Distributed Non-convex Optimization of Multi-agent Systems Using Boosting Functions to Escape Local Optima

Abstract

We address the problem of multiple local optima arising in cooperative multi-agent optimization problems with non-convex objective functions. We propose a systematic approach to escape these local optima using the concept of boosting functions. The essence of boosting functions approach is to temporarily transform a gradient at a local optimum into a "boosted" non-zero gradient. Extending a prior centralized optimization approach, we develop a distributed framework for the use of boosted gradients (called a distributed boosting scheme (DBS)) and show that convergence of this DBS can be attained by employing an optimal variable step size scheme for gradient-based algorithms. However, since the exact forms of the boosted gradients and the boosting scheme depend on the class of problems considered, to set an example, we apply the developed boosting concepts to the class of multi-agent coverage control problems. For that specific application, two new boosted gradient computation techniques (known as boosting function families) are introduced along with a new DBS. Simulation results are provided to compare the performance of different boosting functions families and boosting schemes. Moreover, simulation results are used to show the effectiveness of utilizing boosting functions approach in attaining improved optima (still generally local).

Optimal composition of heterogeneous multi-agent teams for coverage problems with performance bound guarantees

“…In (25), sgn(•) represents the signum function and the subscript x is used to represent the x-component of a two dimensional vector. The second term in ( 25) is due to the linear shaped boundary segments of the sensing region V (s i ) formed due to the obstacle vertices v i j ∈ V (s i ).…”

“…Algorithm 2 is a Projected Gradient Ascent (PGA) algorithm for solving (9) which utilizes the gradients derived in (25) and (26). As seen in Algorithm 2, a gradient ascent update is first implemented in (29), where η…”

“…t > 0 are the step sizes chosen based on standard technical conditions [2] (more application-specific details on the step size selection can be found in [24]). Subsequently, the projection mechanisms are applied to guarantee the satisfaction of all constraints.…”

“…Coverage performance of the PGA solution s PGA For the initialization of the PGA algorithm given in Algorithm 2, we use the greedy solution s (0) = [S G ] obtained from Algorithm 1 using: (i) The pre-specified discretized feasible space F D , and, (ii) The complete set of agents (all N of them). The overall performance bound obtained using (24) under this initial configuration is L 1 ≤ H({s (0) }) H(S * ) . Therefore, L 1 does not convey any information about the coverage performance of the obtained PGA solution.…”

“…On the other hand, on-line algorithms sacrifice potential optimality to achieve efficiency; this includes distributed gradient-based algorithms [6,13,28] and Voronoi-partition-based algorithms [3,9,12] which lead to generally locally optimal solutions. Methods for efficiently escaping such local optima using a "boosting function" approach were proposed in [19,25], while a decentralized control law in [18] seeks a combination of optimal coverage and exploration of the area of interest.…”