Symmetric separable convex resource allocation problems with structured disjoint interval bound constraints

TL;DR

This paper introduces an exact polynomial-time algorithm for a generalized resource allocation problem with structured disjoint interval constraints, motivated by electric vehicle charging scheduling in smart grids.

Contribution

It presents a novel algorithm for symmetric separable convex RAP with disjoint interval bounds, extending classical RAPs and applicable to EV charging scheduling.

Findings

Algorithm efficiently solves special cases with fixed number of intervals.

The method adapts to integer variables without increased complexity.

Computational experiments confirm practical efficiency for small number of intervals.

Abstract

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, where each individual allocation is restricted to a disjoint collection of $m$ intervals. This is a generalization of classical RAPs studied in the literature where in contrast each allocation is only restricted by simple lower and upper bounds, i.e., $m=1$. We propose an exact algorithm that, for four special cases of the problem, returns an optimal solution in $O \left(\binom{n+m-2}{m-2} (n \log n + nF) \right)$ 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 to also 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.