A Contractor Based on Convex Interval Taylor

International audienceInterval Taylor has been proposed in the sixties by the interval analysis community for relaxing continuous non-convex 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 andby Lin and Stadtherr to handle n × n (square) systems of equations. This paper presents an interval Newton-like operator, called X-Newton, 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 X-Newton operator can be rapidly encoded, and produces good speedups in constrained global optimization and constraint satisfaction. First experiments compare X-Newton with affine arithmetic

Upper Bounding in Inner Regions for Global Optimization under Inequality Constraints

International audienceIn deterministic continuous constrained global optimization, upper bounding the objective function generally resorts to local minimization at several nodes/iterations of the branch and bound. We propose in this paper an alternative approach when the constraints are inequalities and the feasible space has a non-null volume. First, we extract an inner region , i.e., an entirely feasible convex polyhedron or box in which all points satisfy the constraints. Second, we select a point inside the extracted inner region and update the upper bound with its cost. We describe in this paper two original inner region extraction algorithms implemented in our interval B&B called IbexOpt. They apply to nonconvex constraints involving mathematical operators like +,x,power,sqrt,exp,log,sin. This upper bounding shows very good performance obtained on medium-sized systems proposed in the COCONUT suite

Growing Hierarchical Trees for Data Stream Clustering and Visualization

International audienc

lsmear : a variable selection strategy for interval branch and bound solvers

International audienceSmear-based variable selection strategies are well-known and commonly used bybranch-and-prune interval-based solvers. They estimate the impact of the variables on eachconstraint of the system by using the partial derivatives and the sizes of the variable domains.Then they aggregate these values, in some way, to estimate the impact of each variable onthe whole system. The variable with the greatest impact is then selected. A problem of thesestrategies is that they, generally, consider all constraints equally important. In this work, wepropose a new variable selection strategy which first weights the constraints by using theoptimal Lagrangian multipliers of a linearization of the original problem. Then, the impactof the variables is computed with a typical smear-based function but taking into accountthe weights of the constraints. The strategy isg tested on a set of well-known benchmarkinstances outperforming significantly the classical variable selection strategie

Vitamin D status of schoolchildren in Northern Algeria, seasonal variations and determinants of vitamin D deficiency

Summary There are no published data on the vitamin D status of children living in North Africa. In 435 healthy Algerian children 5–15 years old, we found that vitamin D insufficiency (serum 25-hydroxyvitamin D (25OHD) <50 nmol/L) was frequent, especially in winter. Low vitamin D status was associated with increased parathyroid hormone (PTH) and leg deformation Introduction As there are no published data on the vitamin D status of children living in North Africa, we evaluated the 25OHD concentration of healthy Algerian children at the end of summer and at the end of winter. As secondary objectives, we studied the various determinants of vitamin D status and the PTH-25OHD relationship in these subjects. Methods Four hundred thirty-five children 5–15 years old were examined and had a blood sample in September 2010. Of them, 408 were sampled again in March 2011. Results Median 25OHD concentration in the whole group was 71.4 nmol/L in September and 52.9 nmol/L in March. In September, 58.4, 29.9, and 8.1 % had a 25OHD concentration below 75, 50, and 30 nmol/L respectively. In March, these percentages increased to 65.2, 41.4, and 17.4 % for the 75, 50, and 30 nmol/L threshold, respectively. In multivariate analysis, older age, darker skin phototype, low daily vitamin D and calcium intake, poor socioeconomic status, and short daily sun exposure remained significantly associated with a 25OHD <50 nmol/L at both visits. In 72 (16.6 %) children, genu varum/valgum was present. Compared to the 363 children without leg deformation, they presented more frequently with the risk factors of vitamin D insufficiency. They also had lower 25OHD concentrations and higher PTH and tALP. Serum PTH and 25OHD concentrations were negatively and significantly correlated (r=−0.43; p<0.001) without a 25OHD threshold above which PTH does not decrease anymore. Conclusion Despite a sunny environment, vitamin D insufficiency is frequent in healthy Algerian children

Adaptive constructive interval disjunction: algorithms and experiments

International audienceAn operator called CID and an efficient variant 3BCID were proposed in 2007. For the numerical CSP handled by interval methods, these operators compute a partial consistency equivalent to Partition-1-AC for the discrete CSP. In addition to the constraint propagation procedure used to refute a given subproblem, the main two parameters of CID are the number of times the main CID procedure is called and the maximum number of sub-intervals treated by the procedure. The 3BCID operator is state-of-the-art in numerical CSP, but not in constrained global optimization, for which it is generally too costly. This paper proposes an adaptive variant of 3BCID called ACID. The number of variables handled is auto-adapted during the search, the other parameters are fixed and robust to modifications. On a representative sample of instances, ACID appears to work efficiently, both with the HC4 constraint propagation algorithm and with the state-of-the-art Mohc algorithm. Experiments also highlight that it is relevant to auto-adapt only a number of handled variables, instead of a specific set of selected variables. Finally, ACID appears to be the best interval constraint programming operator for solving and optimization, and has been therefore added to the default strategies of the Ibex interval solver