Lecture

Objective function, Linearity and nonlinearity

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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

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 (32)

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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 .

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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.

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