Distributed continuous-time approximate projection protocols for shortest distance optimization problems

Abstract

In this paper, we investigate the distributed shortest distance optimization problem for a multi-agent network to cooperatively minimize the sum of the quadratic distances from some convex sets, where each set is only associated with one agent. To deal with the optimization problem with projection uncertainties, we propose a distributed continuous-time dynamical protocol based on a new concept of approximate projection. Here each agent can only obtain an approximate projection point on the boundary of its convex set, and communicate with its neighbors over a time-varying communication graph. First, we show that no matter how large the approximate angle is, the system states are always bounded for any initial condition, and uniformly bounded with respect to all initial conditions if the inferior limit of the stepsize is greater than zero. Then, in the two cases, nonempty intersection and empty intersection of convex sets, we provide stepsize and approximate angle conditions to ensure the optimal convergence, respectively. Moreover, we give some characterizations about the optimal solutions for the empty intersection case and also present the convergence error between agents' estimates and the optimal point in the case of constant stepsizes and approximate angles.

Averaging approach to distributed convex optimization for continuous-time multi-agent systems

“…We have obtained in Theorem 3.3 that the solution of the distributed optimization system (6) can be approximated by its average system (11). We now show for the average system (11) that: (a) The average system (11) can achieve consensus, and (b) The consensus state of the average system (11) is exactly X * = (x * , .…”

“…So far, we obtain two backward equations (9) and (10), as well as two corresponding differential equations (6) and (11). Note that W (t, X, s) is the solution to (9), and the first term W 0 (t, X) in the series for W (t, X, s) is the solution to (10).…”

“…As for other forms of results, the work [12] indicates a zero-gradient-sum algorithm which evolve invariantly on a zero-gradient-sum manifold and converge asymptotically to the unknown optimizer. Also, the group led by Hong report on this problem many effective algorithms such as the approximateprojection-based DCO-CT algorithm [11], the distributed primal-dual continuous-time gradient algorithm [27], the initial-free DCO-CT algorithm [28], internal model based DCO-CT [25], potential game base algorithm [26], nonsmooth analysis based algorithm [29], just to mention a few.…”

Distributed classification learning based on nonlinear vector support machines for switching networks

“…Many distributed optimization algorithms have been proposed with time-varying topologies [16,20,30]. Because the (distributed) binary classification problem can be converted to a distributed optimization problem [5,11], some of distributed SVM training has been constructed using distributed optimization approaches.…”

“…However, our algorithm, which uses stochastic sub-gradient for each agent in iteration and is constructed in time-varying networks, is of relatively low communication cost compared with [11]. Different from [7] and [19] based on random gossip network protocol, our algorithm is based on another class of time-varying network, the jointly-connected network [16,30].…”