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nonsmooth convex optimization to Estimate the Covid-19 Reproduction Number Space-Time Evolution With Robustness Against Low Quality Data

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2022年 70卷 2859-2868页

作者： Pascal, Barbara Abry, Patrice Pustelnik, Nelly Roux, Stephane Gribonval, Remi Flandrin, Patrick Univ Lille CNRS Cent Lille UMR 9189CRIStAL F-59000 Lille France CNRS ENS Lyon Lab Phys F-69342 Lyon France Univ Lyon CNRS EnsL INRIALIP F-69342 Lyon 07 France

Daily pandemic surveillance, often achieved through the estimation of the reproduction number, constitutes a critical challenge for national health authorities to design counter-measures. In an earlier work, we proposed to formulate the estimation of the reproduction number as an optimization problem, combining data-model fidelity and space-time regularity constraints, solved by nonsmooth convex proximal minimizations. Though promising, that first formulation significantly lacks robustness against the Covid-19 data low quality (irrelevant or missing counts, pseudo-seasonalities,...) stemming from the emergency and crisis context, which significantly impairs accurate pandemic evolution assessments. The present work aims to overcome these limitations by carefully crafting a functional permitting to estimate jointly, in a single step, the reproduction number and outliers defined to model low quality data. This functional also enforces epidemiology-driven regularity properties for the reproduction number estimates, while preserving convexity, thus permitting the design of efficient minimization algorithms, based on proximity operators that are derived analytically. The explicit convergence of the proposed algorithm is proven theoretically. Its relevance is quantified on real Covid-19 data, consisting of daily new infection counts for 200+ countries and for the 96 metropolitan France counties, publicly available at Johns Hopkins University and Sante-Publique-France. The procedure permits automated daily updates of these estimates, reported via animated and interactive maps. Open-source estimation procedures will be made publicly available.

关键词： Pandemics COVID-19 Data models Computational modeling Maximum likelihood estimation Biological system modeling Robustness Covid-19 reproduction number space-time evolution nonsmooth convex optimization outlier robustness

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Distributed proximal-gradient algorithms for nonsmooth convex optimization of second-order multiagent systems

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INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL 2020年第17期30卷 7574-7592页

作者： Wang, Qing Chen, Jie Zeng, Xianlin Xin, Bin Beijing Inst Technol Sch Automat 5 South Zhongguancun St Beijing 100081 Peoples R China Beijing Inst Technol State Key Lab Intelligent Control & Decis Complex Beijing Peoples R China Tongji Univ Dept Control Sci & Engn Shanghai Peoples R China Beijing Inst Technol Beijing Adv Innovat Ctr Intelligent Robots & Syst Beijing Peoples R China

This article studies the distributed nonsmooth convex optimization problems for second-order multiagent systems. The objective function is the summation of local cost functions which are convex but nonsmooth. Each agent only knows its local cost function, local constraint set, and neighbor information. By virtue of proximal operator and Lagrangian methods, novel continuous-time distributed proximal-gradient algorithms with derivative feedback are proposed to solve the nonsmooth convex optimization for the consensus and resource allocation of multiagent systems, respectively. With the proposed algorithms, both the consensus and resource allocation problems are solved. Moreover, the system can converge to the optimal solution. Finally, simulation examples are given to illustrate the effectiveness of the proposed algorithms.

关键词： multiagent systems nonsmooth convex optimization optimal consensus proximal-gradient algorithm resource allocation

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A Two-Layer Recurrent Neural Network for nonsmooth convex optimization Problems

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2015年第6期26卷 1149-1160页

作者： Qin, Sitian Xue, Xiaoping Harbin Inst Technol Dept Math Weihai 264209 Peoples R China Harbin Inst Technol Dept Math Harbin 150001 Peoples R China

In this paper, a two-layer recurrent neural network is proposed to solve the nonsmooth convex optimization problem subject to convex inequality and linear equality constraints. Compared with existing neural network models, the proposed neural network has a low model complexity and avoids penalty parameters. It is proved that from any initial point, the state of the proposed neural network reaches the equality feasible region in finite time and stays there thereafter. Moreover, the state is unique if the initial point lies in the equality feasible region. The equilibrium point set of the proposed neural network is proved to be equivalent to the Karush-Kuhn-Tucker optimality set of the original optimization problem. It is further proved that the equilibrium point of the proposed neural network is stable in the sense of Lyapunov. Moreover, from any initial point, the state is proved to be convergent to an equilibrium point of the proposed neural network. Finally, as applications, the proposed neural network is used to solve nonlinear convex programming with linear constraints and L-1-norm minimization problems.

关键词： Global convergence Lyapunov function nonsmooth convex optimization two-layer recurrent neural network

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Globally convergent BFGS method for nonsmooth convex optimization

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JOURNAL OF optimization THEORY AND APPLICATIONS 2000年第3期104卷 539-558页

作者： Rauf, AI Fukushima, M Hamdard Univ Islamabad Pakistan Kyoto Univ Grad Sch Informat Dept Appl Math & Phys Kyoto Japan

We propose an implementable BFGS method for solving a nonsmooth convex optimization problem by converting the original objective function into a once continuously differentiable function by way of the Moreau-Yosida regularization. The proposed method makes use of approximate function and gradient values of the Moreau-Yosida regularization instead of the corresponding exact values. We prove the global convergence of the proposed method under the assumption of strong convexity of the objective function.

关键词： nonsmooth convex optimization Moreau-Yosida regularization strong convexity inexact function and gradient evaluations BFGS method

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Distributed Coordination for nonsmooth convex optimization via Saddle-Point Dynamics

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JOURNAL OF NONLINEAR SCIENCE 2019年第4期29卷 1247-1272页

作者： Cortes, Jorge Niederlaender, Simon K. Univ Calif San Diego Jacobs Sch Engn Dept Mech & Aerosp Engn 9500 Gilman Dr La Jolla CA 92093 USA Univ Stuttgart Inst Syst Theory & Automat Control D-70550 Stuttgart Germany

This paper considers continuous-time coordination algorithms for networks of agents that seek to collectively solve a general class of nonsmooth convex optimization problems with an inherent distributed structure. Our algorithm design builds on the characterization of the solutions of the nonsmooth convex program as saddle points of an augmented Lagrangian. We show that the associated saddle-point dynamics are asymptotically correct but, in general, not distributed because of the presence of a global penalty parameter. This motivates the design of a discontinuous saddle-point-like algorithm that enjoys the same convergence properties and is fully amenable to distributed implementation. Our convergence proofs rely on the identification of a novel global Lyapunov function for saddle-point dynamics. This novelty also allows us to identify mild convexity and regularity conditions on the objective function that guarantee the exponential convergence rate of the proposed algorithms for convex optimization problems subject to equality constraints. Various examples illustrate our discussion.

关键词： Distributed multi-agent coordination nonsmooth convex optimization Saddle-point dynamics Continuous-time optimization algorithms

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Convergence analysis of iterative methods for nonsmooth convex optimization over fixed point sets of quasi-nonexpansive mappings

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MATHEMATICAL PROGRAMMING 2016年第1-2期159卷 509-538页

作者： Iiduka, Hideaki Meiji Univ Dept Comp Sci Tama Ku 1-1-1 Higashimita Kawasaki Kanagawa 2148571 Japan

This paper considers a networked system with a finite number of users and supposes that each user tries to minimize its own private objective function over its own private constraint set. It is assumed that each user's constraint set can be expressed as a fixed point set of a certain quasi-nonexpansive mapping. This enables us to consider the case in which the projection onto the constraint set cannot be computed efficiently. This paper proposes two methods for solving the problem of minimizing the sum of their nondifferentiable, convex objective functions over the intersection of their fixed point sets of quasi-nonexpansive mappings in a real Hilbert space. One method is a parallel subgradient method that can be implemented under the assumption that each user can communicate with other users. The other is an incremental subgradient method that can be implemented under the assumption that each user can communicate with its neighbors. Investigation of the two methods' convergence properties for a constant step size reveals that, with a small constant step size, they approximate a solution to the problem. Consideration of the case in which the step-size sequence is diminishing demonstrates that the sequence generated by each of the two methods strongly converges to the solution to the problem under certain assumptions. Convergence rate analysis of the two methods under certain situations is provided to illustrate the two methods' efficiency. This paper also discusses nonsmooth convex optimization over sublevel sets of convex functions and provides numerical comparisons that demonstrate the effectiveness of the proposed methods.

关键词： Fixed point Incremental subgradient method nonsmooth convex optimization Parallel subgradient method Quasi-nonexpansive mapping

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Incremental subgradient method for nonsmooth convex optimization with fixed point constraints

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optimization METHODS & SOFTWARE 2016年第5期31卷 931-951页

作者： Iiduka, H. Meiji Univ Dept Comp Sci Tama Ku 1-1-1 Higashimita Kawasaki Kanagawa Japan

This paper proposes an incremental subgradient method for solving the problem of minimizing the sum of nondifferentiable, convex objective functions over the intersection of fixed point sets of nonexpansive mappings in a real Hilbert space. The proposed algorithm can work in nonsmooth optimization over constraint sets onto which projections cannot be always implemented, whereas the conventional incremental subgradient method can be applied only when a constraint set is simple in the sense that the projection onto it can be easily implemented. We first study its convergence for a constant step size. The analysis indicates that there is a possibility that the algorithm with a small constant step size approximates a solution to the problem. Next, we study its convergence for a diminishing step size and show that there exists a subsequence of the sequence generated by the algorithm which weakly converges to a solution to the problem. Moreover, we show the whole sequence generated by the algorithm with a diminishing step size strongly converges to the solution to the problem under certain assumptions. We also give examples of real applied problems which satisfy the assumptions in the convergence theorems and numerical examples to support the convergence analyses.

关键词： fixed point incremental subgradient method Krasnosel'ski-Mann algorithm nonexpansive mapping nonsmooth convex optimization subdifferential 65K05 90C25 90C90

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Distributed Multiproximal Algorithm for nonsmooth convex optimization With Coupled Inequality Constraints

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2023年第12期68卷 8126-8133页

作者： Huang, Yi Meng, Ziyang Sun, Jian Ren, Wei Beijing Inst Technol Sch Automat Beijing 100081 Peoples R China Tsinghua Univ Dept Precis Instrument Beijing 100081 Peoples R China Beijing Inst Technol Chongqing Innovat Ctr Chongqing Peoples R China Beijing Inst Technol Sch Automat Natl Key Lab Autonomous Intelligent Unmanned Syst Beijing 100081 Peoples R China Univ Calif Riverside Dept Elect & Comp Engn Riverside CA 92521 USA

This article studies a class of distributed nonsmooth convex optimization problems subject to local set constraints and coupled nonlinear inequality constraints. In particular, each local objective function consists of one differentiable convex function and multiple nonsmooth convex functions. By applying multiple proximal splittings and derivative feedback techniques, a new distributed continuous-time multiproximal algorithm is developed, whose dynamics satisfies Lipschitz continuity even if the considered problem is nonsmooth. Compared with previous results that rely on either the differentiability or strong convexity of local objective functions, the proposed algorithm can be applied to more general functions, which are only convex and not necessarily smooth. Moreover, in contrast to some results that require some specific initial conditions, the developed algorithm is free of initialization. The convergence analysis of the proposed algorithm is conducted by applying Lyapunov stability theory. It is shown that the states of all the agents achieve consensus at an optimal solution. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed algorithm.

关键词： Coupled inequality constraint distributed algorithm nonsmooth convex optimization proximal splitting

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A globally convergent proximal Newton-type method in nonsmooth convex optimization

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MATHEMATICAL PROGRAMMING 2023年第1期198卷 899-936页

作者： Mordukhovich, Boris S. Yuan, Xiaoming Zeng, Shangzhi Zhang, Jin Wayne State Univ Dept Math Detroit MI 48202 USA Univ Hong Kong Dept Math Hong Kong Peoples R China Univ Victoria Dept Math & Stat Victoria BC Canada Southern Univ Sci & Technol Natl Ctr Appl Math Shenzhen Dept Math Shenzhen 518055 Peoples R China

The paper proposes and justifies a new algorithm of the proximal Newton type to solve a broad class of nonsmooth composite convex optimization problems without strong convexity assumptions. Based on advanced notions and techniques of variational analysis, we establish implementable results on the global convergence of the proposed algorithm as well as its local convergence with superlinear and quadratic rates. For certain structured problems, the obtained local convergence conditions do not require the local Lipschitz continuity of the corresponding Hessian mappings that is a crucial assumption used in the literature to ensure a superlinear convergence of other algorithms of the proximal Newton type. The conducted numerical experiments of solving the l(1) regularized logistic regression model illustrate the possibility of applying the proposed algorithm to deal with practically important problems.

关键词： nonsmooth convex optimization Machine learning Proximal Newton methods Global and local convergence Metric subregularity

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Distributed quasi-monotone subgradient algorithm for nonsmooth convex optimization over directed graphs

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AUTOMATICA 2019年 101卷 175-181页

作者： Liang, Shu Wang, Leyi Yin, George Univ Sci & Technol Beijing Sch Automat & Elect Engn Minist Educ Key Lab Knowledge Automat Ind Proc Beijing 100083 Peoples R China Wayne State Univ Dept Elect & Comp Engn Detroit MI 48202 USA Wayne State Univ Dept Math Detroit MI 48202 USA

Distributed optimization is of essential importance in networked systems. Most of the existing distributed algorithms either assume the information exchange over undirected graphs, or require that the underlying directed network topology provides a doubly stochastic weight matrix to the agents. In this brief paper, a distributed subgradient-based algorithm is proposed to solve nonsmooth convex optimization problems. The algorithm applies to directed graphs without using a doubly stochastic weight matrix. Moreover, the algorithm is a distributed generalization and improvement of the quasi-monotone subgradient algorithm. An O(1/root k) convergence rate is achieved. The effectiveness of our algorithm is also illustrated by a numerical example. (C) 2018 Elsevier Ltd. All rights reserved.

关键词： Distributed optimization nonsmooth convex optimization Quasi-monotone subgradient algorithm Directed graph

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