We introduce a new algorithmic framework for solving nonconvex optimization problems, that is called nested alternating minimization, which aims at combining the classical alternating minimization technique with inner...
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We introduce a new algorithmic framework for solving nonconvex optimization problems, that is called nested alternating minimization, which aims at combining the classical alternating minimization technique with inner iterations of any optimization method. We provide a global convergence analysis of the new algorithmic framework to critical points of the problem at hand, which to the best of our knowledge, is the first of this kind for nested methods in the nonconvex setting. Central to our global convergence analysis is a new extension of classical proof techniques in the nonconvex setting that allows for errors in the conditions. The power of our framework is illustrated with some numerical experiments that show the superiority of this algorithmic framework over existing methods.
Motivated by a recent framework for proving global convergence to critical points of nested alternating minimization algorithms, which was proposed for the case of smooth subproblems, we first show here that non-smoot...
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Motivated by a recent framework for proving global convergence to critical points of nested alternating minimization algorithms, which was proposed for the case of smooth subproblems, we first show here that non-smooth subproblems can also be handled within this framework. Specifically, we present a novel analysis of an optimization scheme that utilizes the FISTA method as a nested algorithm. We establish the global convergence of this nested scheme to critical points of non-convex and non-smooth optimization problems. In addition, we propose a hybrid framework that allows to implement FISTA when applicable, while still maintaining the global convergence result. The power of nested algorithms using FISTA in the non-convex and non-smooth setting is illustrated with some numerical experiments that show their superiority over existing methods.
We investigate the convergence of a recently popular class of first-order primal-dual algorithms for saddle point problems under the presence of errors in the proximal maps and gradients. We study several types of err...
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We investigate the convergence of a recently popular class of first-order primal-dual algorithms for saddle point problems under the presence of errors in the proximal maps and gradients. We study several types of errors and show that, provided a sufficient decay of these errors, the same convergence rates as for the error-free algorithm can be established. More precisely, we prove the (optimal) O(1/N) convergence to a saddle point in finite dimensions for the class of non-smooth problems considered in this paper, and prove a O1/N2 or even linear ON convergence rate if either the primal or dual objective respectively both are strongly convex. Moreover we show that also under a slower decay of errors we can establish rates, however slower and directly depending on the decay of the errors. We demonstrate the performance and practical use of the algorithms on the example of nested algorithms and show how they can be used to split the global objective more efficiently.
Force-gradient decomposition methods are used to improve the energy preservation of symplectic schemes applied to Hamiltonian systems. If the potential is composed of different parts with strongly varying dynamics, th...
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Force-gradient decomposition methods are used to improve the energy preservation of symplectic schemes applied to Hamiltonian systems. If the potential is composed of different parts with strongly varying dynamics, this multirate potential can be exploited by coupling force-gradient decomposition methods with splitting techniques for multi-time scale problems to further increase the accuracy of the scheme and reduce the computational costs. In this paper, we derive novel force-gradient nested methods and test them numerically. We apply them on the three-body problem, modified for a better observation of the advantageous properties, needed for the future research. (C) 2014 Elsevier B.V. All rights reserved.
This paper proposes a new nested algorithm (NPL) for the estimation of a class of discrete Markov decision models and studies its statistical and computational properties. Our method is based on a representation of th...
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This paper proposes a new nested algorithm (NPL) for the estimation of a class of discrete Markov decision models and studies its statistical and computational properties. Our method is based on a representation of the solution of the dynamic programming problem in the space of conditional choice probabilities. When the NPL algorithm is initialized with consistent nonparametric estimates of conditional choice probabilities, successive iterations return a sequence of estimators of the structural parameters which we call K-stage policy iteration estimators. We show that the sequence includes as extreme cases a Hotz-Miller estimator (for K = 1) and Rust's nested fixed point estimator (in the limit when K --> infinity). Furthermore, the asymptotic distribution of all the estimators in the sequence is the same and equal to that of the maximum likelihood estimator, We illustrate the performance of our method with several examples based on Rust's bus replacement model. Monte Carlo experiments reveal a trade-off between finite sample precision and computational cost in the sequence of policy iteration estimators.
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