Proximal Point-Type Algorithm Overview

• Proximal point-type algorithms are iterative methods that regularize optimization problems using a proximally defined divergence to guarantee convergence even in nonconvex settings.
• They employ techniques such as abstract Bregman divergences and mirror descent to effectively handle problems in non-Euclidean spaces and variational inequalities.
• Convergence is ensured under strict conditions on step sizes, prox-function properties, and the algebraic structure of abstract convexity frameworks.

A proximal point-type algorithm is an iterative scheme for minimizing (or finding zeros of) a function or operator by successively regularizing the original problem with a proximally defined term, typically involving a divergence or metric. Originally developed for convex minimization, such algorithms have been generalized to abstract convexity frameworks, variational inequalities, monotone inclusions, operator splitting, and constraint satisfaction over nonlinear spaces. The unifying principle is to define and solve at each iteration a proximal (or regularized) subproblem that approximates or regularizes the original objective, yielding robust convergence properties even in settings beyond classical convexity.

1. Abstract Convexity and Proximal Regularization

In the general setting of abstract convex minimization, let $X$ be an arbitrary nonempty set and $L$ a family of "abstract linear" functions on $X$ (e.g., linear or quadratic forms, or other structured functionals). A function $f : X \to \mathbb{R}\cup\{+\infty\}$ is $L$-convex if $f(x) = \sup\{u(x) : u \in U\}$ for some $U\subset L$ for all $x\in X$. The generic problem is

$$\text{Minimize } f(x) \quad \text{subject to} \quad x\in X,$$

with $f$ proper and $L$-convex.

A key tool is the abstract Bregman divergence: for an $L$-convex prox-function $\varphi$ and subgradient $\lambda\in\partial_L\varphi(y)$ at $y$, the divergence is

$$D^\lambda_\varphi(x,y) = \varphi(x) - \varphi(y) - [\lambda(x)-\lambda(y)].$$

This generalizes the Euclidean squared norm and Bregman divergences to nonvectorial or functionally abstract settings. The divergence determines the geometry of the proximal regularization.

In this framework, two principal proximal point-type methods emerge (Millán et al., 2024):

• Bregman proximal-point algorithm: Each iteration solves

$$x_{k+1} \in \arg\min_{x\in X} \big\{ f(x) + \frac{1}{c_k} D^{\lambda_k}_\varphi(x,x_k) \big\},$$

and updates the subgradient $\lambda_{k+1} = \lambda_k - c_k g_{k+1}$ for $g_{k+1} \in \partial_L f(x_{k+1})$.

• Mirror descent algorithm: Each iteration linearizes $f$ and solves

$$x_{k+1} \in \arg\min_{x\in X} \big\{u_k(x) + \frac{1}{c_k} D^{\lambda_k}_\varphi(x_k, x)\big\}$$

for $u_k \in \partial_L f(x_k)$, with a similar subgradient update.

The core assumption for convergence is that $L$ is closed under addition (often a linear space), which allows duality-based update rules.

2. Algorithmic Structure and Convergence Principles

For convergence, the following are crucial:

• The family $L$ of abstract linear functionals must be at least closed under addition, ensuring the algebraic properties necessary for subdifferential calculus and update rules (Millán et al., 2024).
• The stepsize sequence $\{c_k\}$ must satisfy $\sum_k c_k = +\infty$, ensuring sufficient progress.
• The prox-function $\varphi$ must (i) induce bounded sublevel sets for its Bregman divergence, and (ii) be strongly (or nearly strictly) $L$-convex so that $D^\lambda_\varphi(x,y)=0$ implies $x=y$.
• Existence of a minimizer $x^*$ with finite initial divergence ensures feasibility.

The main technical device is a telescoping sum derived from a triangle identity for the divergence, yielding descent properties for the sequence $\{f(x_k)\}$: $$s_N f(x) - \sum_{k=0}^N c_k f(x_{k+1}) \ge - D^{\lambda_0}_\varphi(x, x_0),$$ where $s_N=\sum_{k=0}^N c_k$. This ensures $\lim_{k\to\infty} f(x_k) = \inf_{x\in X} f(x)$. Any cluster point of $\{x_k\}$ is a global minimizer.

3. Abstract Frameworks and Generalizations

The proximal point paradigm can be unified across Hilbert spaces, CAT(0) metric spaces (Hadamard manifolds), convex cones, and spaces of abstract convexity (Leustean et al., 2017, Bacak, 2012). The key algebraic property is firm nonexpansivity: resolvent-type maps $T_k$ must contract with respect to a suitable metric or divergence, typically encoded through a family property such as: $$d(T_n x, T_m y) \le d((1-\alpha)x+\alpha T_n x, (1-\beta)y+\beta T_m y)$$ for jointly firmly nonexpansive maps (Leustean et al., 2017).

This abstraction enables:

• Formulation in non-Euclidean spaces, e.g., geodesic metric settings (Bacak, 2012).
• Use of general Bregman distances or functional divergences (Millán et al., 2024).
• Application beyond convex minimization, to monotone inclusions, equilibrium problems, and saddle point systems.

The convergence theorems then generalize: in CAT(0) complexes or with uniform monotonicity, strong convergence can be established; otherwise, weak (Δ-)convergence is obtained.

4. Nonconvex and Non-Euclidean Applications

Proximal point-type algorithms are not restricted to convex optimization. For $L$-convex but nonconvex functions (such as $f(x)$ being a max of quadratic or piecewise terms), the proximal subproblems remain well-posed if the prox-function ensures strict $L$-convexity and boundedness of sublevel sets. This extends the method to a broad spectrum of variational and optimization problems outside classical convexity.

Examples given in (Millán et al., 2024) demonstrate that for nonconvex, but $L$-convex, univariate problems, both the proximal-point and mirror-descent algorithms with constant or diminishing stepsizes rapidly converge to the global solution, outperforming generic (non-proximal) descent approaches.

5. Connections to Operator Theory and Related Paradigms

Proximal point-type methods underlie diverse algorithmic frameworks:

• In monotone operator theory, the method is synonymous with iterative resolvent evaluation $x_{k+1} = (I + \lambda_k T)^{-1}(x_k)$, where $T$ is maximal monotone.
• In splitting and saddle-point algorithms (Douglas–Rachford, ADMM), the update can be expressed as a proximal point in a product or preconditioned metric.
• Bregman and mirror descent algorithms can be seen as special cases in which the prox-term is linearized, corresponding to asymmetric or non-quadratic divergences.

Many accelerated schemes (such as Nesterov acceleration) can be cast as discretized or linearized approximations of a proximal point method with large (or variable) proximity parameters, elucidating their robustness and stability properties.

6. Illustrative Example: Abstract Bregman Proximal-Point Algorithm

The workflow for the algorithm (Millán et al., 2024) is summarized here:

Step	Operation	Key Condition
1	$$x_{k+1} = \arg\min_x\big\{f(x) + \frac{1}{c_k} D_{\varphi}^{\lambda_k}(x,x_k)\big\}$$	Well-posedness of prox
2	Choose $g_{k+1}\in \partial_L f(x_{k+1})$	$\partial_L f(x)$ nonempty
3	$\lambda_{k+1} = \lambda_k - c_k g_{k+1}$	$L$ closed under $+$
4	If $0\in\partial_L f(x_{k+1})$ then terminate	Optimality

Convergence (to a global minimizer) is guaranteed under the standing hypotheses of $L$-convexity, step size summability, prox-boundedness, and algebraic closure of $L$.

These results extend the reach of classical proximal point ideas to extremely general classes of minimization problems using only the algebraic structure of the subdifferentials and the geometry of the divergence.

7. Practical and Numerical Implications

Numerical results in (Millán et al., 2024) support the theoretical assertions. For a nonconvex, non-Euclidean $L$-convex minimization (with a prox-function $\varphi(x) = -|x|$ and step sizes $c_k \equiv 0.1$ or $1/k$), both algorithms reliably drive $f(x_k)$ to the global minimum within a handful of steps (typically $4$–$6$). Mirror-descent exhibits more initial variation in its iterates but (under $L$-convexity and parameter conditions) ultimately converges.

Observed behaviors:

• Proximal-point iterates track the minimum monotonically.
• Constant stepsizes may stabilize convergence more rapidly than diminishing stepsizes for some problems.
• Choice of $L$-convex prox-function is critical: the induced divergence must ensure boundedness and strictness.

This demonstrates that the abstract proximal point algorithm, and its Bregman-type and mirror-descent variants, offer a practical, robust, and theoretically sound framework for minimizing not just standard convex functions, but mathematical objects defined by much looser convexity and differentiability properties (Millán et al., 2024).