Efficient and Safe Global Constraints for Handling Numerical Constraint Systems

Abstract:Interval Taylor has been proposed in the sixties by the interval analysis community for relaxing non-convex continuous constraint systems. However, it generally produces a non-convex relaxation of the solution set. A simple way to build a convex polyhedral relaxation is to select a corner of the studied domain/box as expansion point of the interval Taylor form, instead of the usual midpoint. The idea has been proposed by Neumaier to produce a sharp range of a single function and by Lin and Stadtherr to handle n × n (square) systems of equations. This paper presents an interval Newton-like operator, called ❳✲◆❡✇t♦♥, that iteratively calls this interval convexification based on an endpoint interval Taylor. This general-purpose contractor uses no preconditioning and can handle any system of equality and inequality constraints. It uses Hansen's variant to compute the interval Taylor form and uses two opposite corners of the domain for every constraint. The ❳✲◆❡✇t♦♥ operator can be rapidly encoded, and produces good speedups in constrained global optimization and non-convex constraint satisfaction. First experiments compare ❳✲◆❡✇t♦♥ with affine arithmetic.

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“…In this sense, it is closer to a reliable convexification method like ◗✉❛❞ [18,17] or affine arithmetic [26].…”

Section: X-newtonmentioning

confidence: 95%

“…The ◗✉❛❞ [18,17] method is an interval reformulation-linearization technique that produces a convex polyhedral approximation of the quadratic terms in the constraints. Affine arithmetic produces a polytope by replacing in the constraint expressions every basic operator by specific affine forms [10,33,3].…”

Section: Contributionsmentioning

confidence: 99%

A Contractor Based on Convex Interval Taylor

Araya

Trombettoni

Neveu

2012

Integration of AI and OR Techniques in Contraint Programming for Combinatorial Optimzation Problems

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“…They have been also used in the Quad system [15] designed to solve constraints over the real numbers. x×y is linearized according to Mc Cormick [18]:…”

Section: Linearization Of Nonlinear Operationsmentioning

confidence: 99%

Boosting Local Consistency Algorithms over Floating-Point Numbers

Belaid¹,

Michel²,

Rueher³

2012

Lecture Notes in Computer Science

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Abstract. Solving constraints over oating-point numbers is a critical issue in numerous applications notably in program veri cation. Capabilities of ltering algorithms over the oating-point numbers (F) have been so far limited to 2b-consistency and its derivatives. Though safe, such ltering techniques su er from the well known pathological problems of local consistencies, e.g., inability to e ciently handle multiple occurrences of the variables. These limitations also have their origins in the strongly restricted oating-point arithmetic. To circumvent the poor properties of oating-point arithmetic, we propose in this paper a new ltering algorithm, called FPLP, which relies on various relaxations over the real numbers of the problem over F. Safe bounds of the domains are computed with a mixed integer linear programming solver (MILP) on safe linearizations of these relaxations. Preliminary experiments on a relevant set of benchmarks are promising and show that this approach can be e ective for boosting local consistency algorithms over F. IntroductionCritical systems are more and more relying on oating-point (FP) computations. For instance, embedded systems are typically controlled by software that store measurements and environment data as oating-point number (F). The initial values and the results of all operations must therefore be rounded to some nearby oat. This rounding process can lead to signi cant changes, and, for example, can modify the control ow of the program. Thus, the veri cation of programs performing FP computations is a key issue in the development of critical systems.Methods for verifying programs performing FP computations are mainly derived from standard program veri cation methods. Bounded model checking (BMC) techniques have been widely used for nding bugs in hardware design [3] and software [11]. SMT solvers are now used in most of the state-of-the-art BMC tools to directly work on high level formula (see [2,9,11]). The bounded model checker CBMC encodes each FP operation of the program with a set of logic This work was partially supported by ANR VACSIM (ANR-11-INSE-0004), ANR AEOLUS (ANR-10-SEGI-0013) and OSEO ISI PAJERO projects.

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“…Cheap Lagrangian probing techniques for doing this are described in Tawarmalani & Sahinidis [302] and are implemented in BARON. Recent results of Lebbah et al [193,194] show that the more expensive approach of minimizing and maximizing each variable with respect to a linear relaxation may give significant speedups on difficult constraint satisfaction problems. [100,262,263,264], appeared that propose the use of semidefinite relaxations or convex conic relaxations to solve polynomial constraint satisfaction and global optimization problems.…”

Section: Linear and Convex Relaxationsmentioning

confidence: 99%

“…First results in this direction are presented by Lebbah et al [193,194], who report rigorous results for a combination of constraint propagation, interval Newton and linear programming methods that significantly outperform other rigorous solvers (and also the general purpose solver BARON) on a number of difficult constraint satisfaction problems.…”

Section: Rigorous Verification and Certificatesmentioning

confidence: 99%

Complete search in continuous global optimization and constraint satisfaction

Neumaier¹

2004

Acta Numerica 2004

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Abstract. This survey covers the state of the art of techniques for solving general purpose constrained global optimization problems and continuous constraint satisfaction problems, with emphasis on complete techniques that provably find all solutions (if there are finitely many). The core of the material is presented in sufficient detail that the survey may serve as a text for teaching constrained global optimization.After giving motivations for and important examples of applications of global optimization, a precise problem definition is given, and a general form of the traditional first order necessary conditions for a solution. Then more than a dozen software packages for complete global search are described.A quick review of incomplete methods for bound constrained problems and recipes for their use in the constrained case follows, an explicit example is discussed, introducing the main techniques used within branch and bound techniques. Sections on interval arithmetic, constrained propagation and local optimization are followed by a discussion of how to avoid the cluster problem. Then a discussion of important problem transformations follows, in particular of linear, convex, and semilinear (= mixed integer linear) relaxations that are important for handling larger problems.Next, reliability issues -centering around rounding error handling and testing methodologyare discussed, and the COCONUT framework for the integration of the different techniques is introduced. A list of challenges facing the field in the near future concludes the survey.

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Efficient and Safe Global Constraints for Handling Numerical Constraint Systems

Cited by 43 publications

References 49 publications

A Contractor Based on Convex Interval Taylor

A Contractor Based on Convex Interval Taylor

Boosting Local Consistency Algorithms over Floating-Point Numbers

Complete search in continuous global optimization and constraint satisfaction

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