## Section: New Results

### Discrete Optimization Algorithms

#### Estimating Satisfiability

Participants : Yacine Boufkhad, Thomas Hugel.

In [4] , 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 this paper 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.

#### Eigenvectors of three term recurrence Toeplitz matrices and Riordan group

Participant : Dominique Fortin.

Eigenvalues of tridiagonal (including main) Toeplitz matrices are analytically known under some regular distance to the main diagonal. Any eigenvector may be easily computed then, through a backward process; instead, in [11] , we give an analytical form for each component through the reciprocation of the underlied trinomial. More generally, the connection to the Riordan group follows some bilinear iterative process.

#### Piecewise Convex Maximization problems and algorithms

Participants : Dominique Fortin, Ider Tseveendorj.

In [14] , we provide a global search algorithm for
maximizing a piecewise convex function $F$ over a compact $D$.
We propose to iteratively refine the function $F$ at local solution
$y$
by a *virtual
cutting* function ${p}_{y}(\xb7)$
and to solve

$max\{min\{F\left(x\right)-F\left(y\right),{p}_{y}\left(x\right)\}\mid x\in D\}$ instead.
We call this function either a patch, when it avoids returning back to
the same local solutions,
or a pseudo patch, when it possibly yields a better point.
It is *virtual* in the sense that the role of cutting constraints is played by additional convex pieces in the objective function.
We report some computational results, that represent an improvement on
previous linearization based techniques.

It is well known that maximization of any difference of convex functions could be turned into a convex maximization; in [13] , we aim at a piecewise convex maximization problem instead. Despite, it may seem harder, sometimes the dimension may be reduced by 1 and the local search improved by using extreme points of the closure of the convex hull of better points. We show that it is always the case for both binary and permutation problems and give, as such instances, piecewise convex formulations for the maximum clique problem and the quadratic assignment problem.

in [12] , 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.