Circumcenter Smoothing Strategy

• Circumcenter smoothing strategy is a geometric acceleration method that replaces exact projections with surrogate circumcenter steps, yielding improved contraction rates.
• The method leverages recent iterates in Hilbert spaces by computing an equidistant circumcenter via a Gram matrix approach, enhancing both efficiency and stability.
• Empirical results, such as on ellipsoid intersections, demonstrate that this strategy consistently outperforms classical projection methods with finite termination guarantees.

The circumcenter smoothing strategy is an acceleration and stabilization technique in projection and reflection algorithms for convex feasibility problems that replaces exact projections with surrogate steps involving the circumcenter of recent iterates. This method leverages geometric properties of circumcenters—points equidistant from a finite set in a Hilbert space—to provide one-step best fits in the affine span of iterates, often yielding improved contraction and finite convergence guarantees compared to traditional projection algorithms. Its formalization and analysis have been advanced in the context of Hilbert spaces and convex feasibility by Bauschke–Ouyang–Wang and, more recently, in the construction of finitely convergent algorithms by using separating halfspaces and perturbation schedules.

1. Definition and Geometric Foundation

Given a real Hilbert space $\mathcal H$ with inner product $\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|$, the circumcenter of a finite set $S = \{x_1, x_2, \dots, x_m\} \subset \mathcal H$ is defined as the unique point $c \in \mathrm{aff}(S)$ (the affine hull of $S$) such that the distances $\|c - x_i\|$ are equal for all $i$. The circumcenter operator $C(S)$ is single-valued and well-defined if and only if the points of $S$ are in general (affine independent) position, i.e., $\langle\cdot,\cdot\rangle$0 (Bauschke et al., 2018). For $\langle\cdot,\cdot\rangle$1, existence of the circumcenter requires the points not be collinear.

Algebraically, the circumcenter $\langle\cdot,\cdot\rangle$2 is obtained by solving the linear system arising from the equidistance constraints: $\langle\cdot,\cdot\rangle$3 Using a basis $\langle\cdot,\cdot\rangle$4 of the differences $\langle\cdot,\cdot\rangle$5 and setting $\langle\cdot,\cdot\rangle$6, the coefficients $\langle\cdot,\cdot\rangle$7 solve

$\langle\cdot,\cdot\rangle$8

where $\langle\cdot,\cdot\rangle$9 and $\|\cdot\|$0.

2. The Circumcenter Smoothing and Acceleration Strategy

The circumcenter smoothing strategy ("CCS", Editor's term) is applied to sequences generated by iterative algorithms in real Hilbert spaces. At each step, a subset of recent iterates (typically 2 or 3) is collected; their circumcenter is computed and used as the next iterate. Specifically, given iterates $\|\cdot\|$1, form $\|\cdot\|$2 and set $\|\cdot\|$3. An explicit formula using the Gram matrix allows this computation efficiently whenever the points are in general position (Bauschke et al., 2018).

This strategy is particularly effective in reflection-projection methods (e.g., Douglas–Rachford, method of alternating projections) where the circumcenter step can yield a contraction in the direction of intersection or feasibility, reducing zig-zagging and accelerating convergence.

3. Classical and Smoothed Circumcenter Methods for Convex Feasibility

In the context of the Convex Feasibility Problem (CFP)—finding $\|\cdot\|$4 with $\|\cdot\|$5 closed convex—the classical circumcentered-reflection method (CRM) applies the circumcenter operator to points produced by compositions of reflections across the constraint sets. In product space reformulation, this reduces multi-set feasibility to a two-set problem in a higher-dimensional space (Behling et al., 2023).

However, CRM requires exact projections and may fail for non-affine sets or inexact arithmetic. To address this, the circumcenter smoothing strategy introduces surrogate sets. At each iteration, instead of projecting onto $\|\cdot\|$6, the algorithm projects onto a separating halfspace $\|\cdot\|$7 containing $\|\cdot\|$8. These halfspaces are constructed using subgradients and a positive perturbation parameter $\|\cdot\|$9, ensuring both computational tractability and that iterates are gradually pulled into the intersection.

4. Algorithmic Structure and Finite Convergence

A prototypical algorithm, referred to as the Perturbed Approximate Circumcenter Algorithm (PACA) (Behling et al., 2023), proceeds as follows:

1. Subgradient Computation: For each constraint $S = \{x_1, x_2, \dots, x_m\} \subset \mathcal H$0 and current iterate $S = \{x_1, x_2, \dots, x_m\} \subset \mathcal H$1, compute a subgradient $S = \{x_1, x_2, \dots, x_m\} \subset \mathcal H$2.
2. Halfspace Construction: Define

$S = \{x_1, x_2, \dots, x_m\} \subset \mathcal H$3

with $S = \{x_1, x_2, \dots, x_m\} \subset \mathcal H$4.

1. Projection and Correction: For each $S = \{x_1, x_2, \dots, x_m\} \subset \mathcal H$5, compute $S = \{x_1, x_2, \dots, x_m\} \subset \mathcal H$6. Average to obtain $S = \{x_1, x_2, \dots, x_m\} \subset \mathcal H$7.
2. Circumcenter Update: Compute the reflection factor

$S = \{x_1, x_2, \dots, x_m\} \subset \mathcal H$8

and set $S = \{x_1, x_2, \dots, x_m\} \subset \mathcal H$9.

The perturbation sequence $c \in \mathrm{aff}(S)$0 is nonincreasing, vanishes asymptotically, and must be chosen such that $c \in \mathrm{aff}(S)$1 to guarantee finite convergence under the Slater condition.

5. Convergence Properties and Theoretical Guarantees

Under a Slater assumption (existence of an interior point for all constraints), the circumcenter smoothing algorithm is Fejér* monotone relative to the feasible set and achieves finite termination if the perturbations decrease slowly enough that $c \in \mathrm{aff}(S)$2 (Behling et al., 2023). In contrast, classic CRM or projection algorithms using exact projections often converge only asymptotically and require strong regularity (such as affine constraints or an error bound).

In the classical setting of two closed subspaces $c \in \mathrm{aff}(S)$3 with nontrivial intersection, the circumcenter smoothing strategy applied to Douglas–Rachford sequences produces iterates that contract at least as fast as the cosine of the Friedrichs angle between $c \in \mathrm{aff}(S)$4 and $c \in \mathrm{aff}(S)$5, matching or improving the theoretical rate of the original method (Bauschke et al., 2018). The updated iterate is always the best approximation of the true solution in the affine hull of recent points, and the strategy generalizes to more than two constraint sets.

6. Practical Implementation and Empirical Observations

Implementation of circumcenter smoothing requires only subgradient oracles and simple vector operations, with projections onto halfspaces being explicit and numerically stable. The method avoids the potential numerical instability and computational cost of exact projections onto complex convex sets.

Empirical tests in the intersection of ellipsoids demonstrate that PACA consistently outperforms both classical circumcenter methods with exact projections and standard subgradient projection methods, achieving robust, rapid, and finite termination as predicted by theory (Behling et al., 2023). Choice of perturbation decay is critical: sequences such as $c \in \mathrm{aff}(S)$6 or $c \in \mathrm{aff}(S)$7 balance steady constraint tightening with eventual containment of a Slater point. Too rapid decay (e.g., $c \in \mathrm{aff}(S)$8) can delay convergence.

7. Comparison with Classical Methods and Limitations

Circumcenter smoothing extends the CRM framework by ensuring global well-posedness, feasibility monotonicity, and practical computability for general convex constraints. It offers provable acceleration and stabilization advantages whenever recent iterates are affinely independent. CRM without smoothing may fail or diverge if certain sets are not affine or exact projections are unavailable.

A limitation is that general-position assumptions are necessary for uniqueness and existence of the circumcenter. In the absence of affine independence or when iterates are nearly collinear, the circumcenter update may become ill-defined; in such cases, the algorithm must detect failure and revert to alternative updates.

Overall, the circumcenter smoothing strategy provides a principled and practical geometric framework for accelerating and regularizing projection-type algorithms for convex feasibility and related optimization problems, with finite convergence and theoretical robustness under mild constraint assumptions (Bauschke et al., 2018, Behling et al., 2023).