Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints
Name
2009.10395.pdf
Description
Submitted version
Size
1.2 MB
Format
Adobe PDF
Checksum (MD5)
44a9c398afd8e6b511271c6796818d96
Author(s)
Bertsimas, Dimitris
Cory-Wright, Ryan
Pauphilet, Jean
Date Issued
2021
Journal
Operations Research
Publisher
Institute for Operations Research and the Management Sciences (INFORMS)
Citation
Bertsimas, Dimitris, Cory-Wright, Ryan and Pauphilet, Jean. 2021. "Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints." Operations Research.
Version
Original manuscript
Abstract
Many central problems throughout optimization, machine learning, and statistics are equivalent to optimizing a low-rank matrix over a convex set. However, although rank constraints offer unparalleled modeling flexibility, no generic code currently solves these problems to certifiable optimality at even moderate sizes. Instead, low-rank optimization problems are solved via convex relaxations or heuristics that do not enjoy optimality guarantees. In “Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints,” Bertsimas, Cory-Wright, and Pauphilet propose a new approach for modeling and optimizing over rank constraints. They generalize mixed-integer optimization by replacing binary variables z that satisfy z2 =z with orthogonal projection matrices Y that satisfy Y2 = Y. This approach offers the following contributions: First, it supplies certificates of (near) optimality for low-rank problems. Second, it demonstrates that some of the best ideas in mixed-integer optimization, such as decomposition methods, cutting planes, relaxations, and random rounding schemes, admit straightforward extensions to mixed-projection optimization.
MIT Department
Sloan School of Management
Massachusetts Institute of Technology. Operations Research Center
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1287/OPRE.2021.2182
