Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Objective function, Linearity and nonlinearity

Description

This lecture delves into the concepts of linear and non-linear functions in optimization. It explains how linear functions can be simplified by expressing them as linear combinations of variables, while non-linear functions present varying levels of difficulty. The lecture also introduces the concept of affine functions, which are linear functions shifted by a constant. It discusses how adding a constant to an objective function in optimization does not impact the outcome. The instructor illustrates the difference between linear and non-linear functions using graphical representations and explains how Lipschitz continuity of the gradient can be used to measure the level of non-linearity in a function. By the end of the lecture, viewers will have a clear understanding of the distinctions between linear and non-linear functions in optimization.

In MOOC

Optimization: principles and algorithms - Unconstrained nonlinear optimization

Official source

Related concepts (32)

Linear form

In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k with addition and scalar multiplication defined pointwise. This space is called the dual space of V, or sometimes the algebraic dual space, when a topological dual space is also considered.

Linear map

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a .

Discontinuous linear map

In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). If the spaces involved are also topological spaces (that is, topological vector spaces), then it makes sense to ask whether all linear maps are continuous. It turns out that for maps defined on infinite-dimensional topological vector spaces (e.g.

Continuous linear operator

In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous function (topology) and Discontinuous linear map Bounded operator Suppose that is a linear operator between two topological vector spaces (TVSs). The following are equivalent: is continuous.

Generalized linear model

In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value. Generalized linear models were formulated by John Nelder and Robert Wedderburn as a way of unifying various other statistical models, including linear regression, logistic regression and Poisson regression.

Related lectures (37)

Linear Homogeneous Second Order Equations

Covers linear homogeneous second order differential equations and their solutions, including the concept of the Wronskian.

Optimization Methods: Theory Discussion

Explores optimization methods, including unconstrained problems, linear programming, and heuristic approaches.

Optimization: Gradient Descent and Subgradients

Explores optimization methods like gradient descent and subgradients for training machine learning models, including advanced techniques like Adam optimization.

Matrix Representations: Linear Maps

Explores matrix representations of linear maps and their invariance, using examples and special cases.

Introduction to Electronics MOOC: Electronics I