Abstract: Global optimization methods in connection with interval arithmetic permit to determine an accurate enclosure of the global optimum, and of all the corresponding optimizers. One of the main features of these algorithms consists in the construction of an interval function which produces an enclosure of the range of the studied function over a box (right parallelepiped). We use here affine arithmetic in global optimization algorithms, in order to elaborate new inclusion functions. These techniques are implemented and then discussed. Three new affine and quadratic forms are introduced. On some polynomial examples, we show that these new tools often yield more efficient lower bounds (and upper bounds) compared to several well-known classical inclusion functions. The three new methods, presented in this paper, are integrated into various Branch and Bound algorithms. This leads to improve the convergence of these algorithms by attenuating some negative effects due to the use of interval analysis and standard affine arithmetic.

The purpose of this article is to show the advantage of a deterministic global optimization method in the optimal design of electromechanical actuators. The numerical methods classically used are found either on Non-linear Programming techniques (Augmented Lagrangian, Sequential Quadratic Programming, ...) or on stochastic approaches which are more satisfactorily adapted to global optimum research (Genetic Algorithm, Simulated Annealing, ...). However, these kinds of methods only guarantee to reach the global optimum with some probabilities. The present paper proposes a deterministic interval Branch and Bound algorithm adapted to the dimensioning of a heterogeneous bimorph bar, working in bending quasi-static mode. This problem is formulated as a optimization problem with constraints and mixed variables: real and categorical types. Original solutions are given and discussed at the end of the paper. Keywords: - Interval Analysis, Deterministic Global Optimization, Branch and Bound algorithms, Design of Piezoelectric Actuators, Mixed Problem, Categorical Variables. 1