## Section: New Results

### Interval analysis

#### Inner Regions and Interval Linearizations for Global Optimization

Participants : Gilles Trombettoni [correspondant] , Bertrand Neveu.

Researchers 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. This interval Branch & Bound significantly outperforms the best reliable global optimizers [22] , [25] , [28] .

#### An Interval Extension Based on Occurrence Grouping

Participants : Bertrand Neveu [correspondant] , Gilles Trombettoni.

We proposed last year a new “occurrence grouping”
interval extension ${\left[f\right]}_{og}$ 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 ${f}^{og}$ which 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. ${x}_{b}$. ${\left[f\right]}_{og}$ is the interval
extension by monotonicity of ${f}^{og}$ and produces a sharper interval
image than the natural extension does.

This year we have improved the linear program and algorithm that minimize a Taylor-based over-estimate of the image diameter of ${\left[f\right]}_{og}$. We have detailed the proofs of correctness and reliability of this occurrence grouping algorithm [8] , [29] .