Lecture

Descent methods and line search: Preconditioned steepest descent

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Description

This lecture covers the concept of preconditioning in optimization problems, introducing a change of variables to simplify the problem. It explains the symmetric positive definite matrix Hk, the steepest descent iteration, and the descent direction in the context of preconditioned steepest descent methods.

In MOOC

Optimization: principles and algorithms - Unconstrained nonlinear optimization

What you'll learn Formulation: you will learn from simple examples how to formulate, transform and characterize an optimization problem. Objective function: you will review the mathematical properties of the objective function that are important in optimization. Optimality conditions: you will learn sufficient and necessary conditions for an optimal solution. Solving equations, Newton: this is a reminder about Newton's method to solve nonlinear equations. Newton's local method: you will see how to interpret

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Related concepts (23)

Gradient descent

In mathematics, gradient descent (also often called steepest descent) is a iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent.

Coordinate descent

Coordinate descent is an optimization algorithm that successively minimizes along coordinate directions to find the minimum of a function. At each iteration, the algorithm determines a coordinate or coordinate block via a coordinate selection rule, then exactly or inexactly minimizes over the corresponding coordinate hyperplane while fixing all other coordinates or coordinate blocks. A line search along the coordinate direction can be performed at the current iterate to determine the appropriate step size.

Optimization problem

In mathematics, computer science and economics, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete: An optimization problem with discrete variables is known as a discrete optimization, in which an object such as an integer, permutation or graph must be found from a countable set.

Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if denotes the entry in the th row and th column then for all indices and Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.

Definite matrix

In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector where is the transpose of . More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector where denotes the conjugate transpose of Positive semi-definite matrices are defined similarly, except that the scalars and are required to be positive or zero (that is, nonnegative).

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