# Variable Metric Methods Along Geodesics

• Tamás Rapcsák
Chapter
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 19)

## Abstract

This part of the book is devoted to the analysis of variable metric methods along geodesies. These methods are iterative, meaning that the algorithms generate a series of points, each point calculated on the basis of the points preceding it, and they are descent which means that each new point is generated by the algorithms and that the corresponding value of some function, evaluated at the most recent point, decreases in value. The theory of iterative algorithms can be divided into three parts. The first is concerned with the creation of the algorithms based on the structure of problems and the efficiency of computers. The second is the verification whether a given algorithm generates a sequence converging to a solution. This aspect is referred to as global convergence analysis, since the question is whether an algorithm starting from an arbitrary initial point, be it far from the solutions, converges to a solution. Thus, an algorithm is said to be globally convergent if for arbitrary starting points , the algorithm is guaranteed to generate a sequence of points converging to a solution. Many of the most important algorithms for solving nonlinear optimization problems are not globally convergent in the purest form, and thus occasionally generate sequences that either do not converge at all, or converge to points that are not solutions. The third is referred to as local convergence analysis and is concerned with the rate at which the generated sequence of points converges to a solution.

## Keywords

Riemannian Manifold Global Convergence Penalty Method Interior Point Method Riemannian Metrics
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