Extended Power Weiszfeld Method

• Extended Power Weiszfeld is a robust optimization algorithm that minimizes q-th power distances, generalizing the geometric median and weighted least squares.
• It introduces a desingularization subgradient strategy to escape singularities, ensuring well-defined iterative updates and global convergence.
• Empirical results, especially in portfolio optimization, show that intermediate q values balance robustness and convergence speed for enhanced performance.

The Extended Power Weiszfeld method is a de-singularity subgradient approach for solving the extended Weber location problem, formulated as the minimization of a sum of $q$-th power distances between a variable point and a collection of weighted data points: $$f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q, \qquad 1 \leq q < 2,\quad x\in\mathbb{R}^d,$$ where $w_i > 0$ are positive weights, $a_i\in\mathbb{R}^d$ are data points, and the objective $f$ is strictly convex under non-collinearity assumptions. This formulation generalizes the geometric median ($q=1$) and weighted least squares ($q=2$) problems. Classical iterative solutions, such as the Weiszfeld algorithm, experience a breakdown at singularities (data points) for $q < 2$. The Extended Power Weiszfeld method introduces systematic desingularization through a subgradient-based remedy, ensuring well-defined iterative progress and global convergence, including at data points.

1. Formulation and Singularity of the Extended Weber Problem

The extended Weber location problem seeks the unique minimizer $x_*$ of

$f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q,$

with $$f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q, \qquad 1 \leq q < 2,\quad x\in\mathbb{R}^d,$$0. For $$f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q, \qquad 1 \leq q < 2,\quad x\in\mathbb{R}^d,$$1 for all $$f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q, \qquad 1 \leq q < 2,\quad x\in\mathbb{R}^d,$$2, the gradient is

$$f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q, \qquad 1 \leq q < 2,\quad x\in\mathbb{R}^d,$$3

The standard "power–Weiszfeld" update is defined as

$$f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q, \qquad 1 \leq q < 2,\quad x\in\mathbb{R}^d,$$4

For $$f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q, \qquad 1 \leq q < 2,\quad x\in\mathbb{R}^d,$$5, if an iterate lands on $$f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q, \qquad 1 \leq q < 2,\quad x\in\mathbb{R}^d,$$6, then $$f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q, \qquad 1 \leq q < 2,\quad x\in\mathbb{R}^d,$$7, causing the update to be undefined ("breakdown"). This singularity obstructs convergence at and near data points.

2. The De-Singularity Subgradient Approach

To address the breakdown, a desingularized subgradient strategy is adopted.

Subgradient at a Data Point

For $$f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q, \qquad 1 \leq q < 2,\quad x\in\mathbb{R}^d,$$8, isolate $$f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q, \qquad 1 \leq q < 2,\quad x\in\mathbb{R}^d,$$9 as $w_i > 0$0, with $w_i > 0$1. Define the $w_i > 0$2-power desingularity subgradient as

$w_i > 0$3

For $w_i > 0$4, the subdifferential at $w_i > 0$5 is the singleton $w_i > 0$6; for $w_i > 0$7 it is the classical $w_i > 0$8-median subgradient set.

Escape from a Singular Point

If $w_i > 0$9 and $a_i\in\mathbb{R}^d$0, move by

$a_i\in\mathbb{R}^d$1

with $a_i\in\mathbb{R}^d$2 chosen small enough to ensure strict decrease in $a_i\in\mathbb{R}^d$3. An initial $a_i\in\mathbb{R}^d$4 can be set as

$a_i\in\mathbb{R}^d$5

and then backtracked ($a_i\in\mathbb{R}^d$6 with $a_i\in\mathbb{R}^d$7) until $a_i\in\mathbb{R}^d$8 decreases.

Unified Iteration Scheme

The overall method operates as follows: $a_i\in\mathbb{R}^d$9 with backtracking ensuring $f$0 at each step (Lai et al., 2024).

3. Convergence Properties

Global Convergence

Under the assumptions $f$1 and non-collinear $f$2, the sequence $f$3 generated by this rule exhibits:

• Monotonic decrease: $f$4 until termination.
• Fixed points: Only possible at the data points $f$5 and the unique minimizer $f$6.
• Any non-optimal data point can be visited at most once; the escape step immediately returns the iterate to the regular region.
• Global convergence: $f$7 from any starting point.

Superlinear Local Convergence at Singular Minima

If the unique minimizer $f$8 coincides with a data point $f$9, then as $q=1$0,

$q=1$1

Thus, near a singular minimizer and for $q=1$2,

$q=1$3

implying a superlinear convergence rate (Lai et al., 2024).

4. Algorithmic Structures

A structured summary of the procedure is as follows:

Scenario	Update Step	Stopping Criterion
$q=1$4	$q=1$5	N/A
$q=1$6, $q=1$7	$q=1$8 (optimum found)	$q=1$9
$q=2$0, $q=2$1	$q=2$2 (with backtracking on $q=2$3 until $q=2$4 decreases)	N/A

Typical parameters are backtracking factor $q=2$5 and tolerance $q=2$6.

5. Comparisons across Exponent Choices

• $q=2$7: Reduces to the classical Weiszfeld algorithm ("geometric median") with the Kuhn–Weiszfeld escape step for singularities.
• $q=2$8: Exact minimizer in one step at the weighted Euclidean mean.
• $q=2$9: Interpolates between robustness ($q < 2$0) and speed ($q < 2$1), achieving faster (superlinear) convergence near singular minima, with an automatically scaled linesearch step.

A plausible implication is that selecting intermediate values $q < 2$2–$q < 2$3 provides a balance between robustness to outliers and iterative efficiency.

6. Empirical Evaluation in Portfolio Optimization

In online portfolio selection:

• At each trading day, compute the $q < 2$4-median of the last $q < 2$5 price-relative vectors via this method and use as forecast in a reversion strategy.
• Evaluated on NYSE(N) ($q < 2$6, $q < 2$7) and CSI300 (Chinese index, $q < 2$8, $q < 2$9), for $x_*$0 or $x_*$1, over $x_*$2 to $x_*$3.
• The algorithm does not stall and escapes singularities efficiently ($x_*$42–3 inner backtrackings).
• Iteration count is modest ($x_*$5), CPU time per median below $x_*$6 ms.
• Empirical rate of convergence decreases from $x_*$7 at $x_*$8 to $x_*$9 at $f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q,$0.
• Using $f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q,$1-median forecasts in reversion portfolios, intermediate $f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q,$2 ($f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q,$3--$f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q,$4) often yields better cumulative returns and Sharpe ratios than both $f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q,$5 and $f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q,$6 (Lai et al., 2024).

7. Practical Recommendations and Significance

• The de-singularity escape step guarantees robustness against iterate stalling and adds negligible computational overhead.
• For online learning, machine learning, and optimization applications that generalize the median/minimum norm paradigm, intermediate values of $f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q,$7 are recommended for an optimal robustness–convergence trade-off.
• Theoretical guarantees include monotonic decrease, avoidance of singularity lock-in, and superlinear local convergence at most problematic cases.

The Extended Power Weiszfeld method thus advances both theoretical analysis and practical computation for the extended Weber location problem in the non-quadratic, nonconvex setting of $f(x) = \sum_{i=1}^n w_i \|x - a_i\|^q,$8 (Lai et al., 2024).