Optimal $(0,1)$-Matrix Completion with Majorization Ordered Objectives (To the memory of Pravin Varaiya)

TL;DR

This paper introduces two optimal binary matrix completion problems with majorization objectives, extending Gale and Ryser's work from feasibility to optimality, with applications in EV charging, portfolio optimization, and data security.

Contribution

It formulates and solves two novel integer partial order programming problems with majorization objectives, proving the uniqueness of optimal values and developing efficient linear-time algorithms.

Findings

Optimal solutions are essentially unique.

Algorithms operate in linear time.

Numerical simulations confirm empirical efficiency.

Abstract

We propose and examine two optimal $(0,1)$-matrix completion problems with majorization ordered objectives. They elevate the seminal study by Gale and Ryser from feasibility to optimality in partial order programming (POP), referring to optimization with partially ordered objectives. We showcase their applications in electric vehicle charging, portfolio optimization, and secure data storage. Solving such integer POP (iPOP) problems is challenging because of the possible non-comparability among objective values and the integer requirements. Nevertheless, we prove the essential uniqueness of all optimal objective values and identify two particular ones for each of the two inherently symmetric iPOP problems. Furthermore, for every optimal objective value, we decompose the construction of an associated optimal~$(0,1)$-matrix into a series of sorting processes, respectively agreeing with the rule of thumb "peak shaving" or "valley filling." We show that the resulting algorithms have linear time complexities and verify their empirical efficiency via numerical simulations compared to the standard order-preserving method for POP.