[Paper Review] A Weiszfeld-like algorithm for a Weber location problem constrained to a closed and convex set

G. Torres|arXiv (Cornell University)|Apr 4, 2012

Computational Geometry and Mesh Generation21 references3 citations

TL;DR

This paper proposes a Weiszfeld-like algorithm for the constrained Weber location problem in a closed and convex set, ensuring convergence even when iterates land on vertices. By combining orthogonal projection with a modified Weiszfeld iteration, the method generates feasible, descent sequences that converge to the unique solution satisfying KKT conditions, outperforming standard nonlinear solvers in numerical tests.

ABSTRACT

The Weber problem consists of finding a point in $\mathbbm{R}^n$ that minimizes the weighted sum of distances from $m$ points in $\mathbbm{R}^n$ that are not collinear. An application that motivated this problem is the optimal location of facilities in the 2-dimensional case. A classical method to solve the Weber problem, proposed by Weiszfeld in 1937, is based on a fixed point iteration. In this work a Weber problem constrained to a closed and convex set is considered. A Weiszfeld-like algorithm, well defined even when an iterate is a vertex, is presented. The iteration function $Q$ that defines the proposed algorithm, is based mainly on an orthogonal projection over the feasible set, combined with the iteration function of a modified Weiszfeld algorithm presented by Vardi and Zhang in 2001. It can be seen that the proposed algorithm generates a sequence of feasible iterates that have descent properties. Under certain hypotheses, the limit of this sequence satisfies the KKT optimality conditions, is a fixed point of the iteration function that defines the algorithm, and is the solution of the constrained minimization problem. Numerical experiments confirmed the theoretical results.

Motivation & Objective

• To address the limitation of the classical Weiszfeld algorithm when iterates land on vertices, which can cause premature convergence.
• To develop a robust, feasible-iterative method for the Weber problem constrained to a closed and convex set.
• To ensure descent properties and convergence to the optimal solution under KKT conditions.
• To provide a well-defined algorithm that remains valid even at vertices, overcoming differentiability issues of the Weber function.

Proposed method

• The algorithm uses a modified Weiszfeld iteration function T₀ for interior points, adjusted via orthogonal projection onto the feasible set Ω.
• When an iterate coincides with a vertex aᵏ, the algorithm defines a new point along the line joining T₀(aᵏ) and aᵏ, ensuring feasibility.
• The iteration function Q combines projection and vertex-handling logic, ensuring all iterates remain within the convex feasible set.
• The method guarantees a non-increasing sequence of objective function values f(x⁽ˡ⁾), with strict decrease when x⁽ˡ⁾ ≠ Q(x⁽ˡ⁾).
• The algorithm is proven to converge to a fixed point of Q, which corresponds to the unique solution satisfying KKT conditions.
• Numerical validation uses MATLAB to compare the algorithm against fmincon on 1000 random instances with m=50 vertices and random weights.

Experimental results

Research questions

• RQ1 Can a Weiszfeld-like algorithm be made robust to vertex iterates in constrained Weber problems?
• RQ2 Does the proposed algorithm maintain descent and feasibility when constrained to a closed and convex set?
• RQ3 Can the algorithm converge to the true solution even when the objective function is non-differentiable at vertices?
• RQ4 How does the performance of the proposed algorithm compare to standard nonlinear programming solvers like fmincon in constrained Weber problems?
• RQ5 Is the fixed point of the iteration function Q equivalent to the KKT solution under the given constraints?

Key findings

• The proposed algorithm generates a sequence of feasible iterates that maintain descent properties, ensuring convergence to the optimal solution.
• The algorithm is well-defined even when an iterate lands exactly on a vertex, avoiding the stagnation problem of the classical Weiszfeld method.
• The unique solution of the constrained Weber problem satisfies the KKT conditions and is a fixed point of the iteration function Q.
• In 1000 numerical experiments, the proposed algorithm found solutions equal to or better than fmincon in all cases, with differences exceeding 0.01 in 35 instances.
• The algorithm's performance advantage stems from its ability to handle non-differentiability at vertices without relying on gradient-based methods.