Proximal Point Methods for Quasiconvex and Convex Functions With Bregman Distances

This paper generalizes the proximal point method using Bregman distances to solve convex and quasiconvex optimization problems on noncompact Hadamard manifolds. We will proved that the sequence generated by our method is well defined and converges to an optimal solution of the problem. Also, we obtain the same convergence properties for the classical proximal method, … Read more

Benchmark of Some Nonsmooth Optimization Solvers for Computing Nonconvex Proximal Points

The major focus of this work is to compare several methods for computing the proximal point of a nonconvex function via numerical testing. To do this, we introduce two techniques for randomly generating challenging nonconvex test functions, as well as two very specific test functions which should be of future interest to Nonconvex Optimization Benchmarking. … Read more

Computing Proximal Points on Nonconvex Functions

The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with … Read more

Joint minimization with alternating Bregman proximity operators

A systematic study of the proximity properties of Bregman distances is carried out. This investigation leads to the introduction of a new type of proximity operator which complements the usual Bregman proximity operator. We establish key properties of these operators and utilize them to devise a new alternating procedure for solving a broad class of … Read more

Convergence rate estimates for the gradient differential inclusion

Let f be a lower semi-continuous convex function in a Euclidean space, finite or infinite dimensional. The gradient differential inclusion is a generalized differential equation which requires that -x'(t) be in the subgradient of f at x. It is the continuous versions of the gradient descent method for minimizing f in case f is differentiable, … Read more

Convex- and Monotone- Transformable Mathematical Programming Problems and a Proximal-Like Point Method

The problem of finding singularities of monotone vectors fields on Hadamard manifolds will be considered and solved by extending the well-known proximal point algorithm. For monotone vector fields the algorithm will generate a well defined sequence, and for monotone vector fields with singularities it will converge to a singularity. It will be also shown how … Read more

An Analysis of the EM Algorithm andEntropy-Like Proximal Point Methods

The EM algorithm is a popular method for maximum likelihood estimation from incomplete data. This method may be viewed as a proximal point method for maximizing the log-likelhood function using an integral form of the Kullback-Leibler distance function. Motivated by this interpretation, we consider a proximal point method using an integral form of entropy-like distance … Read more