Efficient Linearization Techniques for Quadratic Binary Optimization Problems

Core Concepts

This paper presents several new linearization techniques for quadratic binary optimization problems (QBOPs) using selective aggregation of constraints. The proposed models provide strong linear programming relaxations and have practical value.

Abstract

The paper focuses on the linearization of quadratic binary optimization problems (QBOPs). It starts with a review of existing explicit linearization models, including the Dantzig-Watters (DW), Glover-Woolsey (GW), and Fortet (FT) formulations. The authors then introduce a new basic linearization model called PK.
The key contributions of the paper are:

Weighted aggregation of constraints: The authors present a general technique to generate new explicit linearizations of QBOPs by using weighted aggregation of selected constraints from the basic models. This includes aggregation of both type 1 (product constraints) and type 2 (auxiliary constraints) linearization constraints.

Precise and optimality-restricted models: The authors distinguish between precise models, where the linearized variables precisely represent the product of binary variables, and optimality-restricted models, where the linearized variables only guarantee the correct product at optimality. Both types of models are analyzed.

Theoretical and experimental analysis: The paper provides a detailed theoretical analysis of the proposed linearization techniques, including proofs of validity for the new models. Extensive computational experiments are also reported, offering insights into the relative merits of the different explicit linearization approaches.

Compact explicit linearizations: The authors present new explicit linearization models that match the number of constraints in many compact formulations, while still providing strong LP relaxation bounds.

The paper makes significant contributions to the understanding and development of efficient explicit linearization techniques for QBOPs, which have important applications in various optimization problems.

Stats

None.

Quotes

None.

Key Insights Distilled From

Revisiting some classical linearizations of the quadratic binary optimization problem

by Abraham P. P... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.09437.pdf

Deeper Inquiries

How can the proposed weighted aggregation techniques be extended to handle more general quadratic optimization problems beyond the binary case

The proposed weighted aggregation techniques can be extended to handle more general quadratic optimization problems beyond the binary case by adapting the constraints and weights to accommodate continuous variables and non-binary constraints. For quadratic optimization with continuous variables, the aggregation constraints can be formulated to capture the relationships between variables in the quadratic terms. By adjusting the weights and constraints accordingly, the aggregation approach can be applied to quadratic optimization with continuous variables, allowing for the linearization of more general quadratic optimization problems.

What are the potential connections between the weighted aggregation approach and other duality-based techniques, such as surrogate duality, and how can these connections be further explored

The potential connections between the weighted aggregation approach and other duality-based techniques, such as surrogate duality, lie in the concept of strengthening the linearization models through additional constraints and valid inequalities. By exploring the relationships between weighted aggregation and surrogate duality, researchers can potentially enhance the efficiency and effectiveness of the linearization process for quadratic optimization problems. Further research can focus on integrating the principles of surrogate duality into the weighted aggregation framework to develop more robust and powerful linearization models.

Can the insights gained from the polyhedral structure of the Boolean quadric polytope be leveraged to develop even stronger valid inequalities and cutting planes for the proposed explicit linearization models

The insights gained from the polyhedral structure of the Boolean quadric polytope can be leveraged to develop even stronger valid inequalities and cutting planes for the proposed explicit linearization models. By utilizing the geometric properties and structural characteristics of the Boolean quadric polytope, researchers can derive new valid inequalities that tighten the linear relaxation bounds and improve the overall performance of the linearization models. Exploring the connections between the polyhedral structure and the linearization techniques can lead to the development of more efficient algorithms and optimization strategies for quadratic binary optimization problems.