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Lecture

Riemannian gradient descent

In course

MATH-512: Optimization on manifolds

We develop, analyze and implement numerical algorithms to solve optimization problems of the form min f(x) where x is a point on a smooth manifold. To this end, we first study differential and Riemannian geometry (with a focus dictated by computational con ...

Description

This lecture covers Riemannian gradient descent, focusing on Taylor expansions, first-order optimality conditions, algorithm templates, line search, sufficient decrease, regularity conditions, and critical points. It also discusses the behavior of local minimizers and the concept of critical points. The instructor explains the pull back of functions through retractions, the definition of local minimizers, and the necessary conditions for a point to be a critical or stationary point. The lecture concludes with a theorem on the sequence produced by Riemannian gradient descent under specific assumptions.

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Instructor

Nicolas Boumal

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Global optimization is a branch of applied mathematics and numerical analysis that attempts to find the global minima or maxima of a function or a set of functions on a given set. It is usually described as a minimization problem because the maximization of the real-valued function is equivalent to the minimization of the function . Given a possibly nonlinear and non-convex continuous function with the global minima and the set of all global minimizers in , the standard minimization problem can be given as that is, finding and a global minimizer in ; where is a (not necessarily convex) compact set defined by inequalities .

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