Weber Set in Optimization

• The Weber set is a collection of allocation vectors or facility locations that minimizes weighted distances and represents convex combinations of marginal contributions.
• It provides a bridge between classical optimization methods in facility location and cooperative game theory through geometric and variational analyses.
• Extensions into tropical geometry and advanced algorithmic approaches enable robust consensus estimation and efficient computation in non-Euclidean frameworks.

A Weber Set is a central concept in both classical facility location theory and cooperative game theory, as well as in modern variants including tropical geometry and optimization over continuous or combinatorial domains. In its original context, the Weber set refers to the collection of imputations (or allocation vectors) that can be justified as convex combinations of marginal contribution payoffs derived from all possible orders or permutations of agents or locations. The concept generalizes to solution sets of minimization problems for weighted sums (or integrals) of distances, with significant geometric, variational, and computational implications.

1. Classical Definition and Facility Location Interpretation

In the setting of discrete facility location, the Fermat-Weber (or 1-median) problem seeks a point $x^* \in \mathbb{R}^d$ that minimizes the total weighted distance to a finite set of points $p_1, \ldots, p_n$:

$f(x) = \sum_{i=1}^n w_i \|x - p_i\| .$

When generalized to multiple centers (the $k$-median or continuous Fermat-Weber problem), for $k$ centers $x_1, \ldots, x_k$, the objective becomes

$$g(x_1, \ldots, x_k) = \sum_{i=1}^n w_i \min_{j=1,\ldots, k} \|p_i - x_j\| .$$

In continuous spaces with a demand density $\varphi(y)$ over domain $\mathcal{U}$, the objective is

$F(x) = \int_{\mathcal{U}} d(x, y)\,\varphi(y) dy .$

The set of minimizers is often convex and may be identified with the “Weber set” in facility location literature, which, in certain settings, is the convex hull of all possible locations that minimize weighted distances given permutations of assignment or orderings of points [0310027].

2. Weber Set in Cooperative Game Theory

Within cooperative games, the Weber set is defined through marginal vectors corresponding to permutations $T$ of the player set $N$:

$$x^T_i = v(\{T(1), \ldots, T(k)\}) - v(\{T(1), \ldots, T(k-1)\}),$$

for the $i$th player in permutation $T$. The Weber set $W(v)$ is:

$$W(v) = \operatorname{conv}\{ x^T : T \in \Pi(N) \},$$

where $v$ is the characteristic function quantifying payoffs available to coalitions (Adam et al., 2015). This set forms the convex hull of all marginal allocation vectors, providing a “stable” region in which every imputation can be justified by some ordering of coalition formations.

In variational analysis, this is captured as the Clarke superdifferential $\partial^C \hat{u}(0)$ of the Lovász extension $\hat{u}$ of the game, with the core corresponding to the Fréchet superdifferential $\hat{\partial}\hat{u}(0)$, and an intermediate set associated with the limiting (Mordukhovich) superdifferential $\partial_L \hat{u}(0)$, yielding the inclusion $$\text{core} \subseteq \text{intermediate set} \subseteq \text{Weber set}$$ for solution concepts (Adam et al., 2015).

3. Geometric and Computational Characterization

The Weber set is tightly linked to convexity and combinatorial geometry. In classical location problems, it often coincides with the convex hull of anchor points, projected as appropriate onto constraint sets. For instance, in constrained Fermat-Weber problems in Hilbert spaces, the unique minimizer lies in the projection of the classical Weber set onto the feasible set $C$ (Nguyen, 2018).

For continuous demand regions, computational geometry methods such as Voronoi diagrams and analysis of level sets are used to partition space and optimize facility location. For the $L_1$ metric, solutions often decouple to coordinate-wise median computations; for $k$-median problems, iterative algorithms alternate between partitioning demand (using Voronoi diagrams) and updating facility locations. The $k$-median problem is NP-hard for large $k$, which translates to high combinatorial complexity for the corresponding Weber set [0310027].

4. Tropical Geometry Extensions

The Weber set concept generalizes beyond Euclidean frameworks to tropical geometry. In tropical convexity, the tropical Fermat-Weber point minimizes the (max-plus) tropical distance, and the set of all such minimizers forms a tropical polytope (“polytrope”) (Sabol et al., 22 Feb 2024). The weighted tropical Fermat-Weber set coincides with a covector cell of the tropical convex hull, and any cell can arise as the set of weighted Fermat-Weber points for some choice of weights (Cox et al., 2023). Thus, the tropical polytope is the union over all weighted tropical Fermat-Weber sets.

For Bergman fans (matroid-theoretic analogs), the Fermat-Weber set may not be contained in the fan, but tropical projection guarantees at least one Fermat-Weber point in the fan. This provides robustness for consensus estimation in phylogenetics and related fields (Cox et al., 14 May 2025).

5. Algorithmic and Optimization Approaches

Various algorithms exist for locating points in the Weber set or solving Weber-type minimization problems. The Weiszfeld algorithm is a classical fixed-point iteration for unconstrained Weber problems, with extensions involving projection steps ensuring feasibility in constrained domains (Torres, 2012, Nguyen, 2018). Gradient descent and DC (difference of convex functions) methods, often with smoothing techniques (e.g., Nesterov’s smoothing), are deployed for generalized multi-source Weber problems with Minkowski gauge or convex metrics (Long et al., 20 Sep 2024).

In tropical settings, linear programming and network flow duality underpin algorithms for finding Fermat-Weber polytropes, and tropical projections (onto polyhedral fans) secure consensus points with stability guarantees (safety radius) under perturbations (Cox et al., 14 May 2025).

6. Applications and Theoretical Significance

The Weber set has substantial implications for resource allocation, consensus formation, clustering, and facility placement. In logistics, for example, throughput maximization and travel time minimization problems both reduce to Weber-type minimizations, and the optimal location remains robust against operational perturbations (Lange et al., 2022). In cooperative games, the Weber set provides efficient, stable, and Pareto-optimal allocations that serve as an upper bound for the core and intermediate solution concepts.

In tropical data analysis and phylogenetics, the Weber set approach unifies consensus tree computation, multidimensional scaling, and the reconciliation of non-Euclidean summaries. Safety radius results guarantee that consensus structures are robust to noise.

7. Summary and Contemporary Directions

The Weber set unifies methodologies from facility location, cooperative game theory, convex optimization, and tropical geometry. Its geometric convex hull character, connection to marginal contributions and coalition chains, variational interpretations, and extension to non-Euclidean domains make it a central object in both theoretical and applied optimization. Algorithmic advances (Weiszfeld-type iterations, gradient descent on polytropes, boosted DC methods) ensure practical computability, while robustness under noise and stability under projection inform its relevance to data analysis, logistics, and economic applications. Continued research explores computational complexity, generalizations to arbitrary metrics or combinatorial structures, and applications to networked systems and phylogenetics.