Constrained Weiszfeld Problem

• The constrained Weiszfeld problem is a generalization of the Fermat–Weber location problem, minimizing a weighted sum of Euclidean distances subject to closed convex constraints.
• Projected Weiszfeld algorithms employ fixed-point iterations with projection steps to ensure convergence to a unique minimizer even when iterates hit anchor points.
• Extensions include addressing convex sets, signed weights, and d.c. decompositions, enhancing applicability in high-dimensional facility location and real-time optimization.

The constrained Weiszfeld problem generalizes the classical Fermat–Weber location problem by seeking a point that minimizes a weighted sum of Euclidean distances to fixed anchors, subject to closed convex constraints. It is fundamentally connected to location theory, convex analysis, and iterative optimization, with significant implications for facility location in constrained environments and for convex programming techniques in both finite and infinite-dimensional spaces.

1. Mathematical Formulation

Given a real Hilbert space $H$ (in particular, $\mathbb R^n$), the constrained Fermat–Weber problem is specified by:

• A closed convex constraint set $C \subset H$
• Anchor points $a_1, \dots, a_m \in H$, not all collinear
• Strictly positive weights $w_i > 0$ for $i=1,\dots,m$

The objective is to solve: $$\min_{x \in C} \, f(x) = \sum_{i=1}^m w_i \|x - a_i\|$$ An equivalent unconstrained formulation uses the indicator function $I_C$ of $C$: $$\min_{x \in H} \, F(x) = \sum_{i=1}^m w_i \|x - a_i\| + I_C(x)$$ Optimality is characterized variationally: $$0 \in \sum_{i=1}^m w_i \frac{x - a_i}{\|x - a_i\|} + N(x, C)$$ where $N(x, C)$ is the normal cone to $C$ at $x$.

With all weights positive and anchors non-collinear, the functional $f(x)$ is strictly convex on $C$, ensuring uniqueness of the minimizer (Nguyen, 2018, Torres, 2012).

2. Algorithmic Developments

The Weiszfeld algorithm provides a fixed-point method for the unconstrained case. The core mapping is

$$T(x) = \frac{\sum_{i=1}^m \frac{w_i}{\|x - a_i\|} a_i}{\sum_{i=1}^m \frac{w_i}{\|x - a_i\|}}$$

For the constrained problem, the projected Weiszfeld algorithm iterates: $x_{k+1} = P_C \left(T(x_k)\right)$ where $P_C$ is the metric projection onto $C$.

Careful algorithmic treatment is required if an iterate lands on an anchor, necessitating modifications (such as the Vardi–Zhang approach or segment-based steps) to maintain descent and feasibility (Torres, 2012).

In broader settings, replacing points by convex sets and allowing some negative weights leads to difference-of-convex (d.c.) decompositions, solved by the DCA (Pham Dinh–Le Thi’s Difference-of-Convex Algorithm) combined with an inner generalized Weiszfeld solver (An et al., 2014). The iteration is then composed of outer DCA steps and inner projections following a generalized Weiszfeld mapping.

3. Existence, Uniqueness, and Stability

Existence of a constrained minimizer follows from coercivity ($\|x\| \to \infty \implies f(x) \to +\infty$) and lower semicontinuity of $f$. For non-collinear anchors and strictly positive weights, uniqueness is guaranteed by the strict convexity of $f$ on $C$ (Nguyen, 2018, Torres, 2012).

Stability under perturbations of the anchor points is formalized via the solution map $M: \mathcal{A} \to C$, $M(a) = \arg\min_{x \in C} f(x; a)$ on the parameter set $\mathcal{A}$ of non-collinear anchors. $M(\cdot)$ is continuous; the optimal-value function is convex and locally Lipschitz in the anchor positions, which ensures well-posedness of the problem (Nguyen, 2018).

4. Convergence Theory

For the projected Weiszfeld iteration, convergence to the unique minimizer is guaranteed under standard assumptions (non-collinearity, positive weights, appropriate handling at anchors, and closed, convex constraints). Each step monotonically decreases the objective except at fixed points. The limit points of the sequence are characterized as solutions to the subgradient optimality conditions (Nguyen, 2018, Torres, 2012).

For more general d.c. settings (e.g., involving distances to convex sets and possibly negative weights), the inner generalized Weiszfeld steps converge linearly due to strict convexity of the auxiliary subproblems; outer DCA steps provide global objective decrease, and cluster points are critical points of the original nonconvex problem (An et al., 2014).

Constrained Weiszfeld algorithms can be extended to Hilbert spaces, and all projected iterates remain feasible when the projection operator is well defined (Nguyen, 2018).

5. Generalizations and Algorithmic Variants

Classical generalization involves replacing Euclidean anchors $a_i$ by arbitrary closed convex sets $C_i$, minimizing

$\min_{x \in S} \sum_{i=1}^m w_i d(x; C_i)$

where $d(x; C_i)$ denotes the Euclidean distance from $x$ to $C_i$. With only positive weights, this is convex and can be solved by a projected generalized Weiszfeld iteration (An et al., 2014).

Inclusion of negative weights or more general d.c. objectives requires DCA-based schemes, with inner subproblems attacked via the generalized Weiszfeld approach and projection (An et al., 2014). In the limiting case where all sets are singleton points and all weights are positive, these methods specialize to the standard constrained Weiszfeld algorithm.

Table: Algorithmic Approaches for the Constrained Weiszfeld Problem

Setting	Primary Algorithm	Key Features
Points, positive weights	Projected Weiszfeld	Fixed-point, projection
Convex sets, positive weights	Gen. Weiszfeld + Proj.	Subgradient, projection
Signed weights, convex sets	DCA + Gen. Weiszfeld	d.c. split, majorization

6. Numerical and Practical Observations

Practical implementation in finite-dimensional spaces typically yields rapid convergence (to machine precision in dozens of iterations) for standard facility location scenarios (Nguyen, 2018). Numerical experiments confirm the descent and feasibility guarantees, and the superior performance (in terms of optimality) over standard nonlinear programming solvers such as MATLAB’s fmincon has been observed for large-scale two-dimensional instances (Torres, 2012).

A plausible implication is that the projected or generalized Weiszfeld-type methods are preferable in high-dimensional or large-scale instances with strict feasibility and descent requirements.

7. Extensions and Open Directions

Research avenues include relaxing convexity of the constraint set, replacing the Euclidean metric with Bregman or other non-Euclidean divergences, and developing inertial or Nesterov-type acceleration schemes. Dynamic and online scenarios, where the anchor points themselves are time-varying, present additional challenges for real-time optimization. The extension to infinite-dimensional Hilbert spaces opens applications in distributed systems and functional analytic contexts (Nguyen, 2018).

Modeling frameworks with negative weights and set-valued distances confer additional flexibility for complex multi-facility or competitive location environments (An et al., 2014), but introduce nonconvexity necessitating advanced algorithmic techniques.

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References:

• (Nguyen, 2018) S.D. Nguyen, "Constrained Fermat-Torricelli-Weber Problem in real Hilbert Spaces"
• (An et al., 2014) "A D.C. Algorithm via Convex Analysis Approach for Solving a Location Problem Involving Sets"
• (Torres, 2012) "A Weiszfeld-like algorithm for a Weber location problem constrained to a closed and convex set"