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Subgradient method with feasible inexact projections for constrained convex optimization problems

optimization 2022年第12期71卷 3515-3537页

作者： Aguiar, A. A. Ferreira, O. P. Prudente, L. F. Univ Fed Goias IME Goiania Go Brazil

In this paper, we propose a new inexact version of the projected subgradient method to solve nondifferentiable constrained convex optimization problems. The method combine epsilon-subgradient method with a procedure to obtain a feasible inexact projection onto the constraint set. Asymptotic convergence results and iteration-complexity bounds for the sequence generated by the method employing the well-known exogenous stepsizes, Polyak's stepsizes, and dynamic stepsizes are established.

关键词： Subgradient method feasible inexact projection constrained convex optimization

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Proximal alternating penalty algorithms for nonsmooth constrained convex optimization

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COMPUTATIONAL optimization AND APPLICATIONS 2019年第1期72卷 1-43页

作者： Quoc Tran-Dinh Univ North Carolina UNC Chapel Hill Dept Stat & Operat Res 333 Hanes HallCB 3260 Chapel Hill NC 27599 USA

We develop two new proximal alternating penalty algorithms to solve a wide range class of constrained convex optimization problems. Our approach mainly relies on a novel combination of the classical quadratic penalty, alternating minimization, Nesterov's acceleration, adaptive strategy for parameters. The first algorithm is designed to solve generic and possibly nonsmooth constrained convex problems without requiring any Lipschitz gradient continuity or strong convexity, while achieving the best-known -convergence rate in a non-ergodic sense, where k is the iteration counter. The second algorithm is also designed to solve non-strongly convex, but semi-strongly convex problems. This algorithm can achieve the best-known convergence rate on the primal constrained problem. Such a rate is obtained in two cases: (1) averaging only on the iterate sequence of the strongly convex term, or (2) using two proximal operators of this term without averaging. In both algorithms, we allow one to linearize the second subproblem to use the proximal operator of the corresponding objective term. Then, we customize our methods to solve different convex problems, and lead to new variants. As a byproduct, these algorithms preserve the same convergence guarantees as in our main algorithms. We verify our theoretical development via different numerical examples and compare our methods with some existing state-of-the-art algorithms.

关键词： Proximal alternating algorithm Quadratic penalty method Accelerated scheme constrained convex optimization First-order methods Convergence rate

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Primal-dual stochastic distributed algorithm for constrained convex optimization

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JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS 2019年第16期356卷 9763-9787页

作者： Niu, Youcheng Wang, Haijing Wang, Zheng Xia, Dawen Li, Huaqing Southwest Univ Coll Elect & Informat Engn Chongqing Key Lab Nonlinear Circuits & Intelligen Chongqing 400715 Peoples R China Guizhou Minzu Univ Coll Data Sci & Informat Engn Guiyang 550025 Guizhou Peoples R China

This paper investigates distributed convex optimization problems over an undirected and connected network, where each node's variable lies in a private constrained convex set, and overall nodes aim at collectively minimizing the sum of all local objective functions. Motivated by a variety of applications in machine learning problems with large-scale training sets distributed to multiple autonomous nodes, each local objective function is further designed as the average of moderate number of local instantaneous functions. Each local objective function and constrained set cannot be shared with others. A primal-dual stochastic algorithm is presented to address the distributed convex optimization problems, where each node updates its state by resorting to unbiased stochastic averaging gradients and projects on its private constrained set. At each iteration, for each node the gradient of one local instantaneous function selected randomly is evaluated and the average of the most recent stochastic gradients is used to approximate the true local gradient. In the constrained case, we show that with strong-convexity of the local instantaneous function and Lipschitz continuity of its gradient, the algorithm converges to the global optimization solution almost surely. In the unconstrained case, an explicit linear convergence rate of the algorithm is provided. Numerical experiments are presented to demonstrate correctness of the theoretical results. (C) 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

关键词： constrained convex optimization Machine learning Primal-dual algorithm Stochastic averaging gradients Linear convergence

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constrained Quaternion-Variable convex optimization: A Quaternion-Valued Recurrent Neural Network Approach

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2020年第3期31卷 1022-1035页

作者： Liu, Yang Zheng, Yanling Lu, Jianquan Cao, Jinde Rutkowski, Leszek Zhejiang Normal Univ Coll Math & Comp Sci Jinhua 321004 Zhejiang Peoples R China Southeast Univ Sch Math Jiangsu Prov Key Lab Networked Collect Intelligen Nanjing 210096 Peoples R China Nantong Univ Sch Elect Engn Nantong 226000 Peoples R China Shandong Normal Univ Sch Math Sci Jinan 250014 Peoples R China Czestochowa Tech Univ Inst Computat Intelligence PL-42200 Czestochowa Poland Univ Social Sci Informat Technol Inst PL-90113 Lodz Poland

This paper proposes a quaternion-valued one-layer recurrent neural network approach to resolve constrained convex function optimization problems with quaternion variables. Leveraging the novel generalized Hamilton-real (GHR) calculus, the quaternion gradient-based optimization techniques are proposed to derive the optimization algorithms in the quaternion field directly rather than the methods of decomposing the optimization problems into the complex domain or the real domain. Via chain rules and Lyapunov theorem, the rigorous analysis shows that the deliberately designed quaternion-valued one-layer recurrent neural network stabilizes the system dynamics while the states reach the feasible region in finite time and converges to the optimal solution of the considered constrained convex optimization problems finally. Numerical simulations verify the theoretical results.

关键词： Quaternions optimization convex functions Recurrent neural networks Calculus Biological neural networks Collective intelligence constrained convex optimization generalized gradient inclusion generalized Hamilton-real (GHR) calculus Lyapunov function

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Distributed constrained optimization algorithms with linear convergence rate over time-varying unbalanced graphs☆

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AUTOMATICA 2024年 159卷

作者： Liu, Hongzhe Yu, Wenwu Zheng, Wei Xing Nedic, Angelia Zhu, Yanan Southeast Univ Sch Math Nanjing 210096 Peoples R China Western Sydney Univ Sch Comp Data & Math Sci Sydney NSW 2751 Australia Arizona State Univ Sch Elect Comp & Energy Engn Tempe AZ 85281 USA Nanjing Univ Informat Sci & Technol Sch Automat Nanjing 210044 Peoples R China

In this work, the constrained optimization problem is studied, where the global objective function is the sum of N convex functions and a closed convex set constraint is involved. The aim of this work is to design distributed optimization algorithms with linear convergence rate over timevarying unbalanced graphs for solving the studied problem. Towards this end, the Push-DIGing based accelerated distributed constrained optimization algorithm (PDADCO) and the Push-Pull based accelerated distributed constrained optimization algorithm (PPADCO) are developed. Specially, during the algorithm developments, the method of feasible direction, which can be successfully integrated into the Push-DIGing framework and the Push-Pull framework, is newly introduced to deal with the involved closed convex set constraint. Furthermore, based on the small gain theorem, the linear convergence properties of both the PDADCO and the PPADCO are rigorously established, with the feasible regions of the step-sizes being given. Finally, a simulation example is provided to verify the effectiveness of the established theoretical results.

关键词： constrained convex optimization Distributed discrete -time algorithm Linear convergence rate Time -varying unbalanced graphs

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Inexact gradient projection method with relative error tolerance

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COMPUTATIONAL optimization AND APPLICATIONS 2023年第2期84卷 363-395页

作者： Aguiar, A. A. Ferreira, O. P. Prudente, L. F. Univ Fed Goias Inst Matemat & Estat BR-74001970 Goiania Go Brazil

A gradient projection method with feasible inexact projections is proposed in the present paper. The inexact projection is performed using a general relative error tolerance. Asymptotic convergence analysis under quasiconvexity assumption and iteration-complexity bounds under convexity assumption of the method employing constant and Armijo step sizes are presented. Numerical results are reported illustrating the potential advantages of considering inexact projections instead of exact ones in some medium scale instances of a least squares problem over the spectrohedron.

关键词： Gradient method Feasible inexact projection constrained convex optimization

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A new randomized primal-dual algorithm for convex optimization with fast last iterate convergence rates

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optimization METHODS & SOFTWARE 2023年第1期38卷 184-217页

作者： Quoc Tran-Dinh Liu, Deyi Univ North Carolina UNC Chapel Hill Dept Stat & Operat Res 318 Hanes Hall Chapel Hill NC 27599 USA

We develop a novel unified randomized block-coordinate primal-dual algorithm to solve a class of nonsmooth constrained convex optimization problems, which covers different existing variants and model settings from the literature. We prove that our algorithm achieves O(n/k) and O(n(2)/k(2)) convergence rates in two cases: merely convexity and strong convexity, respectively, where k is the iteration counter and n is the number of block-coordinates. These rates are known to be optimal (up to a constant factor) when n = 1. Our convergence rates are obtained through three criteria: primal objective residual and primal feasibility violation, dual objective residual, and primal-dual expected gap. Moreover, our rates for the primal problem are on the last-iterate sequence. Our dual convergence guarantee requires additionally a Lipschitz continuity assumption. We specify our algorithm to handle two important special cases, where our rates are still applied. Finally, we verify our algorithm on two well-studied numerical examples and compare it with two existing methods. Our results show that the proposed method has encouraging performance on different experiments.

关键词： Randomized block coordinate algorithm primal-dual method constrained convex optimization fast last-iterate convergence rate

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Distributed Energy Resource Management: All-Time Resource-Demand Feasibility, Delay-Tolerance, Nonlinearity, and Beyond

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IEEE CONTROL SYSTEMS LETTERS 2023年 7卷 3423-3428页

作者： Doostmohammadian, Mohammadreza Semnan Univ Fac Mech Engn Semnan *** Iran

In this letter, we propose distributed and networked energy management scenarios to optimize the production and reservation of energy among a set of distributed energy nodes. In other words, the idea is to optimally allocate the generated and reserved powers based on nodes' local cost gradient information while meeting the demand energy. One main concern is the all-time (or anytime) resource-demand feasibility, implying that at all iterations of the scheduling algorithm, the balance between the produced power and demand plus reserved power must hold. The other concern is to design algorithms to tolerate communication time-delays and changes in the network. Further, one can incorporate possible model nonlinearity in the algorithm to address both inherent (e.g., saturation and quantization) and purposefully-added (e.g., signum-based) nonlinearities in the model. The proposed optimal allocation algorithm addresses all the above concerns, while it benefits from possible features of the distributed (or networked) solutions such as no-single-node-of-failure and distributed information processing. We show both the all-time feasibility of the proposed scheme and its convergence under certain bound on the step-rate using Lyapunov-type proofs.

关键词： Optimal allocation all-time feasibility constrained convex optimization

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Duality in convex optimization for the hyperbolic augmented Lagrangian

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BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA 2024年第2期30卷 1-24页

作者： Ramirez, Lennin Mallma Maculan, Nelson Xavier, Adilson Elias Xavier, Vinicius Layter Univ Fed Rio de Janeiro Syst Engn & Comp Sci Program COPPE Rio De Janeiro Brazil Univ Fed Rio de Janeiro Syst Engn & Comp Sci Program Appl Math COPPE & IM Rio De Janeiro Brazil Univ Estado Rio De Janeiro Inst Math & Stat Dept Stat Grad Program Computat Sci Rio De Janeiro Brazil

We guarantee the strong duality and the existence of a saddle point of the hyperbolic augmented Lagrangian function (HALF) in convex optimization. To guarantee these results, we assume a set of convexity hypothesis an... 详细信息

关键词： Hyperbolic augmented Lagrangian Nonlinear programming constrained convex optimization Saddle point Duality

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Measure inducing classification and regression trees for functional data

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STATISTICAL ANALYSIS AND DATA MINING-AN ASA DATA SCIENCE JOURNAL 2022年第5期15卷 553-569页

作者： Belli, Edoardo Vantini, Simone Politecn Milan MOX Dept Math Milan Italy

We propose a tree-based algorithm (mu CART) for classification and regression problems in the context of functional data analysis, which allows to leverage measure learning and multiple splitting rules at the node level, with the objective of reducing error while retaining the interpretability of a tree. For each internal node, our main contribution is the idea of learning a weighted functional L-2 space by means of constrained convex optimization, which is then used to extract multiple weighted integral features from the functional predictors, in order to determine the binary split. The approach is designed to manage multiple functional predictors and/or responses, by defining suitable splitting rules and loss functions that can depend on the specific problem and can also be combined with additional scalar and categorical predictors, as the tree is grown with the original greedy CART algorithm. We focus on the case of scalar-valued functional predictors defined on unidimensional domains and illustrate the effectiveness of our method in both classification and regression tasks, through a simulation study and four real-world applications.

关键词： constrained convex optimization decision trees functional data analysis high-dimensional data splitting rule weight function

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