Initialization-Free Distributed Fixed-Time Convergent Algorithms for Optimal Resource Allocation

This article considers the distributed fixed-time resource allocation problem subject to global equality and local inequality constraints. In the case that the optimal solutions are strictly feasible, a projection-gradient-based continuous-time algorithm is first proposed to minimize a team of cost functions in the fixed time. As some optimal solutions locate on the boundary of local constraints, we further provide an alternative suboptimal solution based on the $\epsilon $ -exact penalty function method. Different from the existing distributed optimal allocation results, the (sub)optimal solutions can be obtained in the fixed time and the optimization protocols can be implemented in an initialization-free manner. Thus, the convergence time can be offline preassigned and the initial states can be chosen arbitrarily such that a better robust performance is achieved for the new algorithms. Case studies of the widely discussed economic dispatch problems are performed to validate the effectiveness of the obtained results.

I. Introduction

With the rapid development of parallel and distributed systems, such as smart grids [1]–[9] sensor networks [10], [11], transportation systems [12], and so on, the corresponding distributed resource allocation problem (DRAP) has attracted a great deal of attention. In recent years, the DRAP has been extensively investigated from various perspectives. Based on the Lagrangian multiplier method and the consensus of increasing cost, the early works solve DRAP with quadratic cost functions by applying the distributed consensus method [1]–[3], [6], [7]. As the general convex cost functions are considered, the DRAP can be solved by the distributed gradient method [8], [13], [14]. Based on Karush–Kuhn–Tucker (KKT) conditions of DRAP, the distributed primal–dual algorithms are extensively investigated in recent years [4], [5], [15]–[20]. Moreover, the works in [14] and [16]–[19] take the privacy-guaranteed problem into account. By virtue of random coordinate descent method, the optimization problems with equality constraint are studied in [15], [21], and [37]. The primal–dual method is convenient in dealing with the equality and inequality constraints [24]. The alternating direction method of multipliers (ADMMs) is also used to solve DRAP [23]. Some works investigated the effects of communication constraints on resource allocation, such as time delays, data drops, and time-varying links [6], [22].