + ε � y � 1 inf sup λ λ , ˜ α f | ˜ s.t. M d ( y ) � 0 f α − f α | � ε ˜ y 0 = 1 f − λ ∈ Σ d Victor Magron Two-player games between polynomial optimizers & SDP solvers 9 / 21

Priority to SDP Inequalities: η = 0 Theorem (Lasserre-Magron) Inaccurate SDP relaxations of the robust problem ˜ max min f ( x ) ˜ x f ∈ B ∞ ( f , ε ) Victor Magron Two-player games between polynomial optimizers & SDP solvers 10 / 21

A Two-player Game Interpretation max − min R OBUST O PTIMIZATION Player 1 (solver) picks ˜ f ∈ B ∞ ( f ) � SDP leads Player 2 (optimizer) picks an SOS � User follows Victor Magron Two-player games between polynomial optimizers & SDP solvers 11 / 21