Lecture

Descent methods and line search: Golden section

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Description

This lecture covers the main idea behind descent methods and line search, focusing on the generation of intervals, symmetric reduction, and the calculation of the golden section. The instructor explains the assumptions, unidimensional functions, and the process of generating the next interval using different cases. The lecture also delves into the symmetric reduction technique and the recycling of function evaluations. The calculation of the golden section parameter 'p' is detailed, emphasizing the iterative process and the mathematical operations involved.

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)

Combinatorial optimization

Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.

Convex optimization

Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard.

Mathematical optimization

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.

Unimodality

In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. In statistics, a unimodal probability distribution or unimodal distribution is a probability distribution which has a single peak. The term "mode" in this context refers to any peak of the distribution, not just to the strict definition of mode which is usual in statistics. If there is a single mode, the distribution function is called "unimodal".

Multimodal distribution

In statistics, a multimodal distribution is a probability distribution with more than one mode. These appear as distinct peaks (local maxima) in the probability density function, as shown in Figures 1 and 2. Categorical, continuous, and discrete data can all form multimodal distributions. Among univariate analyses, multimodal distributions are commonly bimodal. When the two modes are unequal the larger mode is known as the major mode and the other as the minor mode. The least frequent value between the modes is known as the antimode.

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