A Structure-Tensor Approach to Integer Matrix Completion with Applications to Differentiated Energy Services
For researchers in matrix completion and smart grid resource allocation, this work offers a theoretical condition for feasibility, but the application is domain-specific and the results are incremental.
The paper provides a necessary and sufficient condition for the existence of (0,1)-matrices with given row/column sums and fixed zeros, expressed via a structure tensor, and extends this to nonnegative integer matrices with bounds. The results are applied to supply/demand matching and minimum purchase profile problems in smart grid differentiated energy services.
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.