arXiv subjects

Pierre-Alexandre Bliman

Publications and source records attributed to Pierre-Alexandre Bliman.

24 records · Page 2Linked to original sources

arXiv2014

Convergence results for continuous-time dynamics arising in ant colony optimization

This paper studies the asymptotic behavior of several continuous-time dynamical systems which are analogs of ant colony optimization algorithms that solve shortest path problems. Local asymptotic stability of the equilibrium corresponding to the shortest path is shown under mild assumptions. A complete study is given for a recently proposed model called EigenAnt: global asymptotic stability is shown, and the speed of convergence is calculated explicitly and shown to be proportional to the difference between the reciprocals of the second shortest and the shortest paths.

math.DS↗

arXiv2012

A cooperative conjugate gradient method for linear systems permitting multithread implementation of low complexity

This paper proposes a generalization of the conjugate gradient (CG) method used to solve the equation $Ax=b$ for a symmetric positive definite matrix $A$ of large size $n$. The generalization consists of permitting the scalar control parameters (= stepsizes in gradient and conjugate gradient directions) to be replaced by matrices, so that multiple descent and conjugate directions are updated simultaneously. Implementation involves the use of multiple agents or threads and is referred to as cooperative CG (cCG), in which the cooperation between agents resides in the fact that the calculation of each entry of the control parameter matrix now involves information that comes from the other agents. For a sufficiently large dimension $n$, the use of an optimal number of cores gives the result that the multithread implementation has worst case complexity $O(n^{2+1/3})$ in exact arithmetic. Numerical experiments, that illustrate the interest of theoretical results, are carried out on a multicore computer.

math.NA↗

arXiv2007

Convergence speed of unsteady distributed consensus: decay estimate along the settling spanning-trees

Results for estimating the convergence rate of non-stationary distributed consensus algorithms are provided, on the basis of qualitative (mainly topological) as well as basic quantitative information (lower-bounds on the matrix entries). The results appear to be tight in a number of instances and are illustrated through simple as well as more sophisticated examples. The main idea is to follow propagation of information along certain spanning trees which arise in the communication graph.

math.OC↗

arXiv2007

Tight estimates for convergence of some non-stationary consensus algorithms

The present paper is devoted to estimating the speed of convergence towards consensus for a general class of discrete-time multi-agent systems. In the systems considered here, both the topology of the interconnection graph and the weight of the arcs are allowed to vary as a function of time. Under the hypothesis that some spanning tree structure is preserved along time, and that some nonzero minimal weight of the information transfer along this tree is guaranteed, an estimate of the contraction rate is given. The latter is expressed explicitly as the spectral radius of some matrix depending upon the tree depth and the lower bounds on the weights.

math.OC↗

arXiv2005

Average consensus problems in networks of agents with delayed communications

The present paper is devoted to the study of average consensus problems for undirected networks of dynamic agents having communication delays. The accent is put here on the study of the time-delays influence: both constant and time-varying delays are considered, as well as uniform and non uniform repartitions of the delays in the network. The main results provide sufficient conditions (also necessary in most cases) for existence of average consensus under bounded, but otherwise unknown, communication delays. Simulations are provided that show adequation with these results.

math.OC↗

arXiv2004

Stability of leaderless multi-agent systems. Extension of a result by Moreau

The paper presents a result which relates connectedness of the interaction graphs in a multi-agent systems with the capability for global convergence to a common equilibrium of the system. In particular we extend a previously known result by Moreau by including the possibility of arbitrary bounded time-delays in the communication channels and relaxing the convexity of the allowed regions for the state transition map of each agent.

math.OC↗