Symmetry Breaking in Numeric Constraint Problems

Goldsztejn, Alexandre; Jermann, Christophe; Angulo, Vicente Ruiz de; Torras, Carme

doi:10.1007/978-3-642-23786-7_25

Cited by 3 publications

(9 citation statements)

References 17 publications

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“…The 3-cyclic roots problem consists in finding all (x 1 , x 2 , x 3 ) ∈ R 3 satisfying (x 1 + x 2 + x 3 = 0) ∧ (x 1 x 2 + x 2 x 3 + x 3 x 1 = 0) ∧ (x 1 x 2 x 3 = 1). This problem has six variable symmetries (including identity): , 3], [1,3,2], [2,1,3], [2,3,1], [3,1,2], [3, 2, 1]},…”

Section: Principles Of Variable Symmetry Breakingmentioning

confidence: 99%

“…When n = 2 the only possible symmetry is the interchangeability of x 1 and x 2 , i.e., permutation [2,1]. It yields the lex constraint…”

Section: Lex Intractability and Impracticality For Ncspsmentioning

confidence: 99%

“…Things get more complicated when n = 3. Consider for instance permutation [2,3,1] and the corresponding lex SBC…”

Section: Lex Intractability and Impracticality For Ncspsmentioning

confidence: 99%

“…L C3 for the cyclic symmetry group in 3D is the intersection of two subspaces corresponding to lex [2,3,1] and lex [3,1,2] , as depicted in Figure 3.…”

Section: Lex Intractability and Impracticality For Ncspsmentioning

confidence: 99%

“…The rlex constraint for the cyclic symmetry groups C 2 is just rlex [2,1] := x 1 ≤ x 2 . That for C 3 are rlex [2,3,1] := x 1 ≤ x 2 and rlex [3,1,2] := x 1 ≤ x 3 . The corresponding domains are depicted in Figure 4, which illustrates their relation to the corresponding lex constraints (see Figures 1 and 3 above).…”

Section: Definition Of the Rlex Partial Sbcsmentioning

confidence: 99%

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Variable symmetry breaking in numerical constraint problems

Goldsztejn¹,

Jermann

Angulo

et al. 2015

Artificial Intelligence

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Symmetry breaking has been a hot topic of research in the past years, leading to many theoretical developments as well as strong scaling strategies for dealing with hard applications. Most of the research has however focused on discrete, combinatorial, problems, and only few considered also continuous, numerical, problems. While part of the theory applies in both contexts, numerical problems have specificities that make most of the technical developments inadequate.In this paper, we present the rlex constraints, partial symmetry-breaking inequalities corresponding to a relaxation of the famous lex constraints extensively studied in the discrete case. They allow (partially) breaking any variable symmetry and can be generated in polynomial time. Contrarily to lex constraints that are impractical in general (due to their overwhelming number) and inappropriate in the continuous context (due to their form), rlex constraints can be efficiently handled natively by numerical constraint solvers. Moreover, we demonstrate their pruning power on continuous domains is almost as strong as that of lex constraints, and they subsume several previous work on breaking specific symmetry classes for continuous problems. Their experimental behavior is assessed on a collection of standard numerical problems and the factors influencing their impact are studied. The results confirm rlex constraints are a dependable counterpart to lex constraints for numerical problems.

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Section: Principles Of Variable Symmetry Breakingmentioning

confidence: 99%

“…When n = 2 the only possible symmetry is the interchangeability of x 1 and x 2 , i.e., permutation [2,1]. It yields the lex constraint…”

Section: Lex Intractability and Impracticality For Ncspsmentioning

confidence: 99%

“…Things get more complicated when n = 3. Consider for instance permutation [2,3,1] and the corresponding lex SBC…”

Section: Lex Intractability and Impracticality For Ncspsmentioning

confidence: 99%

“…L C3 for the cyclic symmetry group in 3D is the intersection of two subspaces corresponding to lex [2,3,1] and lex [3,1,2] , as depicted in Figure 3.…”

Section: Lex Intractability and Impracticality For Ncspsmentioning

confidence: 99%

Section: Definition Of the Rlex Partial Sbcsmentioning

confidence: 99%

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Variable symmetry breaking in numerical constraint problems

Goldsztejn¹,

Jermann

Angulo

et al. 2015

Artificial Intelligence

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Constraint Based Computation of Periodic Orbits of Chaotic Dynamical Systems

Goldsztejn¹,

Granvilliers

Jermann

2013

Lecture Notes in Computer Science

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The chaos theory emerged at the end of the 19th century, and it has given birth to a deep mathematical theory in the 20th century, with a strong practical impact (e.g., weather forecast, turbulence analysis). Periodic orbits play a key role in understanding chaotic systems. Their rigorous computation provides some insights on the chaotic behavior of the system and it enables computer assisted proofs of chaos related properties (e.g., topological entropy). In this paper, we show that the (numerical) constraint programming framework provides a very convenient and efficient method for computing periodic orbits of chaotic dynamical systems: Indeed, the flexibility of CP modeling allows considering various models as well as including additional constraints (e.g., symmetry breaking constraints). Furthermore, the richness of the different solving techniques (tunable local propagators, search strategies, etc.) leads to highly efficient computations. These strengths of the CP framework are illustrated by experimental results on classical chaotic systems from the literature.

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Estimating the robust domain of attraction for non-smooth systems using an interval Lyapunov equation

Goldsztejn

Chabert

2019

Automatica

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The Lyapunov equation allows finding a quadratic Lyapunov function for an asymptotically stable fixed point of a linear system. Applying this equation to the linearization of a nonlinear system can also prove the exponential stability of its fixed points. This paper proposes an interval version of the Lyapunov equation, which allows investigating a given Lyapunov candidate function for non-smooth nonlinear systems inside an explicitly given neighborhood, leading to rigorous estimates of the domain of attraction (EDA) of exponentially stable fixed points. These results are developed in the context of uncertain systems. Experiments are presented, which show the interest of the approach including with respect to usual approaches based on sum-of-squares for the computation of EDA.

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Symmetry Breaking in Numeric Constraint Problems

Cited by 3 publications

References 17 publications

Variable symmetry breaking in numerical constraint problems

Variable symmetry breaking in numerical constraint problems

Constraint Based Computation of Periodic Orbits of Chaotic Dynamical Systems

Estimating the robust domain of attraction for non-smooth systems using an interval Lyapunov equation

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