{ "id": "epf3gth3nd", "slug": "proximal-point-algorithm", "title": "Proximal point algorithm", "updated": "2026-10-15", "topic_path": [ { "id": "technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/topics/technology" }, { "id": "technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing" }, { "id": "technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai" }, { "id": "technology.computing.ai.algorithms", "label": "Algorithms and computational methods", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.algorithms" }, { "id": "technology.computing.ai.optimization_dp", "label": "Optimization and dynamic programming", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.optimization_dp" }, { "id": "technology.computing.ai.optimization_dp.mathematical-programming-methods", "label": "Mathematical programming methods", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.optimization_dp.mathematical-programming-methods" } ], "geo": [ { "id": "geo.nongeo.t1946.technology.computing.ai.optimization_dp", "label": "Non-geographic · 1946 to 2000: Optimization and dynamic programming", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.optimization_dp", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology" }, { "id": "geo.nongeo.t1946.technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing" }, { "id": "geo.nongeo.t1946.technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai" }, { "id": "geo.nongeo.t1946.technology.computing.ai.algorithms", "label": "Algorithms and computational methods", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.algorithms" }, { "id": "geo.nongeo.t1946.technology.computing.ai.optimization_dp", "label": "Optimization and dynamic programming", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.optimization_dp" } ] }, { "id": "geo.nongeo.t2001.technology.computing.ai.optimization_dp", "label": "Non-geographic · 2001 to 2020: Optimization and dynamic programming", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.optimization_dp", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t2001", "label": "Non-geographic · 2001 to 2020", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001" }, { "id": "geo.nongeo.t2001.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology" }, { "id": "geo.nongeo.t2001.technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing" }, { "id": "geo.nongeo.t2001.technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai" }, { "id": "geo.nongeo.t2001.technology.computing.ai.algorithms", "label": "Algorithms and computational methods", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.algorithms" }, { "id": "geo.nongeo.t2001.technology.computing.ai.optimization_dp", "label": "Optimization and dynamic programming", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.optimization_dp" } ] } ], "excerpt": "The proximal point algorithm, or proximal point method, is an iterative optimization method that repeatedly minimizes a convex objective plus a quadratic penalty centered at the current iterate.", "snippet": "The proximal point algorithm, or proximal point method, is an iterative optimization method that repeatedly minimizes a convex objective plus a quadratic penalty centered at the current iterate.", "node": "technology.computing.ai.optimization_dp.mathematical-programming-methods", "markdown": "# Proximal point algorithm\n\nThe proximal point algorithm (PPA) is an iterative method for minimizing a convex function or, more generally, finding a zero of a maximal monotone operator, by repeatedly minimizing the objective plus a quadratic penalty centered at the current iterate. One exact step takes \\( x_{k+1} = \\operatorname{argmin}_{z}\\, f(z) + \\tfrac{1}{2c_k}\\|z - x_k\\|^2 \\)\n\nor equivalently applies the resolvent when \\( T = \\partial f \\).[1](https://doi.org/10.1137/0314056) PPA is among the fundamental algorithms for monotone zero-finding and serves as a general framework for analyzing the convergence of many other algorithms, while remaining efficient on some structured problems in its own right.[2](https://encyclopediaofmath.org/wiki/Proximal_point_methods_in_mathematical_programming) • [3](https://www.jorsc.shu.edu.cn/EN/10.1007/s40305-021-00352-x)\n\n| Key fact | Detail |\n|---|---|\n| One iteration | Minimize \\( f(z) + \\tfrac{1}{2c_k}\\|z - z_k\\|^2 \\), a strongly convex subproblem with a unique solution[1](https://doi.org/10.1137/0314056) • [4](https://proceedings.mlr.press/v242/liao24a/liao24a.pdf) |\n| Operator view | the resolvent of a maximal monotone operator \\( T \\)[1](https://doi.org/10.1137/0314056) |\n| Step-size condition | For a proper, lower semicontinuous convex objective with a minimizer, PPA converges in general if and only if \\( \\sigma_n = \\sum_{k=1}^{n} \\lambda_k \\to \\infty \\)[5](https://epubs.siam.org/doi/10.1137/0329022) |\n| General rate | Sublinear \\( O(1/k) \\) on the function-value gap with constant step size[4](https://proceedings.mlr.press/v242/liao24a/liao24a.pdf) |\n| Linear rate | Under strong monotonicity, PL, error bound, or quadratic growth conditions[4](https://proceedings.mlr.press/v242/liao24a/liao24a.pdf) • [2](https://encyclopediaofmath.org/wiki/Proximal_point_methods_in_mathematical_programming) |\n| Origin | Proximity operator: Moreau (1962); algorithm: Martinet; general monotone-operator theory: Rockafellar (1976)[6](https://ar5iv.labs.arxiv.org/html/0912.3522) • [7](https://www.math.uni-frankfurt.de/~baumeist/Nonex-Kap8.pdf) • [1](https://doi.org/10.1137/0314056) |\n| Main practical drawback | Each iteration requires minimizing \\( f \\) plus a quadratic[8](https://stanford.edu/~boyd/papers/pdf/prox_algs.pdf) |\n\n## How it works\n\nThe proximal operator of \\( f \\) is \\( \\operatorname{prox}_{f}(v) = \\arg\\min_{x}\\, f(x) + \\tfrac{1}{2}\\|x - v\\|^{2} \\), with the scaled version \\( \\operatorname{prox}_{\\lambda \\cdot f}(v) = \\arg\\min_{x}\\, f(x) + \\tfrac{1}{2\\lambda}\\|x - v\\|^{2} \\).[8](https://stanford.edu/~boyd/papers/pdf/prox_algs.pdf) The quadratic term makes the subproblem strongly convex, so a unique solution always exists even when \\( f \\) itself is nonsmooth.[4](https://proceedings.mlr.press/v242/liao24a/liao24a.pdf) For \\( f \\in \\Gamma_{0}(\\mathbb{R}^N) \\), the solution \\( p = \\operatorname{prox}_{f}x \\) is characterized by the inclusion \\( x - p \\in \\partial f(p) \\), which reduces to \\( x - p = \\nabla f(p) \\) when \\( f \\) is differentiable.[6](https://ar5iv.labs.arxiv.org/html/0912.3522)\n\nA minimizer \\( x^{\\star} \\) of \\( f \\) is exactly a fixed point of \\( \\operatorname{prox}_{f} \\), and the proximal operator is firmly nonexpansive, a property sufficient for fixed point iteration to converge.[8](https://stanford.edu/~boyd/papers/pdf/prox_algs.pdf) • [7](https://www.math.uni-frankfurt.de/~baumeist/Nonex-Kap8.pdf) The Moreau envelope, or Moreau–Yosida regularization, is the scalar-valued function \\( e_s f(x) = \\inf_{y}\\, f(y) + \\tfrac{1}{2s}\\|x - y\\|^{2} \\); for convex \\( f \\) it is related to the proximal map by \\( \\nabla e_s f(x) = (x - \\operatorname{prox}_{s,f}(x))/s \\), and it is a regularized version of \\( f \\) that approximates it from below and shares its minimizing values.[7](https://www.math.uni-frankfurt.de/~baumeist/Nonex-Kap8.pdf) • [9](https://arxiv.org/pdf/1502.03175v3.pdf) In monotone operator theory, the resolvent \\( (I + c \\cdot T)^{-1} \\) converts the zero-finding problem \\( 0 \\in T(z) \\) into a fixed point equation, which is how the proximal point method is introduced.[10](https://web.stanford.edu/~boyd/papers/pdf/monotone_primer.pdf)\n\n## How it is done\n\nThe basic iteration is \\( x_{k+1} := \\operatorname{prox}_{\\lambda_k \\cdot f}(x_k) \\).[8](https://stanford.edu/~boyd/papers/pdf/prox_algs.pdf) The practitioner must choose the step-size sequence: convergence is guaranteed for \\( \\lambda_k > 0 \\) with \\( \\sum_k \\lambda_k = \\infty \\),[8](https://stanford.edu/~boyd/papers/pdf/prox_algs.pdf) and Güler showed that convergence in general holds if and only if \\( \\sigma_n = \\sum_{k=1}^{n} \\lambda_k \\to \\infty \\), with global rate estimates for the residual \\( f(x_n) - f(u) \\).[5](https://epubs.siam.org/doi/10.1137/0329022)\n\nThe inner subproblem need not be solved exactly. Convergence is preserved when the computed point satisfies \\( \\|x^{k+1} - \\widetilde{x}^{k}\\| \\le \\varepsilon_k \\) with \\( \\sum_k \\varepsilon_k < \\infty \\),[2](https://encyclopediaofmath.org/wiki/Proximal_point_methods_in_mathematical_programming) and Rockafellar's convergence theorem likewise allows approximate resolvent evaluations so long as the sum of all errors is finite.[11](https://faculty.engineering.asu.edu/bertsekas/wp-content/uploads/sites/129/2020/03/Eckstein_Bertsekas_D-R.pdf) For the realization of the iteration itself, all methods of fixed point iteration and alternating projection methods may be applied.[7](https://www.math.uni-frankfurt.de/~baumeist/Nonex-Kap8.pdf) When the proximal step is computed by an iterative subroutine, convergence to a neighborhood is preserved, with the radius increased by at least \\( 1/(1-c) \\) and the allowable step size reduced by \\( (1-c) \\).[12](https://opt-ml.org/papers/2025/paper94.pdf)\n\n## Origin\n\nThe proximity operator is an extension of the notion of a projection operator, and proximal operators took their current name and form in 1960s work.[6](https://ar5iv.labs.arxiv.org/html/0912.3522) • [8](https://stanford.edu/~boyd/papers/pdf/prox_algs.pdf) The algorithm was later generalized.[7](https://www.math.uni-frankfurt.de/~baumeist/Nonex-Kap8.pdf) who extended proximal minimization to finding the zero of an arbitrary maximal monotone operator in his 1976 paper *Monotone Operators and the Proximal Point Algorithm* in the SIAM Journal on Control and Optimization; Rockafellar notes that Martinet had already obtained convergence results for constant step sizes, with weak convergence to a zero of \\( T \\).[1](https://doi.org/10.1137/0314056) The method's origin also traces back to the study of regularization of ill-posed problems and is closely related to Moreau–Yosida convolution.[2](https://encyclopediaofmath.org/wiki/Proximal_point_methods_in_mathematical_programming)\n\n## Variants\n\n**Proximal gradient.** For \\( f = l + \\phi \\) with \\( l \\) smooth, one takes a gradient step \\( v_t = x_t - \\gamma \\nabla l(x_t) \\) followed by \\( x_{t+1} = \\operatorname{prox}_{\\gamma \\cdot \\phi}(v_t) \\); with fixed \\( \\gamma = 1/\\lambda_l \\) the rate is \\( 1/t \\).[9](https://arxiv.org/pdf/1502.03175v3.pdf) This scheme converges at \\( O(1/k) \\) for \\( \\lambda \\in (0, 1/L] \\) when \\( \\nabla l \\) is \\( L \\)-Lipschitz.[8](https://stanford.edu/~boyd/papers/pdf/prox_algs.pdf) Widely studied as iterative shrinkage thresholding (IST), it was accelerated by Beck and Teboulle's FISTA, motivated by the slow convergence of standard IST methods;[9](https://arxiv.org/pdf/1502.03175v3.pdf) • [13](https://doi.org/10.1137/080716542) accelerated schemes add an extrapolation step \\( y_{k+1} := x_k + \\omega_k \\cdot (x_k - x_{k-1}) \\) and achieve the optimal \\( O(1/n^2) \\) objective rate.[8](https://stanford.edu/~boyd/papers/pdf/prox_algs.pdf) • [6](https://ar5iv.labs.arxiv.org/html/0912.3522)\n\n**Splitting and duality.** The Douglas–Rachford iteration \\( x_{k+1} := \\operatorname{prox}_{\\lambda \\cdot f}(z_k - u_k) \\), \\( z_{k+1} := \\operatorname{prox}_{\\lambda \\cdot g}(x_{k+1} + u_k) \\), \\( u_{k+1} := u_k + x_{k+1} - z_{k+1} \\) is connected to PPA for maximal monotone operators by the Eckstein–Bertsekas analysis;[8](https://stanford.edu/~boyd/papers/pdf/prox_algs.pdf) • [14](https://doi.org/10.1007/bf01581204) the Peaceman–Rachford algorithm is its limiting case with relaxation parameter \\( \\lambda_n \\equiv 2 \\), where the iteration becomes the composition of the two reflections \\( y^{k+1} = R_{\\gamma F} R_{\\gamma G} y^k \\).[6](https://ar5iv.labs.arxiv.org/html/0912.3522) • [26](https://ar5iv.labs.arxiv.org/html/1301.0542) The method of multipliers is a dual application of PPA, and ADMM is also an application of PPA via the Douglas–Rachford connection.[15](https://ar5iv.labs.arxiv.org/html/1905.04537) When \\( T \\) is the subdifferential of a dual objective, the proximal sequence coincides with the dual sequence of the augmented Lagrangian method,[2](https://encyclopediaofmath.org/wiki/Proximal_point_methods_in_mathematical_programming) and Rockafellar's 1976 *Mathematics of Operations Research* paper derives generalized rate-of-convergence results for the method of multipliers.[16](https://doi.org/10.1287/moor.1.2.97)\n\n**Proximal Newton and stochastic forms.** Proximal Newton methods use second-order or Hessian approximations and can handle some nonconvex problems.[9](https://arxiv.org/pdf/1502.03175v3.pdf) In stochastic settings, the variance-reduced update is \\( x^{k+1} = \\operatorname{prox}_{\\alpha \\cdot f_{i_k}}(x^k + \\alpha \\cdot e^k) \\),[17](https://link.springer.com/article/10.1007/s10957-024-02502-6) and a 2024 Journal of Optimization Theory and Applications paper gives the first unified study of variance reduction for stochastic proximal point algorithms, proving \\( O(1/k) \\) convergence for convex functions with constant step size and linear convergence under the Polyak–Łojasiewicz condition.[17](https://link.springer.com/article/10.1007/s10957-024-02502-6) The inexact stochastic PPA analyzed at ICLR 2025 converges almost surely without smoothness or strong convexity, and with constant steps \\( \\alpha_0 \\) it converges linearly to an \\( O(\\alpha_0) \\) neighborhood of the solution set.[18](https://proceedings.iclr.cc/paper_files/paper/2025/file/b481e303ee0107017826e3493f3ddf10-Paper-Conference.pdf)\n\n**Federated forms.** S-DANE, a stabilized distributed proximal-point method, keeps DANE's best-known communication complexity among non-accelerated methods with a milder subproblem accuracy condition and supports partial client participation.[19](https://proceedings.neurips.cc/paper_files/paper/2024/file/b4b75092bb44a14815be33d052aa47f5-Paper-Conference.pdf) FedExProx analyzes extrapolated FedProx-type methods, proving convergence for any \\( 0 < \\gamma < +\\infty \\) with optimal extrapolation parameter \\( \\alpha_{\\gamma,\\tau} = 1/(\\gamma \\cdot L_{\\gamma,\\tau}) > 1 \\), and includes FedProx as the special case \\( \\alpha = 1 \\).[20](https://proceedings.neurips.cc/paper_files/paper/2024/file/e0b6f389739496e363a89155c9448a8a-Paper-Conference.pdf)\n\n**Broximal point.** The 2025 Ball-Proximal (\"Broximal\") Point Method replaces the quadratic distance penalty with a ball constraint; in the nonsmooth convex regime, where PPM is sublinear, BPM converges linearly and in a finite number of steps, retaining global guarantees under the weaker assumption of ball-convexity.[21](https://arxiv.org/html/2502.02002v1)\n\n## Applications\n\nProximal operators underpin nuclear norm and max norm problems, sparse inverse covariance selection, MAP inference in undirected graphical models, and loss minimization in machine learning.[8](https://stanford.edu/~boyd/papers/pdf/prox_algs.pdf) Numerical experiments on linear SVM, lasso, and elastic-net problems confirm the predicted linear convergence of PPM under the relevant regularity conditions.[4](https://proceedings.mlr.press/v242/liao24a/liao24a.pdf) The proximal point iteration is also used to derive faster stochastic empirical risk minimization methods.[22](https://proceedings.mlr.press/v37/frostig15.pdf)\n\n## Limitations and alternatives\n\nThe basic proximal point method has found few applications because each iteration requires minimizing \\( f \\) plus a quadratic; it is useful mainly when minimizing \\( f \\) alone is hard but minimizing \\( f \\) plus a quadratic is easier.[8](https://stanford.edu/~boyd/papers/pdf/prox_algs.pdf) Although originally designed for global reach in the monotone setting, versions of the algorithm have limitations that later work addresses with localized, inexact, trust-region, and variable-metric versions for nonconvex problems.[23](https://sites.math.washington.edu/~rtr/papers/rtr257-ConvergencePPA.pdf) For nonconvex composite problems, the prox-linear method's guarantee reduces, in the simplest setting, to the convergence guarantee of gradient descent, which is black-box optimal for \\( C^1 \\)-smooth nonconvex optimization.[24](https://sites.math.washington.edu/~ddrusv/proxpoint_arxiv.pdf) On a benchmark test problem, CVX took 15 iterations, proximal gradient 127, the accelerated method 23, and ADMM 20, illustrating that per-iteration cost and iteration count trade off across methods.[25](https://web.stanford.edu/~boyd/papers/pdf/prox_slides.pdf) No published head-to-head benchmark quantifies per-iteration cost against Newton-type or dual methods on ill-conditioned problems.\n\n## References\n\n1. [R. Tyrrell Rockafellar (1976). Monotone Operators and the Proximal Point Algorithm. SIAM Journal on Control and Optimization.](https://doi.org/10.1137/0314056)\n2. [Proximal point methods in mathematical programming (Encyclopedia of Mathematics)](https://encyclopediaofmath.org/wiki/Proximal_point_methods_in_mathematical_programming)\n3. [The Developments of Proximal Point Algorithms (Journal of the Operations Research Society of China, 2021/2022)](https://www.jorsc.shu.edu.cn/EN/10.1007/s40305-021-00352-x)\n4. [Error bounds, PL condition, and quadratic growth for weakly convex functions, and linear convergences of proximal point methods (ICML 2023)](https://proceedings.mlr.press/v242/liao24a/liao24a.pdf)\n5. [On the Convergence of the Proximal Point Algorithm for Convex Minimization (Güler, SIAM J. Control Optim.)](https://epubs.siam.org/doi/10.1137/0329022)\n6. [Proximal Splitting Methods in Signal Processing (Combettes & Pesquet)](https://ar5iv.labs.arxiv.org/html/0912.3522)\n7. [The proximal point algorithm (Baumeister, Nonex book manuscript chapter)](https://www.math.uni-frankfurt.de/~baumeist/Nonex-Kap8.pdf)\n8. [Proximal Algorithms (Parikh & Boyd, Foundations and Trends in Machine Learning)](https://stanford.edu/~boyd/papers/pdf/prox_algs.pdf)\n9. [Proximal Algorithms in Statistics and Machine Learning](https://arxiv.org/pdf/1502.03175v3.pdf)\n10. [A Monotone Operator Primer (Ryu & Boyd)](https://web.stanford.edu/~boyd/papers/pdf/monotone_primer.pdf)\n11. [On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators (Eckstein & Bertsekas)](https://faculty.engineering.asu.edu/bertsekas/wp-content/uploads/sites/129/2020/03/Eckstein_Bertsekas_D-R.pdf)\n12. [Revisiting Stochastic Proximal Point Methods: Generalized Smoothness and Similarity (OPT 2025 workshop)](https://opt-ml.org/papers/2025/paper94.pdf)\n13. [Amir Beck, Marc Teboulle (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences.](https://doi.org/10.1137/080716542)\n14. [Jonathan Eckstein, Dimitri P. Bertsekas (1992). On the Douglas, Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming.](https://doi.org/10.1007/bf01581204)\n15. [On the optimal linear convergence factor of the relaxed proximal point algorithm for monotone inclusion problems](https://ar5iv.labs.arxiv.org/html/1905.04537)\n16. [R. T. Rockafellar (1976). Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming. Mathematics of Operations Research.](https://doi.org/10.1287/moor.1.2.97)\n17. [Variance Reduction Techniques for Stochastic Proximal Point Algorithms (JOTA, 2024)](https://link.springer.com/article/10.1007/s10957-024-02502-6)\n18. [Tight Convergence Analysis of Inexact Stochastic Proximal Point Algorithm for Stochastic Composite Optimization (ICLR 2025)](https://proceedings.iclr.cc/paper_files/paper/2025/file/b481e303ee0107017826e3493f3ddf10-Paper-Conference.pdf)\n19. [Stabilized Proximal-Point Methods for Federated Optimization (NeurIPS 2024)](https://proceedings.neurips.cc/paper_files/paper/2024/file/b4b75092bb44a14815be33d052aa47f5-Paper-Conference.pdf)\n20. [FedExProx: Extrapolation and Adaptive Stepsizes for Federated Proximal Point (NeurIPS 2024)](https://proceedings.neurips.cc/paper_files/paper/2024/file/e0b6f389739496e363a89155c9448a8a-Paper-Conference.pdf)\n21. [The Ball-Proximal (\"Broximal\") Point Method: a New Algorithm, Convergence Theory, and Applications (arXiv, Feb 2025)](https://arxiv.org/html/2502.02002v1)\n22. [Un-regularizing: approximate proximal point and faster stochastic algorithms for empirical risk minimization (Frostig et al., ICML 2015)](https://proceedings.mlr.press/v37/frostig15.pdf)\n23. [Convergence of the proximal point algorithm with local maximal monotonicity (Rockafellar)](https://sites.math.washington.edu/~rtr/papers/rtr257-ConvergencePPA.pdf)\n24. [Proximal point method revisited (Drori, Drusvyatskiy et al.)](https://sites.math.washington.edu/~ddrusv/proxpoint_arxiv.pdf)\n25. [Proximal Algorithms (slides, Boyd)](https://web.stanford.edu/~boyd/papers/pdf/prox_slides.pdf)\n26. [ar5iv.labs.arxiv.org](https://ar5iv.labs.arxiv.org/html/1301.0542)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Algorithms and computational methods › Optimization and dynamic programming › Mathematical programming methods*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://stanford.edu/~boyd/papers/pdf/prox_algs.pdf", "https://web.stanford.edu/~boyd/papers/pdf/monotone_primer.pdf", "https://sites.math.washington.edu/~rtr/papers/rtr257-ConvergencePPA.pdf", "https://sites.math.washington.edu/~ddrusv/proxpoint_arxiv.pdf", "https://web.stanford.edu/~boyd/papers/pdf/prox_slides.pdf" ], "url": "https://www.edgepedia.org/proximal-point-algorithm", "markdown_url": "https://www.edgepedia.org/proximal-point-algorithm.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. 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Edgepedia Community License 1.0.", "speakable": "The proximal point algorithm, or proximal point method, is an iterative optimization method that repeatedly minimizes a convex objective plus a quadratic penalty centered at the current iterate." }