## Section: New Results

### Discrete Optimization Algorithms

#### Estimating satisfiability

Participants : Yacine Boufkhad, Thomas Hugel.

The problem of estimating the proportion of satisfiable instances of a given CSP (constraint satisfaction problem) can be tackled through weighting. It consists in putting onto each solution a non-negative real value based on its neighborhood in a way that the total weight is at least 1 for each satisfiable instance. We define in [3] , a general weighting scheme for the estimation of satisfiability of general CSPs. First we give some sufficient conditions for a weighting system to be correct. Then we show that this scheme allows for an improvement on the upper bound on the existence of non-trivial cores in 3-SAT obtained by Maneva and Sinclair (2008) to $4.419$. Another more common way of estimating satisfiability is ordering. This consists in putting a total order on the domain, which induces an orientation between neighboring solutions in a way that prevents circuits from appearing, and then counting only minimal elements. We compare ordering and weighting under various conditions.

#### Attractive force search algorithm for piecewise convex maximization problems

Participants : Dominique Fortin, Ider Tseveendorj.

In [8] , we consider mathematical programming problems with the so-called piecewise convex objective functions. A solution method for this interesting and important class of nonconvex problems is presented. This method is based on Newton's law of universal gravitation, multicriteria optimization and Helly's theorem on convex bodies. Numerical experiments using well known classes of test problems on piecewise convex maximization, convex maximization as well as the maximum clique problem show the efficiency of the approach.

#### B-spline interpolation: Toeplitz inverse under corner perturbations

Participant : Dominique Fortin.

For Toeplitz matrices associated with degree 3 and 4 uniform B-spline interpolation, the inverse may be analytically known [7] , saving the standard inverse calculations. It generalizes to any degree as a row of the Eulerian numbers triangle.