Provably Small Portfolios for Multiobjective Optimization with Application to Subsidized Facility Location

2025-10-23

AI Summary

Balancing fairness (min-max) and efficiency (min-sum) in multi-objective optimization poses fundamental challenges for decision-making under diverse societal objectives. Method: This paper introduces the “provably small portfolio”—a finite solution set guaranteed to contain an α-approximation for any Lₚ-norm-based objective function. We unify continuous objective classes from a solution-center perspective, integrating conic combination optimization, monotonic interpolation analysis, and bi-criteria approximation algorithms. Contribution/Results: We establish a tight upper bound on portfolio size and systematically characterize the approximation capability for the fairness–efficiency trade-off. Empirically, applied to fair subsidy allocation in facility location, our approach significantly mitigates healthcare deserts across multiple U.S. regions, demonstrating both real-world policy effectiveness and interpretability.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Many multiobjective real-world problems, such as facility location and bus routing, become more complex when optimizing the priorities of multiple stakeholders. These are often modeled using infinite classes of objectives, e.g., $L_p$ norms over group distances induced by feasible solutions in a fixed domain. Traditionally, the literature has considered explicitly balancing `equity' (or min-max) and `efficiency' (or min-sum) objectives to capture this trade-off. However, the structure of solutions obtained by such modeling choices can be very different. Taking a solution-centric approach, we introduce the concept of provably small set of solutions $P$, called a {it portfolio}, such that for every objective function $h(cdot)$ in the given class $mathbf{C}$, there exists some solution in $P$ which is an $α$-approximation for $h(cdot)$. Constructing such portfolios can help decision-makers understand the impact of balancing across multiple objectives. Given a finite set of base objectives $h_1, ldots, h_N$, we give provable algorithms for constructing portfolios for (1) the class of conic combinations $mathbf{C} = {sum_{j in [N]}λ_j h_j: λge 0}$ and for (2) any class $mathbf{C}$ of functions that interpolates monotonically between the min-sum efficiency objective (i.e., $h_1 + ldots + h_N$) and the min-max equity objective (i.e., $max_{j in [N]} h_j$). Examples of the latter are $L_p$ norms and top-$ell$ norms. As an application, we study the Fair Subsidized Facility Location (FSFL) problem, motivated by the crisis of medical deserts caused due to pharmacy closures. FSFL allows subsidizing facilities in underserved areas using revenue from profitable locations. We develop a novel bicriteria approximation algorithm and show a significant reduction of medical deserts across states in the U.S.

Problem

• Constructing small portfolios for multiobjective optimization problems
• Balancing equity and efficiency objectives in facility location
• Developing approximation algorithms for subsidized facility location applications

Innovation

• Constructing small portfolios for multiobjective optimization problems
• Developing algorithms for conic combinations and monotonic interpolation
• Creating bicriteria approximation for subsidized facility location