Tensor methods for finding approximate stationary points of convex functions

Complexity of finding near-stationary points of convex functions stochastically

In the recent paper [3], it was shown that the stochastic subgradient method applied to a weakly convex problem, drives the gradient of the Moreau envelope to zero at the rate O(k−1/4). In this supplementary note, we present a stochastic subgradient method for minimizing a convex function, with the improved rate Õ(k−1/2).

Lower Bounds for Finding Stationary Points of Non-Convex, Smooth High-Dimensional Functions∗

We establish lower bounds on the complexity of finding -stationary points of smooth, non-convex, high-dimensional functions. For functions with Lipschitz continuous pth derivative, we show that all algorithms—even randomized algorithms observing arbitrarily high-order derivatives—have worst-case iteration count Ω( −(p+1)/p). Our results imply that the O( −2) convergence rate of gradient descent...

Iterative methods for finding nearest common fixed points of a countable family of quasi-Lipschitzian mappings

We prove a strong convergence result for a sequence generated by Halpern's type iteration for approximating a common fixed point of a countable family of quasi-Lipschitzian mappings in a real Hilbert space. Consequently, we apply our results to the problem of finding a common fixed point of asymptotically nonexpansive mappings, an equilibrium problem, and a variational inequality problem for co...

Finding Approximate Convex Shapes in RGBD Images

We propose a novel method to find approximate convex 3D shapes from single RGBD images. Convex shapes are more general than cuboids, cylinders, cones and spheres. Many real-world objects are nearconvex and every non-convex object can be represented using convex parts. By finding approximate convex shapes in RGBD images, we extract important structures of a scene. From a large set of candidates ...

Some Properties of Certain Subclasses of Close-to-Convex and Quasi-convex Functions with Respect to 2k-Symmetric Conjugate Points