Duality-driven optimization in energy management: offline and online algorithms for resource allocation problems

Abstract

This research has been conducted within the SIMPS project (project number 647.002.003). This research is supported by the Dutch Research Council (NWO).

A Class of Convex Quadratic Nonseparable Resource Allocation Problems with Generalized Bound Constraints

“…This means that the objective value of x 2 is smaller than that of x 1 for ε < L n−1 ( 1 2 n−1) and thus in that case x 1 is not optimal. This negative result also partially answers an open question posed in earlier work [22] that asks if and how the reduction result in Lemma 3 can be extended to optimization problems with non-convex feasible regions, in particular to extensions of the simple RAP. The constructed counterexample is an instance of the RAP with semi-continuous variables, meaning that already for this simpler extension the reduction result does not apply anymore.…”

“…Our approach also provides a partial answer to an open question in [22]. The simple RAP is known to have a nice reduction property, namely that there exists a solution to the problem that is simultaneously optimal for any choice of continuous convex function ϕ.…”

“…The simple RAP is known to have a nice reduction property, namely that there exists a solution to the problem that is simultaneously optimal for any choice of continuous convex function ϕ. The open question posed in the conclusions of [22] asks whether this property also holds for variants of the simple RAP, in particular for problems with semi-continuous variables. We demonstrate that this is unfortunately not the case by constructing an instance of RAP-DIBC as a counterexample.…”

“…• Lemma 3 can be extended from continuous convex functions to Schur-convex functions (see, e.g., Theorem 5 of [22]).…”

Resource allocation problems with expensive function evaluations

“…We refer to Bretthauer and Shetty (2002), Patriksson (2008), Katoh et al (2013) and Patriksson and Strömberg (2015) for reviews of problem formulations, algorithms and applications. Recent applications also include vaccine allocation in epidemiology (Duijzer et al, 2018) and decentralized energy management (Schoot Uiterkamp, 2021). The latter also provides an overview of algorithms and complexity results for various problem formulations.…”