Thin Partitions: Isoperimetric Inequalities and Sampling Algorithms for some Nonconvex Families

Iterative algorithms for families of variational inequalities fixed points and equilibrium problems

Recent Progress and Open Problems in Algorithmic Convex Geometry

This article is a survey of developments in algorithmic convex geometry over the past decade. These include algorithms for sampling, optimization, integration, rounding and learning, as well as mathematical tools such as isoperimetric and concentration inequalities. Several open problems and conjectures are discussed on the way.

Geometric Random Walks: A Survey∗

The developing theory of geometric random walks is outlined here. Three aspects — general methods for estimating convergence (the “mixing” rate), isoperimetric inequalities in Rn and their intimate connection to random walks, and algorithms for fundamental problems (volume computation and convex optimization) that are based on sampling by random walks — are discussed.

Quantitative isoperimetric inequalities for a class of nonconvex sets

Quantitative versions (i.e., taking into account a suitable “distance” of a set from being a sphere) of the isoperimetric inequality are obtained, in the spirit of [17, 18], for a class of not necessarily convex sets called φ-convex sets. Our work is based on geometrical results on φ-convex sets, obtained using methods of both nonsmooth analysis and geometric measure theory.

Linear Time Varying MPC Based Path Planning of an Autonomous Vehicle via Convex Optimization

In this paper a new method is introduced for path planning of an autonomous vehicle. In this method, the environment is considered cluttered and with some uncertainty sources. Thus, the state of detected object should be estimated using an optimal filter. To do so, the state distribution is assumed Gaussian. Thus the state vector is estimated by a Kalman filter at each time step. The estimation...