Complexity of finding near-stationary points of convex functions stochastically

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

Sweep Line Algorithm for Convex Hull Revisited

Convex hull of some given points is the intersection of all convex sets containing them. It is used as primary structure in many other problems in computational geometry and other areas like image processing, model identification, geographical data systems, and triangular computation of a set of points and so on. Computing the convex hull of a set of point is one of the most fundamental and imp...

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...

Non-monotone Continuous DR-submodular Maximization: Structure and Algorithms

DR-submodular continuous functions are important objectives with wide real-world applications spanning MAP inference in determinantal point processes (DPPs), and mean-field inference for probabilistic submodular models, amongst others. DR-submodularity captures a subclass of non-convex functions that enables both exact minimization and approximate maximization in polynomial time. In this work w...

Convexity and Geodesic Metric Spaces

In this paper, we first present a preliminary study on metric segments and geodesics in metric spaces. Then we recall the concept of d-convexity of sets and functions in the sense of Menger and study some properties of d-convex sets and d-convex functions as well as extreme points and faces of d-convex sets in normed spaces. Finally we study the continuity of d-convex functions in geodesic metr...