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Geometric constraints on the domain for a class of minimum problems

ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2003年第6期9卷 125-133页

作者： Crasta, G Malusa, A Univ Roma La Sapienza Dipartimento Matemat I-00185 Rome Italy

We consider minimization problems of the form [GRAPHICS] where Omega subset of or equal to R-N is a bounded convex open set, and the Borel function f : R-N --> [0, + infinity] is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of Omega and the zero level set of f, we prove that the viscosity solution of a related Hamilton-Jacobi equation provides a minimizer for the integral functional.

关键词： Calculus of Variations existence non-convex problems non-coercive problems viscosity solutions

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Stochastic normalized gradient descent with momentum for large-batch training

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Science China(Information Sciences) 2024年第11期67卷 77-91页

作者： Shen-Yi ZHAO Chang-Wei SHI Yin-Peng XIE Wu-Jun LI National Key Laboratory for Novel Software Technology Department of Computer Science and TechnologyNanjing University

Stochastic gradient descent(SGD) and its variants have been the dominating optimization methods in machine learning. Compared with SGD with small-batch training, SGD with large-batch training can better utilize the computational power of current multi-core systems such as graphics processing units(GPUs)and can reduce the number of communication rounds in distributed training settings. Thus, SGD with large-batch training has attracted considerable attention. However, existing empirical results showed that large-batch training typically leads to a drop in generalization accuracy. Hence, how to guarantee the generalization ability in large-batch training becomes a challenging task. In this paper, we propose a simple yet effective method, called stochastic normalized gradient descent with momentum(SNGM), for large-batch training. We prove that with the same number of gradient computations, SNGM can adopt a larger batch size than momentum SGD(MSGD), which is one of the most widely used variants of SGD, to converge to an?-stationary point. Empirical results on deep learning verify that when adopting the same large batch size,SNGM can achieve better test accuracy than MSGD and other state-of-the-art large-batch training methods.

关键词： non-convex problems large-batch training stochastic normalized gradient descent momentum

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Recent advances in multiparametric nonlinear programming

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COMPUTERS & CHEMICAL ENGINEERING 2010年第5期34卷 707-716页

作者： Dominguez, Luis F. Narciso, Diogo A. Pistikopoulos, Efstratios N. Univ London Imperial Coll Sci Technol & Med Ctr Proc Syst Engn Dept Chem Engn London SW7 2AZ England

In this paper, we present recent developments in multiparametric nonlinear programming. For the case of convex problems, we highlight key issues regarding the full characterization of the parametric solution space and we discuss, through an illustrative example problem, four alternative state-of-the-art multiparametric nonlinear programming algorithms. We also identify a number of main challenges for the non-convex case and highlight future research directions. (C) 2009 Elsevier Ltd. All rights reserved.

关键词： Multiparametric programming Multiparametric nonlinear programming convex problems non-convex problems

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On a result by Boccardo-Ferone-Fusco-Orsina

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RENDICONTI LINCEI-MATEMATICA E APPLICAZIONI 2011年第4期22卷 505-511页

作者： Squassina, Marco Univ Verona Dipartimento Informat I-37134 Verona Italy

Via a symmetric version of Ekeland's principle recently obtained by the author we improve, in a ball or an annulus, a result of Boceardo-Ferone-Fusco-Orsina on the properties of minimizing sequences of functionals... 详细信息

关键词： Symmetry non-convex problems regularity of minimizing sequences

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Asynchronous Optimization Over Heterogeneous Networks Via Consensus ADMM

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IEEE TRANSACTIONS ON SIGNAL AND INFORMATION PROCESSING OVER NETWORKS 2017年第1期3卷 114-129页

作者： Kumar, Sandeep Jain, Rahul Rajawat, Ketan Indian Inst Technol Dept Elect Engn Kanpur 208016 Uttar Pradesh India Qualcomm India Pvt Ltd Bangalore 560066 Karnataka India

This paper considers the distributed optimization of a sum of locally observable, nonconvex functions. The optimization is performed over a multiagent networked system, and each local function depends only on a subset of the variables. An asynchronous and distributed alternating directions method of multipliers (ADMM) method that allows the nodes to defer or skip the computation and transmission of updates is proposed in the paper. The proposed algorithm utilizes different approximations in the update step, resulting in proximal and majorized ADMM variants. Both variants are shown to converge to a local minimum under certain regularity conditions. The proposed asynchronous algorithms are also applied to the problem of cooperative localization in wireless ad hoc networks, where it is shown to outperform the other state-of-the-art localization algorithms.

关键词： ADMM asynchronous algorithms distributed optimization majorization non-convex problems

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An Improved Convergence Analysis for Decentralized Online Stochastic non-convex Optimization

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2021年 69卷 1842-1858页

作者： Xin, Ran Khan, Usman A. Kar, Soummya Carnegie Mellon Univ ECE Dept Pittsburgh PA 15213 USA Tufts Univ Elect & Comp Engn Dept Medford MA 02155 USA

In this paper, we study decentralized online stochastic non-convex optimization over a network of nodes. Integrating a technique called gradient tracking in decentralized stochastic gradient descent, we show that the resulting algorithm, GT-DSGD, enjoys certain desirable characteristics towards minimizing a sum of smooth non-convex functions. In particular, for general smooth non-convex functions, we establish non-asymptotic characterizations of GT-DSGD and derive the conditions under which it achieves network-independent performances that match the centralized minibatch SGD. In contrast, the existing results suggest that GT-DSGD is always network-dependent and is therefore strictly worse than the centralized minibatch SGD. When the global non-convex function additionally satisfies the Polyak-Lojasiewics (PL) condition, we establish the linear convergence of GT-DSGD up to a steady-state error with appropriate constant step-sizes. Moreover, under stochastic approximation step-sizes, we establish, for the first time, the optimal global sublinear convergence rate on almost every sample path, in addition to the asymptotically optimal sublinear rate in expectation. Since strongly convex functions are a special case of the functions satisfying the PL condition, our results are not only immediately applicable but also improve the currently known best convergence rates and their dependence on problem parameters.

关键词： Convergence Stochastic processes Steady-state Signal processing algorithms Optimization Transient analysis Linear programming Decentralized optimization stochastic gradient methods non-convex problems multi-agent systems

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nonconvex stochastic programming problems-formulations, sample approximations and stability

Nonconvex stochastic programming problems-formulations, samp...

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作者： Branda, Martin Charles University of Prague

Title: nonconvex stochastic programming problems - formulations, sample approximations and stability Author: RNDr. Martin Branda Authors e-mail address: branda@*** Supervisor: Doc. RNDr. Petr Lachout, CSc. Supervisors e-mail address: lachout@*** Abstract: We deal with problems where integer variables may appear, hence no assumptions on convexity are made throughout this thesis. The goal of Chapter 2 is to introduce stochastic programming problems and to outline the most important tasks connected with solving the problems. In Chapter 3, we compare basic formulations of static stochastic programming problems with chance constraints, with integrated chance constraints and with penalties in the objective function. We show that the problems are asymptotically equivalent under mild conditions. We discuss solving the problems using sample approximation techniques and extend some results on rates of convergence. All the formulations and corresponding sample approximations are compared on an investment problem with real features with Value at Risk constraint, integer allocations and transaction costs. Then, stability of financial decision models where two-stage mixed-integer value function appears as a loss variable is studied. In Chapter 4, we study qualitative properties of the...

关键词： Stochastic programming formulations sample approximations stability non-convex problems

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Uncovered bargaining solutions

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INTERNATIONAL JOURNAL OF GAME THEORY 2009年第4期38卷 601-610页

作者： Lombardi, Michele Mariotti, Marco Univ St Andrews Sch Econ & Finance St Andrews KY16 9AL Fife Scotland Univ Surrey Dept Econ Guildford GU2 7XH Surrey England Univ Maastricht Dept Quantitat Econ NL-6200 MD Maastricht Netherlands

An uncovered bargaining solution is a bargaining solution for which there exists a complete and asymmetric relation (tournament) such that, for each feasible set, the bargaining solution set coincides with the uncover... 详细信息

关键词： Bargaining Tournaments Uncovered set non-convex problems

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On the existence and uniqueness of minimizers for a class of integral functionals

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NODEA-nonLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS 2005年第2期12卷 129-150页

作者： Crasta, G Malusa, A Univ Roma 1 Dipartimento Matemat I-00185 Rome Italy

We study the solvability of the minimization problem [GRAPHICS] where K-alpha is a subset of AC(loc)[0,T] depending on the weight function alpha. Neither the convexity nor the superlinearity of f are required. The main application concerns the existence and uniqueness of minimizers to integral functionals on convex domains Omega subset of R-N, defined in the class of functions in W-0(1,1)(Omega) depending only on the distance from the boundary of Omega. As a corollary, when Q is a ball we obtain the existence of radially symmetric solutions to nonconvex and noncoercive functionals.

关键词： calculus of variations existence uniqueness non-convex problems non-coercive problems

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On the convergence and improvement of stochastic normalized gradient descent

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Science China(Information Sciences) 2021年第3期64卷 105-117页

作者： Shen-Yi ZHAO Yin-Peng XIE Wu-Jun LI National Key Laboratory for Novel Software Technology Department of Computer Science and TechnologyNanjing University

non-convex models, like deep neural networks, have been widely used in machine learning applications. Training non-convex models is a difficult task owing to the saddle points of models. Recently,stochastic normalized gradient descent(SNGD), which updates the model parameter by a normalized gradient in each iteration, has attracted much attention. Existing results show that SNGD can achieve better performance on escaping saddle points than classical training methods like stochastic gradient descent(SGD).However, none of the existing studies has provided theoretical proof about the convergence of SNGD for non-convex problems. In this paper, we firstly prove the convergence of SNGD for non-convex ***, we prove that SNGD can achieve the same computation complexity as SGD. In addition, based on our convergence proof of SNGD, we find that SNGD needs to adopt a small constant learning rate for convergence guarantee. This makes SNGD do not perform well on training large non-convex models in practice. Hence, we propose a new method, called stagewise SNGD(S-SNGD), to improve the performance of SNGD. Different from SNGD in which a small constant learning rate is necessary for convergence guarantee,S-SNGD can adopt a large initial learning rate and reduce the learning rate by stage. The convergence of S-SNGD can also be theoretically proved for non-convex problems. Empirical results on deep neural networks show that S-SNGD achieves better performance than SNGD in terms of both training loss and test accuracy.

关键词： non-convex problems stochastic normalized gradient descent computation complexity

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