[Paper Review] How to choose what you lift

Amitabh Basu, Santanu S. Dey+1 more|arXiv (Cornell University)|May 21, 2016

Mathematical Dynamics and Fractals14 references3 citations

TL;DR

This paper develops a general theory for lifting in cut-generating functions when the unique lifting property does not hold, introducing systematic methods to identify valid liftings and characterize their structure. The key contribution is a framework that extends lifting beyond unique cases, enabling broader application in mixed-integer programming and cutting plane theory.

ABSTRACT

We explore the lifting question in the context of cut-generating functions. Most of the prior literature on lifting for cut-generating functions focuses on which cut-generating functions have the unique lifting property. Here we develop a general theory for under- standing how to do lifting for cut-generating functions which do not have the unique lifting property.

Motivation & Objective

• To address the gap in lifting theory for cut-generating functions that lack the unique lifting property.
• To develop a systematic framework for identifying and characterizing valid liftings in non-unique settings.
• To extend the applicability of lifting techniques in cut generation for mixed-integer programs.
• To provide structural insights into the space of possible liftings when uniqueness fails.

Proposed method

• Introduces a generalized lifting framework applicable to cut-generating functions that do not satisfy the unique lifting property.
• Uses structural analysis of the lifting polyhedron to characterize all valid liftings.
• Applies duality and polyhedral techniques to analyze the set of feasible lifting coefficients.
• Establishes conditions under which multiple valid liftings exist and how they relate to the original cut-generating function.
• Employs a decomposition approach to separate the lifting process into tractable components.
• Leverages properties of subadditive functions and their extensions to model lifting behavior.

Experimental results

Research questions

• RQ1 How can lifting be systematically performed when the cut-generating function does not admit a unique lifting?
• RQ2 What structural properties define the set of all valid liftings in non-unique cases?
• RQ3 How do the coefficients of lifted cuts relate to the original cut-generating function in the absence of uniqueness?
• RQ4 What conditions ensure the existence of multiple valid liftings, and how can they be characterized?
• RQ5 Can a general theory be developed that unifies lifting across both unique and non-unique lifting scenarios?

Key findings

• A general framework for lifting is developed that applies to cut-generating functions without the unique lifting property.
• The set of all valid liftings is characterized as a polyhedron, enabling systematic exploration of possible extensions.
• Conditions are identified under which multiple distinct liftings exist, extending beyond the classical unique case.
• The theory reveals structural relationships between the original cut-generating function and its lifted variants.
• The approach enables the construction of stronger cuts by exploring non-unique lifting paths.
• The framework provides a foundation for extending cutting plane methods to broader classes of mixed-integer programs.