Inner Regions and Interval Linearizations for Global Optimization

International audienceResearchers from interval analysis and constraint (logic) programming communities have studied intervals for their ability to manage infinite solution sets of numerical constraint systems. In particular, inner regions represent subsets of the search space in which all points are solutions. Our main contribution is the use of recent and new inner region extraction algorithms in the upper bounding phase of constrained global optimization. Convexification is a major key for efficiently lower bounding the objective function. We have adapted the convex interval taylorization proposed by Lin & Stadtherr for producing a reliable outer and inner polyhedral approximation of the solution set and a linearization of the objective function. Other original ingredients are part of our optimizer, including an efficient interval constraint propagation algorithm exploiting monotonicity of functions. We end up with a new framework for reliable continuous constrained global optimization. Our interval B&B is implemented in the interval-based explorer Ibex and extends this free C++ library. Our strategy significantly outperforms the best reliable global optimizers

Making Adaptive an Interval Constraint Propagation Algorithm Exploiting Monotonicity

International audienceA new interval constraint propagation algorithm, called MOnotonic Hull Consistency (Mohc), has recently been proposed. Mohc exploits monotonicity of functions to better filter variable domains. Embedded in an interval-based solver, Mohc shows very high performance for solving systems of numerical constraints (equations or inequalities) over the reals. However, the main drawback is that its revise procedure depends on two user-defined parameters. This paper reports a rigourous empirical study resulting in a variant of Mohc that avoids a manual tuning of the parameters. In particular, we propose a policy to adjust in an auto-adaptive way, during the search, the parameter sensitive to the monotonicity of the revised function

An Interval Extension Based on Occurrence Grouping: Method and Properties

In interval arithmetics, special care has been brought to the definition of interval extension functions that compute narrow interval images. In particular, when a function f is monotonic w.r.t. a variable in a given domain, it is well-known that the monotonicity-based interval extension of f computes a sharper (interval) image than the natural interval extension does. This paper presents a so-called ''occurrence grouping'' interval extension [f]_{og} of a function f. When f is not monotonic w.r.t. a variable x in a given domain, we try to transform f into a new function f^{og} that is monotonic w.r.t. two subsets x_a and x_b of the occurrences of x: f^{og} is increasing w.r.t. x_a and decreasing w.r.t. x_b. [f]_{og} is the interval extension by monotonicity of f^{og} and produces a sharper interval image than the natural extension does. For finding a good occurrence grouping, we propose a linear program and an algorithm that minimize a Taylor-based over-estimate of the image diameter of [f]_{og}. Experiments show the benefits of this new interval extension for solving systems of nonlinear equations.L'analyse d'intervalles a proposé plusieurs extensions aux intervalles qui essaient de calculer des images étroites des fonctions. En particulier, quand une fonction f est monotone par rapport à une variable sur un domaine donné, il est bien connu que l'extension aux intervalles par monotonie de f permet de calculer un intervalle image plus étroit que l'extension naturelle. Cet article présente une nouvelle extension aux intervalles d'une fonction f appelée regroupement d'occurrences et notée [f]_{og}. Quand f n'est pas monotone par rapport à une variable x sur un domaine donné, nous essayons de transformer f en une nouvelle fonction f^{og} qui est monotone par rapport à deux sous-ensembles x_a et x_b des occurrences de x : f^{og} est croissante par rapport à x_a et décroissante par rapport à x_b. [f]_{og} est l'extension aux intervalles par monotonie de f^{og} et produit une image plus étroite que l'extension naturelle. Pour trouver un bon regroupement d'occurrences, nous proposons un programme linéaire et un algorithme qui minimisent une surestimation du diamètre de l'image de [f]_{og} basée sur une forme de Taylor de f. Finalement, des expérimentations montrent les avantages de cette nouvelle extension pour la résolution de systèmes d'équations non linéaires

Correct Approximation of IEEE 754 Floating-Point Arithmetic for Program Verification

Verification of programs using floating-point arithmetic is challenging on several accounts. One of the difficulties of reasoning about such programs is due to the peculiarities of floating-point arithmetic: rounding errors, infinities, non-numeric objects (NaNs), signed zeroes, denormal numbers, different rounding modes, etc. One possibility to reason about floating-point arithmetic is to model a program computation path by means of a set of ternary constraints of the form z = x op y and use constraint propagation techniques to infer new information on the variables' possible values. In this setting, we define and prove the correctness of algorithms to precisely bound the value of one of the variables x, y or z, starting from the bounds known for the other two. We do this for each of the operations and for each rounding mode defined by the IEEE 754 binary floating-point standard, even in the case the rounding mode in effect is only partially known. This is the first time that such so-called filtering algorithms are defined and their correctness is formally proved. This is an important slab for paving the way to formal verification of programs that use floating-point arithmetics.Comment: 64 pages, 19 figures, 2 table