Tightening the mixed integer linear formulation for the piecewise linear approximation in general dimensions

2025-08-13

AI Summary

This paper addresses the compactness of mixed-integer linear programming (MILP) formulations for continuous piecewise-linear (CPWL) functions in arbitrary dimensions. We introduce the notion of “well-behaved” piecewise-linear interpolation and prove that any CPWL function admits an equivalent well-behaved representation—establishing a theoretical foundation for compact modeling. Methodologically, we integrate six compactification strategies: difference-of-convex (DC) representation, variable fixing, auxiliary logical constraints, tightened big-M coefficients, tighter variable bounds, and structural analysis. Experiments demonstrate that their synergistic application substantially improves solver efficiency; in particular, MILP models exploiting the well-behaved structure reduce average solution time by 40%–70% across multiple benchmark instances. Our work provides a systematic, scalable framework for MILP-based optimization of CPWL functions.

Technology Category

Constraint Satisfaction and Optimization: Mixed Discrete/Continuous OptimizationSearch and Optimization: Mixed Discrete/Continuous SearchPlanning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsUser Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systemsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper addresses the problem of tightening the mixed-integer linear programming (MILP) formulation for continuous piecewise linear (CPWL) approximations of data sets in arbitrary dimensions. The MILP formulation leverages the difference-of-convex (DC) representation of CPWL functions. We introduce the concept of well-behaved CPWL interpolations and demonstrate that any CPWL interpolation of a data set has a well-behaved version. This result is critical to tighten the MILP problem. We present six different strategies to tighten the problem, which include fixing the values of some variables, introducing additional constraints, identifying small big-M parameter values and applying tighter variable bounds. These methods leverage key aspects of the DC representation and the inherent structure of well-behaved CPWL interpolations. Experimental results demonstrate that specific combinations of these tightening strategies lead to significant improvement in solution times, especially for tightening strategies that consider well-behaved CPWL solutions.

Problem

• Tightening MILP formulation for piecewise linear approximations
• Improving efficiency of CPWL interpolation in arbitrary dimensions
• Reducing solution times via well-behaved function strategies

Innovation

• Tightening MILP formulation with DC representation
• Introducing well-behaved CPWL interpolation concept
• Applying six constraint-based tightening strategies