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

TL;DR

This work introduces a unified optimization-based framework to compute a wide range of ordered measures, including $L$-measures, nested $L$-measures, and a novel class of quadratic $Q$-measures, and to embed these metrics within optimization models. It develops linear formulations ($\ell_1$, $\ell_2$, $\ell_3$) and bilevel reformulations for nesting, plus a mixed-integer linear model for $\mathcal{Q}$-measures, enabling efficient computation and integration into decision problems. The authors validate the approach with computational experiments showing competitive performance against standard methods and demonstrate practical use through scenario analysis in linear programming, the traveling salesman problem, and the weighted multicover set problem. The results suggest broad applicability and potential for flexible, robust decision-making where the objective is defined by ordered or interaction-based summaries of decision-relevant values.

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.

Figures (7)

• Figure 1: Polyhedron considered in the Example of Section \ref{['sec:polyhedron']} (left) and solution obtained (right).
• Figure 2: Graph used to illustrate the TSP problem under our measures.
• Figure 3: Solutions for the TSP example that minimize different $\ell$ ordered measures.
• Figure 4: Solutions for the TSP example that minimize different $\mathfrak{b}$ ordered measures.
• Figure 5: Solutions for the TSP example that minimize different $\mathcal{Q}$ ordered measures..

Theorems & Definitions (17)

• Definition 1
• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3