A Structure-Tensor Approach to Integer Matrix Completion with Applications to Differentiated Energy Services

TL;DR

This paper introduces a structure-tensor method for solving specific integer matrix completion problems with applications to energy management in smart grids, providing necessary and sufficient conditions for feasibility.

Contribution

It develops a novel structure-tensor approach to characterize the feasibility of (0,1)-matrix completion with fixed zeros and extends it to nonnegative integer matrices with bounds, applied to energy services.

Findings

Derived a necessary and sufficient condition for (0,1)-matrix completion.

Extended the approach to nonnegative integer matrices with bounds.

Applied the method to demand response problems in smart grids.

Abstract

Efficient resource allocation is one of the main driving forces of human civilizations. Of the many existing approaches to resource allocation, matrix completion is one that is frequently applied. In this paper, we investigate a special type of matrix completion problem concerning the class of $(0,1)$-matrices with given row/column sums and certain zeros prespecified. We provide a necessary and sufficient condition under which such a class is nonempty. The condition is stated in the form of the nonnegativity of a structure tensor constructed from the information regarding the given row/column sums and fixed zeros. Moreover, we show that a more general matrix completion problem can be studied in a similar manner, namely that involving the class of nonnegative integer matrices with prescribed row/column sums, predetermined zeros, and different bounds across the rows. To illustrate the utility of our results, we apply them to demand response applications in smart grids. Specifically, we address two adequacy problems in differentiated energy services, namely, the problems of supply/demand matching and minimum purchase profile.