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This paper proposes a path-based approach for the minimization of a continuously differentiable function over sparse symmetric sets, which is a hard problem that exhibits a restrictiveness-hierarchy of necessary optimality conditions. To achieve the more restrictive conditions in the hierarchy, state-of-the-art algorithms require a support optimization oracle that must exactly solve the problem in smaller dimensions. The path-based approach developed in this study produces a path-based optimality condition, which is placed well in the restrictiveness-hierarchy, and a method to achieve it that does not require a support optimization oracle and, moreover, is projection-free. In the development process, new results are derived for the regularized linear minimization problem over sparse symmetric sets, which give additional means to identify optimal solutions for convex and concave objective functions. We complement our results with numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we study smooth stochastic multilevel composition optimization problems, where the objective function is a nested composition of T functions. We assume access to noisy evaluations of the functions and their gradients, through a stochastic first-order oracle. For solving this class of problems, we propose two algorithms using moving-average stochastic estimates, and analyze their convergence to an e-stationary point of the problem. We show that the first algorithm, which is a generalization of [S. Ghadimi, A. Ruszczynski, and M. Wang, SIAM J. Optim., 30 (2020), pp. 960-979] to the T level case, can achieve a sample complexity of OT(1/ε6) by using minibatches of samples in each iteration, where OT hides constants that depend on T. By modifying this algorithm using linearized stochastic estimates of the function values, we improve the sample complexity to OT(1/ε4). This modification not only removes the requirement of having a minibatch of samples in each iteration, but also makes the algorithm parameter-free and easy to implement. To the best of our knowledge, this is the first time that such an online algorithm designed for the (un)constrained multilevel setting obtains the same sample complexity of the smooth single-level setting, under standard assumptions (unbiasedness and boundedness of the second moments) on the stochastic first-order oracle. [ABSTRACT FROM AUTHOR]

The Broyden–Fletcher–Goldfarb–Shanno (BFGS) method plays an important role among the quasi-Newton algorithms for nonconvex and unconstrained optimization problems. However, in the proof of global convergence, BFGS-type methods generally need to assume that the gradient of the objective function is Lipschitz continuous. This issue prompts us to try to find quasi-Newton method for gradient non-Lipschitz continuous and nonconvex optimization based on the classical BFGS formula. In this paper, we propose an adaptive scaling damped BFGS method for gradient non-Lipschitz continuous and nonconvex problems. With Armijo or Weak Wolfe–Powell (WWP) line search, global convergence can be obtained. Under suitable conditions the convergence rate is superlinear with WWP-type line search. Applications of the given algorithms include the tested optimization problems, which turn out the proposed method is powerful and promising. • We develop a type of damped BFGS methods that solves nonconvex problems. • The global convergence is established without gradient Lipschitz continuity. • In the case of weaker assumptions, correct the structure of the eigenvalues. • The method convergence to a stationary point under Armijo or WWP line search. • The convergence rate is superlinear under suitable conditions. [ABSTRACT FROM AUTHOR]