This paper examines the convergence and computational complexity of a proximal stochastic gradient algorithm that adaptively incorporates sampling techniques for solving large-scale, non-convex, and non-smooth problems, with a particular emphasis on problems that involve the combination of two non-convex functions.This an area that has been scarcely explored by current methods.By adjusting adaptively the sampling size (or mini-batch size) throughout the algorithm's iterations, this method aims to balance the trade-off between stochastic gradient noise and convergence stability.It maintains a convergence rate similar to that of the proximal gradient method.Moreover, when the objective function is a Kurdyka-ojasiewicz (KL) function, we demonstrate the convergence rate of the expected function value on a case-by-case basis, achieving linear convergence under optimal conditions.Finally, some preliminary numerical results validate the effectiveness and robustness of the proposed method.

Preliminary results of our investigations on solving indefinite quadratic programs by dynamical systems are given.First, dynamical systems corresponding to two fundamental DC programming algorithms to deal with indefinite quadratic programs are considered.Second, the existence and the uniqueness of the global solution of the dynamical system are proved by using some theorems from nonsmooth analysis and the theory of ordinary differential equations.Third, the strong pseudomonotonicity of the restriction of an affine operator on a closed convex set is analyzed in a special case.Finally, for a parametric indefinite quadratic program related to that special case, convergence of the trajectories of the dynamical system to the Karush-Kuhn-Tucker points is established.The elementary direct proofs in the third and fourth topics would be useful for understanding the meaning and significance of several open problems proposed in this paper.

In this paper, we aim to investigate the existence and boundedness of the normalized solutions of the following nonlinear Schrdinger equation by using a non-variational fixed point theorem:where the potential function V (x) : R is measurable and can change sign, h 0 is the perturbation term, the nonlinearity f is measurable and satisfies critical or subcritical growth for N 3, exponential critical growth for N = 2.We first consider as the whole space R N and establish the existence of normalized solutions to the problem for N = 2 and N 3, respectively.While is considered as a bounded domain, we establish the existence and L -boundedness of the normalized solution of the problem for N 3, where the nonlinearity f satisfies the subcritical growth condition.In the bounded domain , we can still obtain the existence of normalized solutions to the above problem with the critical growth condition instead of the L -boundedness of normalized solutions.As far as we know, our results are more general and novel on this topic, and can also provide a new approach for the study of nonlinear Schrdinger equation with prescribed L 2 -norm.

The class of convex sets that admit approximations as Minkowski sum of a compact convex set and a closed convex cone in the Hausdorff distance is introduced.These sets are called approximately Motzkin-decomposable and generalize the notion of Motzkin-decomposability, i.e., the representation of a set as the sum of a compact convex set and a closed convex cone.We characterize these sets in terms of their support functions and show that they coincide with hyperbolic sets, i.e., convex sets contained in the sum of their recession cone and a compact convex set if their recession cones are polyhedral but are more restrictive in general.In particular, we prove that a set is approximately Motzkin-decomposable if and only if its support function has a closed domain relative to which it is continuous.