JZUS - Journal of Zhejiang University SCIENCE

Frontiers of Information Technology & Electronic Engineering 2023 Vol.24 No.9 P.1239-1252

http://doi.org/10.1631/FITEE.2200381

Matrix-valued distributed stochastic optimization with constraints

Author(s): Zicong XIA, Yang LIU, Wenlian LU, Weihua GUI
Affiliation(s): Key Laboratory of Intelligent Education Technology and Application of Zhejiang Province, Zhejiang Normal University, Jinhua 321004, China; more
Corresponding email(s): 201531700128@zjnu.edu.cn, liuyang@zjnu.edu.cn, wenlian@fudan.edu.cn, gwh@csu.edu.cn
Key Words: Distributed optimization, Matrix-valued optimization, Stochastic optimization, Penalty method, Gossip model

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Abstract: In this paper, we address matrix-valued distributed stochastic optimization with inequality and equality constraints, where the objective function is a sum of multiple matrix-valued functions with stochastic variables and the considered problems are solved in a distributed manner. A penalty method is derived to deal with the constraints, and a selection principle is proposed for choosing feasible penalty functions and penalty gains. A distributed optimization algorithm based on the gossip model is developed for solving the stochastic optimization problem, and its convergence to the optimal solution is analyzed rigorously. Two numerical examples are given to demonstrate the viability of the main results.

带约束的矩阵值分布式随机优化

夏子聪^1,2，刘洋^1,2，卢文联³，桂卫华⁴
¹浙江师范大学浙江省智能教育技术与应用重点实验室，中国金华市，321004
²浙江师范大学数学科学学院，中国金华市，321004
³复旦大学数学科学学院，中国上海市，200433
⁴中南大学自动化学院，中国长沙市，410083
摘要：本文研究带有不等式约束和等式约束的矩阵值分布随机优化问题。其中，问题的目标函数是具有随机变量的多个矩阵值函数的和，并以分布式方式解决了该问题。本文推导了处理约束的惩罚方法，并提出选择可行惩罚函数和惩罚增益的原则。针对随机优化问题，提出一种基于gossip模型的分布式优化算法，并对其收敛性进行证明和分析。最后，为验证所提算法的可行性，本文提供了两个数值示例。

关键词：分布式优化；矩阵值优化；随机优化；罚函数法；Gossip模型

Reference

[1]Bin SQ, Xia YS, 2014. Fast multi-channel image reconstruction using a novel two-dimensional algorithm. Multimed Tools Appl, 71(3):2015-2028.

[2]Bouhamidi A, Jbilou K, 2012. A kronecker approximation with a convex constrained optimization method for blind image restoration. Optim Lett, 6(7):1251-1264.

[3]Boyd S, Vandenberghe L, 2004. Convex Optimization. Cambridge University Press, Cambridge, UK.

[4]Boyd S, Ghosh A, Prabhakar B, et al., 2006. Randomized gossip algorithms. IEEE Trans Inform Theory, 52(6):2508-2530.

[5]Deng ZH, Liang S, Hong YG, 2018. Distributed continuous-time algorithms for resource allocation problems over weight-balanced digraphs. IEEE Trans Cybern, 48(11):3116-3125.

[6]Huang LM, Xia YS, Huang LQ, et al., 2021. Two matrix-type projection neural networks for matrix-valued optimization with application to image restoration. Neur Process Lett, 53(3):1685-1707.

[7]Jakovetic D, Xavier J, Moura JMF, 2011. Cooperative convex optimization in networked systems: augmented Lagrangian algorithms with directed gossip communication. IEEE Trans Signal Process, 59(8):3889-3902.

[8]Jiang XR, Qin ST, Xue XP, 2021. Continuous-time algorithm for approximate distributed optimization with affine equality and convex inequality constraints. IEEE Trans Syst Man Cybern Syst, 51(9):5809-5818.

[9]Koloskova A, Stich SU, Jaggi M, 2019. Decentralized stochastic optimization and gossip algorithms with compressed communication. Proc 36^th Int Conf on Machine Learning, p.3478-3487.

[10]Li H, Fang C, Lin ZC, 2020. Accelerated first-order optimization algorithms for machine learning. Proc IEEE, 108(11):2067-2082.

[11]Li JF, Li W, Huang R, 2016. An efficient method for solving a matrix least squares problem over a matrix inequality constraint. Comput Optim Appl, 63(2):393-423.

[12]Li XX, Xie LH, Hong YG, 2020. Distributed continuous-time nonsmooth convex optimization with coupled inequality constraints. IEEE Trans Contr Netw Syst, 7(1):74-84.

[13]Liang S, Zeng XL, Hong YG, 2018a. Distributed nonsmooth optimization with coupled inequality constraints via modified Lagrangian function. IEEE Trans Autom Contr, 63(6):1753-1759.

[14]Liang S, Zeng XL, Hong YG, 2018b. Distributed sub-optimal resource allocation over weight-balanced graph via singular perturbation. Automatica, 95:222-228.

[15]Liu QS, Wang J, 2013. A one-layer projection neural network for nonsmooth optimization subject to linear equalities and bound constraints. IEEE Trans Neur Netw Learn Syst, 24(5):812-824.

[16]Liu QS, Wang J, 2015. A second-order multi-agent network for bound-constrained distributed optimization. IEEE Trans Autom Contr, 60(12):3310-3315.

[17]Liu QS, Yang SF, Wang J, 2017. A collective neurodynamic approach to distributed constrained optimization. IEEE Trans Neur Netw Learn Syst, 28(8):1747-1758.

[18]Lu J, Tang CY, Regier PR, et al., 2011. Gossip algorithms for convex consensus optimization over networks. IEEE Trans Autom Contr, 56(12):2917-2923.

[19]Lv YW, Yang GH, Shi CX, 2020. Differentially private distributed optimization for multi-agent systems via the augmented Lagrangian algorithm. Inform Sci, 538:39-53.

[20]Nedic A, Ozdaglar A, 2009. Distributed subgradient methods for multi-agent optimization. IEEE Trans Autom Contr, 54(1):48-61.

[21]Rakhlin A, Shamir O, Sridharan K, 2012. Making gradient descent optimal for strongly convex stochastic optimization. Proc 29^th Int Conf on Machine Learning, p.1571-1578.

[22]Ruszczyński A, 2006. Nonlinear Optimization. Princeton University Press, Princeton, USA.

[23]Shamir O, Srebro N, 2014. Distributed stochastic optimization and learning. Proc 52^nd Annual Allerton Conf on Communication, Control, and Computing, p.850-857.

[24]Shi XL, Cao JD, Wen GH, et al., 2019. Finite-time consensus of opinion dynamics and its applications to distributed optimization over digraph. IEEE Trans Cybern, 49(10):3767-3779.

[25]Wan P, Lemmon MD, 2009. Event-triggered distributed optimization in sensor networks. Proc Int Conf on Information Processing in Sensor Networks, p.49-60.

[26]Wang D, Wang Z, Wen CY, 2021. Distributed optimal consensus control for a class of uncertain nonlinear multiagent networks with disturbance rejection using adaptive technique. IEEE Trans Syst Man Cybern Syst, 51(7):4389-4399.

[27]Wang XY, Wang GD, Li SH, 2021. Distributed finite-time optimization for disturbed second-order multiagent systems. IEEE Trans Cybern, 51(9):4634-4647.

[28]Xia YS, Chen TP, Shan JJ, 2014. A novel iterative method for computing generalized inverse. Neur Comput, 26(2):449-465.

[29]Xia ZC, Liu Y, Lu JQ, et al., 2021. Penalty method for constrained distributed quaternion-variable optimization. IEEE Trans Cybern, 51(11):5631-5636.

[30]Xia ZC, Liu Y, Kou KI, et al., 2022. Clifford-valued distributed optimization based on recurrent neural networks. IEEE Trans Neur Netw Learn Syst, early access.

[31]Xia ZC, Liu Y, Qiu JL, et al., 2023. An RNN-based algorithm for decentralized-partial-consensus constrained optimization. IEEE Trans Neur Netw Learn Syst, 34(1):534-542.

[32]Xiao L, Boyd S, 2004. Fast linear iterations for distributed averaging. Syst Contr Lett, 53(1):65-78.

[33]Xing CW, Wang S, Chen S, et al., 2021. Matrix-monotonic optimization – part I: single-variable optimization. IEEE Trans Signal Process, 69:738-754.

[34]Yang SF, Liu QS, Wang J, 2017. Distributed optimization based on a multiagent system in the presence of communication delays. IEEE Trans Syst Man Cybern Syst, 47(5):717-728.

[35]Yang T, Yi XL, Wu JF, et al., 2019. A survey of distributed optimization. Annu Rev Contr, 47:278-305.

[36]Yuan DM, 2014. Gossip-based gradient-free method for multi-agent optimization: constant step size analysis. Proc 33^rd Chinese Control Conf, p.1349-1353.

[37]Yue DD, Baldi S, Cao JD, et al., 2022. Distributed adaptive optimization with weight-balancing. IEEE Trans Autom Contr, 67(4):2068-2075.

[38]Zeng XL, Yi P, Hong YG, 2017. Distributed continuous-time algorithm for constrained convex optimizations via nonsmooth analysis approach. IEEE Trans Autom Contr, 62(10):5227-5233.

[39]Zeng XL, Chen J, Hong YG, 2022. Distributed optimization design of iterative refinement technique for algebraic Riccati equations. IEEE Trans Syst Man Cybern Syst, 52(5):2833-2847.

[40]Zhang SC, Xia YH, Xia YS, et al., 2022. Matrix-form neural networks for complex-variable basis pursuit problem with application to sparse signal reconstruction. IEEE Trans Cybern, 52(7):7049-7059.

[41]Zhou HB, Zeng XL, Hong YG, 2017. An exact penalty method for constrained distributed optimization. Proc 36^th Chinese Control Conf, p.8827-8832.

[42]Zhou HB, Zeng XL, Hong YG, 2019. Adaptive exact penalty design for constrained distributed optimization. IEEE Trans Autom Contr, 64(11):4661-4667.

[43]Zhu MH, Martínez S, 2012. On distributed convex optimization under inequality and equality constraints. IEEE Trans Autom Contr, 57(1):151-164.

[44]Zhu ZH, Li QW, Tang GG, et al., 2021. The global optimization geometry of low-rank matrix optimization. IEEE Trans Inform Theory, 67(2):1308-1331.

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