Abstract In deterministic global optimization, techniques for linear relaxation of a non-convex program are used in the lower bound calculation phase. To achieve this phase, most deterministic global optimization codes use reformulation-linearization techniques. However, there exist also two interval-based polyhedral relaxation techniques which produce reliable bounds without adding new auxiliary variables, and which can take into account mathematical operations and most transcendental functions: (i) the affine relaxation technique, used in the IBBA code, based on affine forms and affine arithmetic, and (ii) the extremal Taylor technique, used in the Ibex-Opt code, which is based on a specific interval-based Taylor form. In this paper, we describe how these two interval-based linear relaxation techniques can be hybridized. These two approaches appear to be complementary, and such a hybrid method performs well on a representative sample of constrained global optimization instances.

We propose a global optimization approach to solve ℓ0-norm penalized least-squares problems, using a dedicated branch-and-bound methodology. A specific tree search strategy is built, with branching rules inspired from greedy exploration techniques. We show that the subproblem involved at each node can be evaluated via ℓ1-norm-based optimization problems with box constraints, for which an active-set algorithm is built. Our method is able to solve exactly moderate-size, yet difficult, sparse approximation problems, without resorting to mixed-integer programming (MIP) optimization. In particular, it outperforms the generic MIP solver CPLEX.

Hyperspectral sensors provides informative data in many fields related to Earth observation. On the coastal zone, the inversion of radiative transfer models of the light has shown the ability to estimate parameters characterizing the water column. Particularly the water column depth, its concentration in non-algal particles, in phytoplankton, the bottom reflectance can be retrieved. Nevertheless the reliability of these estimations are difficult to clearly assess. In this paper a methodology based on interval arithmetic is presented to estimate the likelihood function. This allows to describe the behavior of this function around its maximum and therefore to evaluate the reliability of the solution. This methodology is applied to describe the influence of the noise level.