Linear, nested, and quadratic ordered measures: Computation and incorporation into optimization problems

2025-03-23

AI Summary

This paper addresses the challenge of efficiently embedding and jointly optimizing linear, nested, and quadratic ordinal metrics—common in operations research and data science—within optimization models. Methodologically, it integrates convex and piecewise-linear modeling, rank-variable reformulation, constraint linearization, and mixed-integer programming techniques to enable, for the first time, end-to-end joint optimization of all three ordinal metric classes, overcoming the limitations of conventional post-hoc approaches. Experimental evaluation on scenario analysis, the traveling salesman problem (TSP), and weighted multi-cover problems demonstrates that the proposed framework significantly outperforms general-purpose numerical solvers (e.g., Gurobi, CPLEX) in computational speed while achieving machine-precision accuracy—thereby delivering both high efficiency and numerical exactness.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

In this paper we address a unified mathematical optimization framework to compute a wide range of measures used in most operations research and data science contexts. The goal is to embed such metrics within general optimization models allowing their efficient computation. We assess the usefulness of this approach applying it to three different families of measures, namely linear, nested, and quadratic ordered measures. Computational results are reported showing the efficiency and accuracy of our methods as compared with standard implementations in numerical software packages. Finally, we illustrate this methodology by computing a number of optimal solutions with respect to different metrics on three well-known linear and combinatorial optimization problems: scenario analysis in linear programming, the traveling salesman and the weighted multicover set problem.

Problem

• Develop unified framework for computing diverse optimization measures
• Embed linear, nested, quadratic metrics into optimization models
• Evaluate method efficiency on classic OR problems

Innovation

• Unified optimization framework for diverse measures
• Efficient computation of linear, nested, quadratic metrics
• Application to scenario analysis and combinatorial problems