Symmetric Separable Convex Resource Allocation Problems with Structured Disjoint Interval Bound Constraints

Motivated by the problem of scheduling electric vehicle (EV) charging with a minimum charging threshold in smart distribution grids, we introduce the resource allocation problem (RAP) with a symmetric separable convex objective function and disjoint interval bound constraints. In this RAP, the aim is to allocate an amount of resource over a set of n activities, in which each individual allocation is restricted to a disjoint collection of m intervals. This is a generalization of classic RAPs studied in the literature in which, in contrast, each allocation is only restricted by simple lower and upper bounds, that is, m = 1. We propose an exact algorithm that, for four special cases of the problem, returns an optimal solution in O ( ( n + m − 2 m − 2 ) ( n log n + n F ) ) time, where the term nF represents the number of flops required for one evaluation of the separable objective function. In particular, the algorithm runs in polynomial time when the number of intervals m is fixed. Moreover, we show how this algorithm can be adapted also to output an optimal solution to the problem with integer variables without increasing its time complexity. Computational experiments demonstrate the practical efficiency of the algorithm for small values of m and, in particular, for solving EV charging problems.