8 research outputs found

    Forward-backward truncated Newton methods for convex composite optimization

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    This paper proposes two proximal Newton-CG methods for convex nonsmooth optimization problems in composite form. The algorithms are based on a a reformulation of the original nonsmooth problem as the unconstrained minimization of a continuously differentiable function, namely the forward-backward envelope (FBE). The first algorithm is based on a standard line search strategy, whereas the second one combines the global efficiency estimates of the corresponding first-order methods, while achieving fast asymptotic convergence rates. Furthermore, they are computationally attractive since each Newton iteration requires the approximate solution of a linear system of usually small dimension

    A Semismooth Newton Stochastic Proximal Point Algorithm with Variance Reduction

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    We develop an implementable stochastic proximal point (SPP) method for a class of weakly convex, composite optimization problems. The proposed stochastic proximal point algorithm incorporates a variance reduction mechanism and the resulting SPP updates are solved using an inexact semismooth Newton framework. We establish detailed convergence results that take the inexactness of the SPP steps into account and that are in accordance with existing convergence guarantees of (proximal) stochastic variance-reduced gradient methods. Numerical experiments show that the proposed algorithm competes favorably with other state-of-the-art methods and achieves higher robustness with respect to the step size selection

    Global and Quadratic Convergence of Newton Hard-Thresholding Pursuit

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    Algorithms based on the hard thresholding principle have been well studied with sounding theoretical guarantees in the compressed sensing and more general sparsity-constrained optimization. It is widely observed in existing empirical studies that when a restricted Newton step was used (as the debiasing step), the hard-thresholding algorithms tend to meet halting conditions in a significantly low number of iterations and are very efficient. Hence, the thus obtained Newton hard-thresholding algorithms call for stronger theoretical guarantees than for their simple hard-thresholding counterparts. This paper provides a theoretical justification for the use of the restricted Newton step. We build our theory and algorithm, Newton Hard-Thresholding Pursuit (NHTP), for the sparsity-constrained optimization. Our main result shows that NHTP is quadratically convergent under the standard assumption of restricted strong convexity and smoothness. We also establish its global convergence to a stationary point under a weaker assumption. In the special case of the compressive sensing, NHTP effectively reduces to some of the existing hard-thresholding algorithms with a Newton step. Consequently, our fast convergence result justifies why those algorithms perform better than without the Newton step. The efficiency of NHTP was demonstrated on both synthetic and real data in compressed sensing and sparse logistic regression

    Globally Convergent Coderivative-Based Generalized Newton Methods in Nonsmooth Optimization

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    This paper proposes and justifies two globally convergent Newton-type methods to solve unconstrained and constrained problems of nonsmooth optimization by using tools of variational analysis and generalized differentiation. Both methods are coderivative-based and employ generalized Hessians (coderivatives of subgradient mappings) associated with objective functions, which are either of class C1,1\mathcal{C}^{1,1}, or are represented in the form of convex composite optimization, where one of the terms may be extended-real-valued. The proposed globally convergent algorithms are of two types. The first one extends the damped Newton method and requires positive-definiteness of the generalized Hessians for its well-posedness and efficient performance, while the other algorithm is of {the regularized Newton type} being well-defined when the generalized Hessians are merely positive-semidefinite. The obtained convergence rates for both methods are at least linear, but become superlinear under the semismooth∗^* property of subgradient mappings. Problems of convex composite optimization are investigated with and without the strong convexity assumption {on smooth parts} of objective functions by implementing the machinery of forward-backward envelopes. Numerical experiments are conducted for Lasso problems and for box constrained quadratic programs with providing performance comparisons of the new algorithms and some other first-order and second-order methods that are highly recognized in nonsmooth optimization.Comment: arXiv admin note: text overlap with arXiv:2101.1055

    Numerical Analysis of Algorithms for Infinitesimal Associated and Non-Associated Elasto-Plasticity

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    The thesis studies nonlinear solution algorithms for problems in infinitesimal elastoplasticity and their numerical realization within a parallel computing framework. New algorithms like Active Set and Augmented Lagrangian methods are proposed and analyzed within a semismooth Newton setting. The analysis is often carried out in function space which results in stable algorithms. Large scale computer experiments demonstrate the efficiency of the new algorithms

    Global and quadratic convergence of Newton hard-thresholding pursuit

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    Algorithms based on the hard thresholding principle have been well studied with sounding theoretical guarantees in the compressed sensing and more general sparsity-constrained optimization. It is widely observed in existing empirical studies that when a restricted Newton step was used (as the debiasing step), the hard-thresholding algorithms tend to meet halting conditions in a significantly low number of iterations and are very efficient. Hence, the thus obtained Newton hard-thresholding algorithms call for stronger theoretical guarantees than for their simple hard-thresholding counterparts. This paper provides a theoretical justification for the use of the restricted Newton step. We build our theory and algorithm, Newton Hard-Thresholding Pursuit (NHTP), for the sparsity-constrained optimization. Our main result shows that NHTP is quadratically convergent under the standard assumption of restricted strong convexity and smoothness. We also establish its global convergence to a stationary point under a weaker assumption. In the special case of the compressive sensing, NHTP effectively reduces to some of the existing hard-thresholding algorithms with a Newton step. Consequently, our fast convergence result justifies why those algorithms perform better than without the Newton step. The efficiency of NHTP was demonstrated on both synthetic and real data in compressed sensing and sparse logistic regression

    Minimization of SC1 functions and the Maratos effect

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