Comparing q-ary relations on a set $\cal O$ of elementary objects is one of the most fundamental problems ofclassification and combinatorial data analysis. In this paper the specific comparison task that involves classificationtree structures (binary or not) is considered in this context. Two mathematical representations are proposed. One isdefined in terms of a weighted binary relation; the second uses a 4-ary relation. The most classical approaches totree comparison are discussed in the context of a set theoretic representation of these relations. Formal andcombinatorial computing aspects of a construction method for a very general family of association coeficients betweenrelations are presented. The main purpose of this article is to specify the components of this construction, based on apermutational procedure, when the structures to be compared are classification trees.

Proximal Point Methods (PPM) can be traced to the pioneer works of Moreau [16], Martinet [14,15] and Rockafellar [19, 20] who used as regularization function the square of the Euclideannorm. In this work, we study PPM in the context of optimization and we derive a class of suchmethods which contains Rockafellar's result. We also present a less stringent criterion to theacceptance of an approximate solution to the subproblems that arise in the inner loops of PPM.Moreover, we introduce a new family of augmented Lagrangian methods for convex constrainedoptimization, that generalizes the PE+ class presented in [2].

We show in this paper that timed Petri nets, with one resource shared by all the transitions, are directly connected tothe modelling of integer linear programs (ILP). To an ILP can be automatically associated an equivalent Petri net. Theoptimal reachability delay is an optimal solution of the ILP. We show next that a net can model any ILP. I order to dothis, we give a new sufficient reachability condition for the marking equation, which also holds for general Petri netswithout timed transitions.

In this paper we study bi-directional nearness in a network based on AHP (Analytic Hierarchy Process) and ANP(Analytic Network Process). Usually we use forward (one-dimensional) direction nearness based on Euclidean distance.Even if the nearest point to i is point j, the nearest point to j is not necessarily point i. Sowe propose theconcept of bi-directional nearness defined by AHP'ssynthesizing of weights “for” direction and “from” direction.This concept of distance is a relative distance based on the configuration ofthe set of points located on a plane ornetwork. In order to confirm the usefulness of our study we apply the proposed nearness to solving methods of TSP(Traveling Salesman Problem), where to find an approximate solution of TSP we improved Nearest-Neighbor Method.Some numerical experiments of TSP were carried out. To decide a nearest point we used two kind of nearness, forwarddirection nearness and bi-directional nearness. As a result, by using bi-directional nearness,we obtained goodapproximate solution of TSP. Moreover, the relation between AHP and ANP, through an example, is considered.

In this paper we study the well definedness of the central path associated to agiven nonlinear (convex) semidefinite programming problem. Under standard assumptions,we establish that the existence of the central path is equivalent to the nonemptiness andboundedness of the optimal set. Other equivalent conditions are given, such as the existenceof a strictly dual feasible point or the existence of a single central point.The monotonicbehavior of the logarithmic barrier and the objective function along the trajectory is alsodiscussed. Finally, the existence and optimality of cluster points are established.

Sequential scoring rules are multi-stage social choice tules that work as follows: at each stage of the process, a scoreis computed for each alternative by taking into account its position in the individual preference rankings, and thealternative with the lowest score is eliminated. The current paper studies the ability of these rules for choosing theCondorcet winner (or the strong Condorcet winner) when individual preferences are single-peaked.

A cooperative game is defined as a set of players and a cost function. The distribution of the whole cost between theplayers can be done using the core concept, that is the set of all undominated cost allocations which prevent playersfrom grouping. In this paper we study a game whose cost function comes from the optimal solution of a linear integercovering problem. We give necessary and sufficient conditions for the core to be nonempty and characterize itsallocations using linear programming duality. We also discuss a special allocation, called the nucleolus. Wecharacterize that allocation and show that it can be computed in polynomial time using a column generation method.