Solving ill-posed conic optimization problems

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K20217
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 60020:Mathematical informatics-related
• Research Institution: The Institute of Statistical Mathematics (2020-2023); The University of Tokyo (2019)
• Principal Investigator: Lourenco Bruno F. 統計数理研究所, 統計基盤数理研究系, 准教授 (80778720)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: conic optimization / error bounds / amenable cones / p-cones / exponential cone / facial reduction

Outline of Final Research Achievements

We proved several results on the geometry of general convex cones, including a characterization of self-duality of polyhedral cones and a study on automorphisms of certain hyperbolicity cones. We also proposed the notion of "amenable cones", proved its properties and compared it to other classical notions of facial exposedness. We developed new techniques based on facial reduction for regularizing conic linear programs. One of the main new results of this project was a new framework for computing error bounds based on the notion of facial residual functions and facial reduction. New error bounds were computed for symmetric cones, exponential cones, p-cones and power cones. We proposed a new framework for obtaining convergence rate results for algorithms for convex feasibility problems. All the aforementioned results were published in peer-reviewed journals.

Free Research Field

Continuous optimization

Academic Significance and Societal Importance of the Research Achievements

Conic optimization is a class of problems that is quite useful in practice, with applications in many fields. Through this project, we helped to elucidate some aspects of conic optimization that will be helpful when solving certain challenging instances.