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Many problems of substantial current interest in machine learning, statistics, and data science can be formulated as sparse and low-rank optimization problems. In this paper, we present the nonconvex exterior-point optimization solver (NExOS)—a first-order algorithm tailored to sparse and low-rank optimization problems. We consider the problem of minimizing a convex function over a nonconvex constraint set, where the set can be decomposed as the intersection of a compact convex set and a nonconvex set involving sparse or low-rank constraints. Unlike the convex relaxation approaches, NExOS finds a locally optimal point of the original problem by solving a sequence of penalized problems with strictly decreasing penalty parameters by exploiting the nonconvex geometry. NExOS solves each penalized problem by applying a first-order algorithm, which converges linearly to a local minimum of the corresponding penalized formulation under regularity conditions. Furthermore, the local minima of the penalized problems converge to a local minimum of the original problem as the penalty parameter goes to zero. We then implement and test NExOS on many instances from a wide variety of sparse and low-rank optimization problems, empirically demonstrating that our algorithm outperforms specialized methods. [ABSTRACT FROM AUTHOR]

Applying parallel-in-time algorithms to multiscale Hamiltonian systems to obtain stable long-time simulations is very challenging. In this paper, we present novel data-driven methods aimed at improving the standard parareal algorithm developed by Lions et al. in 2001, for multiscale Hamiltonian systems. The first method involves constructing a correction operator to improve a given inaccurate coarse solver through solving a Procrustes problem using data collected online along parareal trajectories. The second method involves constructing an efficient, high-fidelity solver by a neural network trained with offline generated data. For the second method, we address the issues of effective data generation and proper loss function design based on the Hamiltonian function. We show proof-of-concept by applying the proposed methods to a Fermi-Pasta-Ulam (FPU) problem. The numerical results demonstrate that the Procrustes parareal method is able to produce solutions that are more stable in energy compared to the standard parareal. The neural network solver can achieve comparable or better runtime performance compared to numerical solvers of similar accuracy. When combined with the standard parareal algorithm, the improved neural network solutions are slightly more stable in energy than the improved numerical coarse solutions. [ABSTRACT FROM AUTHOR]

Much recent research effort has been directed to the development of efficient algorithms for solving minimax problems with theoretical convergence guarantees due to the relevance of these problems to a few emergent applications. In this paper, we propose a unified single-loop alternating gradient projection (AGP) algorithm for solving smooth nonconvex-(strongly) concave and (strongly) convex–nonconcave minimax problems. AGP employs simple gradient projection steps for updating the primal and dual variables alternatively at each iteration. We show that it can find an ε -stationary point of the objective function in O ε - 2 (resp. O ε - 4 ) iterations under nonconvex-strongly concave (resp. nonconvex–concave) setting. Moreover, its gradient complexity to obtain an ε -stationary point of the objective function is bounded by O ε - 2 (resp., O ε - 4 ) under the strongly convex–nonconcave (resp., convex–nonconcave) setting. To the best of our knowledge, this is the first time that a simple and unified single-loop algorithm is developed for solving both nonconvex-(strongly) concave and (strongly) convex–nonconcave minimax problems. Moreover, the complexity results for solving the latter (strongly) convex–nonconcave minimax problems have never been obtained before in the literature. Numerical results show the efficiency of the proposed AGP algorithm. Furthermore, we extend the AGP algorithm by presenting a block alternating proximal gradient (BAPG) algorithm for solving more general multi-block nonsmooth nonconvex-(strongly) concave and (strongly) convex–nonconcave minimax problems. We can similarly establish the gradient complexity of the proposed algorithm under these four different settings. [ABSTRACT FROM AUTHOR]

Strongly motivated from applications in various fields including machine learning, the methodology of sparse optimization has been developed intensively so far. Especially, the advancement of algorithms for solving problems with nonsmooth regularizers has been remarkable. However, those algorithms suppose that weight parameters of regularizers, called hyperparameters hereafter, are pre-fixed, but it is a crucial matter how the best hyperparameter should be selected. In this paper, we focus on the hyperparameter selection of regularizers related to the ℓp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ell _p$$\end{document} function with 0

In order to solve the problems of low learning efficiency and slow convergence speed when mobile robot uses reinforcement learning method for path planning in complex environment, a reinforcement learning method based on each round path planning result is proposed. Firstly, the algorithm adds obstacle learning matrix to improve the success rate of path planning; and introduces heuristic reward to speed up the learning process by reducing the search space; then proposes a method of dynamically adjusting the exploration factor to balance the exploration and utilization in path planning, so as to further improve the performance of the algorithm. Finally, the simulation experiment in grid environment shows that compared with Q-learning algorithm, the improved algorithm not only shortens the average path length of the robot to reach the target position, but also speeds up the learning efficiency of the algorithm, so that the robot can find the optimal path more quickly. The code of EPRQL algorithm proposed in this paper has been published to GitHub: https://github.com/panpanpanguoguoqian/mypaper1.git. [ABSTRACT FROM AUTHOR]