Lecture

RTR practical aspects + tCG

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

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 practical aspects of Riemannian trust-region (RTR) optimization, including the choice of the trust-region radius and the subproblem solver, along with the truncated conjugate gradient (tCG) method for solving optimization problems. The instructor explains the key concepts, algorithms, and conditions for successful optimization using RTR and tCG.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructor

Nicolas Boumal

Teaching & PhD PhD Students Andreea-Alexandra Musat Past EPFL PhD Students Christopher Criscitiello, Quentin Rebjock Courses Continuous optimization MATH-329 This course introduces students to continuous, nonlinear optimization. We study the theory of optimization with continuous variables (with full proofs), and we analyze and implement important algorithms to solve constrained and unconstrained problems. Linear Algebra MATH-111(e) The purpose of the course is to introduce

Official source

Related concepts (32)

Riemannian manifold

In differential geometry, a Riemannian manifold or Riemannian space (M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p. The family gp of inner products is called a Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take g to be smooth, which means that for any smooth coordinate chart (U, x) on M, the n2 functions are smooth functions.

Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike.

Curvature of Riemannian manifolds

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry of surfaces and other objects. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

Iterative method

In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of the iterative method.

Related lectures (64)

Truncated Conjugate Gradients for Trust-Region Subproblem

Explores truncated conjugate gradients for solving the trust-region subproblem in optimization on manifolds efficiently.

Trust Region Methods: Why, with an Example

Introduces trust region methods and presents an example of Max-Cut Burer-Monteiro rank 2 optimization.

Optimization on Manifolds: Context and Applications MOOC: Introduction to optimization on smooth manifolds: first order methods

Introduces optimization on manifolds, covering classical and modern techniques in the field.

Gradient Descent

Covers the concept of gradient descent in scalar cases, focusing on finding the minimum of a function by iteratively moving in the direction of the negative gradient.

Linear convergence with Polyak-Łojasiewicz: Mechanical proof